Browse Function problems Studdy Solved!

Problem 2701

(1 point)
Evaluate the limit, using L'Hôpital's Rule. Enter INF for $\infty$ , -INF for $-\infty$ , or DNE if the limit does not exist, but is neither $\infty$ nor $-\infty$ . $\lim _{x \rightarrow 0^{+}}(1+3 x)^{3 / x}=$ $\square$
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Problem 2702

Let $g(y)=7 \sin (y)$ Determine the average value, $g(c)$ , of $g$ over $\left[0, \frac{3 \pi}{4}\right]$ . $g(c)=$ $\square$ Determine the value(s) of $c$ in $\left[0, \frac{3 \pi}{4}\right]$ guaranteed by the Mean Value Theorem. Round the solution(s) to four decimal places, if necessary. $c=$ $\square$

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Problem 2703

Question Watch Video
The table below lists the masses and volumes of several pieces of the same type of metal. There is a proportional relationship between the mass and the volume of the pieces of metal. \begin{tabular}{|c|c|} \hline \begin{tabular}{c} Volume \\ (cubic centimeters) \end{tabular} & Mass (grams) \\ \hline 2.4 & 25.008 \\ \hline 3.3 & 34.386 \\ \hline 3.4 & 35.428 \\ \hline \end{tabular}
Determine the mass, in grams, of a piece of metal that has a volume of 12.7 cubic centimeters. Round your answer to the nearest tenth of a gram.

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Problem 2704

$\qquad$ 1. How many $x$ -intercepts does the exponential function $f(x)=2(10)^{x}$ have? a. 0 b. 1 c. 2 d. 3 $\qquad$ 2. Match the following graph with its function. 0 a. $y=3(0.5)^{x}$ b. $\quad y=2(1.25)^{x}$ c. $y=0.5(3)^{x}$ d. $y=2(0.75)^{x}$ $\qquad$ 3. Determine the $y$ -intercept of the exponential function $f(x)=4\left(\frac{1}{2}\right)^{x}$ a. 0 b. 1 c. 2 d. 4

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Problem 2705

$\begin{array}{l} \qquad g(x)=\frac{1}{\sqrt{2 \pi}} e^{-(x-9)^{2} / 2} \\ \text { maximum } \quad(x, y)=(\square) x) \\ \text { minimum } \quad(x, y)=(\square) \\ \text { Inflection points } \\ \text { smatler } x \text {-value } \quad(x, y)=(\square) \\ \text { larger } x \text {-value } \quad(x, y)=(\square) \end{array}$ Need Help? Resiti Welch 8 Sutamit Ansiver

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Problem 2706

List the value of all three trigonometric functions for the given angle $\pi / 3$ .

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Problem 2707

$\begin{aligned} f^{r}(x) & =\left(\frac{e^{x^{2}}}{\sqrt[3]{x}}\right)^{\prime} \\ & =\left(e^{x^{2}} \cdot x^{1 / 5}\right)^{\prime}\end{aligned}$

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Problem 2708

The rational function $r$ is given by $r(x)=\frac{x^{3}-4 x+3}{x^{4}+2 x-4}$ . For what values of $x$ does $r(x)=0 ?$
A $x=-2.303$ and $x=1.000$ only (B) $x=-1.643$ and $x=1.144$ only
C $x=-2.303, x=1.000$ , and $x=1.303$ only (D) $x=-2.303, x=-1.643, x=1.000, x=1.303$ , and $x=1.144$

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Problem 2709

3. The function $f$ and $g$ are given by $f(x)=e^{2 x}+1$ and $g(x)=\frac{1}{2} \ln (x-1)$ . (a) Show that $f(x)$ is a one-to-one function. [2 marks] (b) Determine the inverse function of $g(x)$ .
Hence, state the relationship between $f(x)$ and $g(x)$ . [4 marks] (c) Find the function $h(x)$ if $(f \circ h)(x)=e^{4 x^{2}}+1$ . [3 marks]

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Problem 2710

(3) Question \#18
Reference Q. 11243 For each pair of functions, write a formula for $g(f(x))$ . a. $f(x)=2-x, g(x)=|x+2|$ b. $f(x)=2 x+1, g(x)=x^{4}$ c. $f(x)=3^{x}, g(x)=x-1$

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Problem 2711

9. Consider the functions $f(x)=3 x, g(x)=x^{3}$ , and $h(x)=3^{x}$ . a) Graph each function. b) Make a list of the key features for each function, as in Investigate 2, step 2 b). Organize the information in a table. c) Identify key features that are common to each function. d) Identify key features that are different for each function. e) How do the instantaneous rates of change compare for these three functions?
10. An influenza virus is spreading through a school according to the function $N=10(2)^{t}$ , where $N$ is the number of people infected and $t$ is the time, in days. a) How many people have the virus at each time? i) initially, when $t=0$ ii) after 1 day iii) after 2 days iv) after 3 days b) Graph the function. Does it appear to be exponential? Explain your answer. c) Determine the average rate of change between day 1 and day 2 . d) Estimate the instantaneous rate of change after i) 1 day ii) 2 days e) Explain why the answers to parts c) and d) are different. Use the functions $f(x)=4^{x}$ and $g(x)=\left(\frac{1}{2}\right)^{x}$ to answer questions 11 to 18.
11. a) Sketch a graph of $f$ . b) Graph the line $y=x$ on the same grid. c) Sketch the inverse of $f$ on the same grid by reflecting $f$ in the line $y=x$ .
12. Identify the key features of $f$ . a) domain and range b) $x$ -intercept, if it exists c) $y$ -intercept, if it exists

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Problem 2712

Consider the function, $f(x)=\frac{2 x-10}{-x+2}$ This function has: 1) A $y$ -intercept at the point $\square$ 2) $x$ -intercept(s) at the point(s) $\square$ 3) Equation of the vertical asymptote: $\square$ 4) Equation of the horizontal asymptote: $\square$

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Problem 2713

Use the graph to find (a) the $x$ -intercept(s) and (b) the zero(s) of the function. (a) The $x$ -intercept(s) is/are $\square$ . (Type an ordered pair. Use a comma to separate answers as needed.)

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Problem 2714

Find the value of $(f \circ g)(5)$ given that $f(x) = 2\sqrt{x}$ and $g(x) = 2$ .

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Problem 2715

Question 10 of 10, Step 1 of 1 $7 / 10$ Correct 2
Find the total area bounded by the $x$ -axis and the curve $y=f(x)$ on the indicated interval. Enter your answer in exact form or as a decimal number rounded to the nearest thousandth. $f(x)=6 \sqrt{2 x-1} ;[4,7]$
Answer Tables Keypad Keyboard Shortcuts $\square$ BETA 6 Al Tutor

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Problem 2716

Find all zeros 1) $f(x)=x^{6}+8 x^{5}-114 x^{4}-428 x^{3}+3149 x^{2}+4356 x-16380$
HINT: $2,5,7$ and -6 are zeros 2) $g(x)=2 x^{4}-21 x^{3}+155 x^{2}-44 x-222$
HINT: $3 / 2$ and -1 are zeros 3) $h(x)=x^{5}+7 x^{4}+10 x^{3}-27 x^{2}-189 x-270$
HINT:-5 and -2 are zeros 4) $k(x)=x^{5}-5 x^{4}-3 x^{3}+15 x^{2}-28 x+140$
HINT: 5 is a zero

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Problem 2717

deposited in it a particular savings account is compounded continuously. The account initially had $\$ 2,200$ $A(t)=2200 e^{05}$ (a) By what percent will the worth of the account increase per year? Round to the nearest hundredth of a percent. $\begin{array}{l} \text { e.0s(1) } \\ =1.05127-1 \\ =0.05127(100) \\ =5.127 \\ =5.13 \% \end{array}$ (b) To the nearest tenth of a year, how long will it take for the worth of the account to triple? (c) If another investment began with a principal of $\$ 2,500$ and earned simplest interest of $3.8 \%$ applied once per year, which investment would be worth more after 10 years? Justify.

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Problem 2718

Find the limit. Use l'Hospital's rule where appropriate. If there is a more elementary method, consider using it. $\lim _{x \rightarrow(\pi / 2)^{+}} \frac{\cos (x)}{1-\sin (x)}$

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Problem 2719

Points: 0 of 1
If $f(x)=\sqrt{x}$ and $g(x)=-2 x-2$ , find $(f \circ g)(x)$ and $(g \circ f)(x)$ . $(f \circ g)(x)=$ $\square$ (Simplify your answer. Type an exact answer, using radicals as needed.)

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Problem 2720

Points: 0
Find $(f \circ g)(x)$ and $(g \circ f)(x)$ and the domain of each. $f(x)=7 x-5, g(x)=\frac{x+5}{7}$ $(f \circ g)(x)=$ $\square$ (Simplify your answer.)

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Problem 2721

If $f(x)=x^{2}-6 x+2$ and $g(x)=-2 x$ , find the following composition. $\begin{array}{r} (g \circ f)(2) \\ (g \circ f)(2)=\square \end{array}$

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Problem 2722

Find $(f \circ g)(x)$ and $(g \circ f)(x)$ . $f(x)=-3 x^{2}-1, g(x)=2 x$ $(f \circ g)(x)=$ $\square$ (Simplify your answer.)

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Problem 2723

Solve by factoring: $f(x)=p^{2}-5 p-14$

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Problem 2724

ChallenGE YOURSELF!!!
Which of the following will be the $x$ -intercept of $y=\log _{b}(x-k)$ based on the constants $b$ and $k$ ? (1) $x=k+1$ (2) $x=b+1$ (3) $x=b^{k}+1$ (4) $x=k^{b}+1$

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Problem 2725

2 ties let, Please try again.
The photography club is selling hot chocolate at soccer games to raise money for new cameras. The table shows their proti per game for the fist five games. \begin{tabular}{|c|c|} \hline Game & Profiti (5) \\ \hline 1 & -1250 \\ \hline 2 & -10.15 \\ \hline 3 & 18.65 \\ \hline 4 & 25.90 \\ \hline 5 & 45.75 \\ \hline \end{tabular}
Based on the average proft per game, how much total money can the club expect to eam oy the end of the to-game season? $\qquad$ dollarts)

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Problem 2726

Two ants crawl around a circle of radius $r=7$ with both $x$ and $y$ measured in inches. Both start at the point ( 7,0 ) at the same time. One bug is moving at a rate of $2 \mathrm{in} / \mathrm{sec}$ . The other is moving twice as fast. When will one ant be directly above the other ant as shown below? (Give the first time this happens assuming they start at $t=0$ .)

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Problem 2727

The $y$ -intercept of the function $y=0.5 x^{3}-2.5 x^{2}+0.1 x+10$ is Select one: a. -2.5 b. 0.5 c. 10 d. 0.1

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Problem 2728

Consider the following function. (If an answer does not exist, enter DNE.) $f(x)=e^{-x^{2}}$ (a) Find the vertical asymptote(s). (Enter your answers as a comma-separated list.) $x=$ $\square$
Find the horizontal asymptote(s). (Enter your answers as a comma-separated list.) $\square$ $y=\square$ (b) Find the interval where the function is increasing. (Enter your answer using interval notation.) $\square$
Find the interval where the function is decreasing. (Enter your answer using interval notation.) $\square$ (c) Find the local maximum and minimum values. local maximum value $\square$ local minimum value $\square$ (d) Find the interval where the function is concave up. (Enter your answer using interval notation.) $\square$

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Problem 2729

Find the average value of the following function over the given interval. Draw a graph of the function and indicate the average value. $f(x)=\frac{8}{x} \text { on }[3,3 e]$

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Problem 2730

3. A company estimates that its sales will grow continuously at a rate given by the function $S^{\prime}(t)=$ $20 e^{t}$ where $S^{\prime}(t)$ is the rate at which sales are increasing in dollars per day, on day $t$ . Find the accumulated sales for the first 5 days.

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Problem 2731

Find $f$ . (Use $C$ for the constant of the first antiderivative and $D$ for the constant of the second antiderivative.) $\begin{array}{l} f(x)=24 x^{3}-18 x^{2}+8 x \\ f(x)=\square \end{array}$ Need Help? Read It Watch it

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Problem 2732

Suppose $f(x, y, z)=\frac{1}{\sqrt{x^{2}+y^{2}+z^{2}}}$ and $W$ is the bottom half of a sphere of radius 4. Enter $\rho$ as rho, $\phi$ as phi, and $\theta$ as theta. (a) As an iterated integral, $\iiint_{W} f d V=\int_{A}^{B} \int_{C}^{D} \int_{E}^{F} \square d \rho d \phi d \theta$ with limits of integration $A=$ $\square$ $B=$ $\square$ $c=$ $\square$ $D=$ $\square$ $E=$ $\square$ $F=$ $\square$ (b) Evaluate the integral. $\square$

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Problem 2733

option. Which of the following will give the most accurate approximation of the definite integral $\int_{0}^{1} g(x) d x$ by Simpson's rule? intervals of width 0.1 unit each intervals of width 0.25 unit each intervals of width 0.2 unit each intervals of width 0.5 unit each

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Problem 2734

Graph the exponential function. $f(x)=2^{x}$
Plot five points on the graph of the function, and also draw the asymptote. Then click on the graph-afunction button.

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Problem 2735

19
Use the following infornation to answer the next queations. The exponential function $\mathrm{g}(\mathrm{x})$ is shewn below. out of
The graph of $g(x)$ has the domain
Select one: a. $\{x>0, x \in R\}$ b. $\{x \in R\}$ c. $\{x \geq 0, x \in R\}$ d. $\{x=1\}$

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Problem 2736

Using the rational root theorem, list out all possible/candidate rational roots of $f(x)=-6 x^{4}+23 x^{3}-4 x^{2}-12 x+9$ . Express your answer as integers or as fractions in simplest form. Use commas to separate.

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Problem 2737

Find the real zeros of $f$ and state the multiplicity for each zero. State whether the graph of $f$ crosses or touches the $x$ -axis at each zero. $f(x)=\left(x^{2}+6\right)(4 x-3)^{2}$
Select the correct choice below and fill in the answer boxes to complete your choice. (Type integers or simplified fractions.) A. The real zero of $f$ is $\square$ with multiplicity $\square$ and the graph of f crosses the x -axis at this zero. B. The real zero of $f$ is $\square$ with multiplicity $\square$ and the graph of $f$ touches but does not cross the $x$ -axis at this zero. C. The smallest real zero of $f$ is $\square$ with multiplicity $\square$ and the graph of $f$ touches but does not cross the $x$ -axis at this zero. The largest real zero of $f$ is $\square$ with multiplicity $\square$ and the graph of $f$ touches but does not cross the $x$ -axis at this zero. D. The smallest real zero of $f$ is $\square$ with multiplicity $\square$ and the graph of $f$ crosses the $x$ -axis at this zero. The largest real zero of $f$ is $\square$ with multiplicity $\square$ and the graph of $f$ touches but does not cross the $x$ -axis at this zero.

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Problem 2738

Let $f(x)=x^{4}+2 x^{3}-72 x^{2}+60 x+72$ . Determine all intervals on which $f(x)$ is concave up. Possibilities: (a) $(-3,4)$ (b) $(-\infty, \infty)$ (c) $(-\infty,-3) \cup(4, \infty)$ (d) $(-4,3)$ (e) $(-\infty,-4) \cup(3, \infty)$

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Problem 2739

In Exercises 1-12, graph the function. Label the vertex and axis of symmetry.
1. $f(x)=-3(x-2)^{2}-4$
2. $f(x)=3(x+1)^{2}+5$
3. $g(x)=-\frac{1}{2}(x+3)^{2}+2$
4. $h(x)=\frac{1}{2}(x-2)^{2}-1$
5. $y=0.6(x-2)^{2}$
6. $f(x)=0.25 x^{2}-1$
7. $y=-x^{2}+8$
8. $y=7 x^{2}+2$
9. $y=1.5 x^{2}-6 x+3$
10. $f(x)=0.5 x^{2}+3 x-1$
11. $y=\frac{5}{2} x^{2}-5 x+1$
12. $f(x)=-\frac{3}{2} x^{2}-6 x-4$
13. A quadratic function is decreasing to the left of $x=3$ and increasing to the right of $x=3$ . Will the vertex be the highest or lowest point on the graph of the parabola? Explain.
14. The graph of which function has the same axis of symmetry as the graph of $y=2 x^{2}-8 x+3$ ? Explain your reasoning. A. $y=-4 x^{2}+16 x-5$ B. $y=2 x^{2}+8 x+7$ C. $y=3 x^{2}-6 x+7$ D. $y=-6 x^{2}+10 x-1$
In Exercises 15-18, find the minimum or maximum value of the function. Describe the domain and range of the function, and where the function is increasing and decreasing.
15. $y=3 x^{2}+12$
16. $y=-x^{2}-6 x$
17. $y=-\frac{1}{3} x^{2}-2 x+3$
18. $f(x)=\frac{1}{2} x^{2}+3 x+7$
19. The height of a bridge is given by $y=-3 x^{2}+x$ , where $y$ is the height of the bridge (in miles) and $x$ is the number of miles from the base of the bridge. a. How far from the base of the bridge does the maximum height occur? b. What is the maximum height of the bridge?

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Problem 2740

Graph the exponential function. $f(x)=\left(\frac{3}{2}\right)^{x}$
Plot five points on the graph of the function, and also draw the asymptote. Then click on the graph-afunction button.

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Problem 2741

Explain why the vertex for $f(x)=a|x-h|+k$ is unaffected by $a$ .

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Problem 2742

Wibek of: Secion b. I-wriam venuns
14. For the polynomial function $f(x)=-2 x^{2}\left(x^{2}-5\right)$ inswer the following questions. (a) List each real zero and its mulliplicily. (b) Determine whether the graph crosses or louches the $x$ -axis at each $x$ -intercepl. (c) Delermine the maximum number of turning poinis on the graph. (d) Determine the end behavior; that is, find the power function that tho graph of fresembles for farge values of |al. (a) Find any real zeros of t . Select the correct choice below and, if necessang, fill in the answer bax(es) to complate your choice. O. The real zero of $f$ is with multiplicily (Type an exact answer, using radicals as needed. Type iniegers or fractions.) B. The smallest zero of II with muliplicity $\qquad$ The largesizero of tis $\qquad$ with multiplicity (Type an exaci ansiver, using radicals as needed. Type inlegers or fraclions.) C. The smallest zero of I is with multiplicily
The midale zero of its $\qquad$ muliplicily .The largesi zero of tis with muliplicity with (Type an exach answer, using radicals as needed. Type inlegers or fraclions.) D. There are no real zeros. (b) Select the correct cholce below and, it necessary, fill in the answer box(es) to complete your choice. A. The graph fouches the x-axis at (Type an exact answer, using radicals as needed. Type an inleger or a simplified fraction. Use a comma to separale answers as needed.) B. The graph crosses the $x$ -axis at (Type an exacl answer, using radicals as needed. Type an inleger or a simplified fraction. Use a comma to separate answers as needed.) C. The graph louches the $x$ -axis at and crosses at (Type integers or simplified fractions. Use a comma lo separale answers as needed.) D. The graph neither crosses nor fouches the $x$ -axis. (c) The maximum number of furning points on the graph is $\square$ (Type a whole number.) (d) The power function that the graph of f resembles for large values of $|x|$ is $y=\mid$ $\square$

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Problem 2743

Graph the exponential function. $f(x)=-\frac{1}{4}(4)^{x}$
Plot five points on the graph of the function, and also draw the asymptote. Then click on the graph-afunction button.

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Problem 2744

1. Alvaro is studying the extinction of the bear population of Siberia over time. The following function gives the number of bears $t$ years since Alvaro started tracking it. $B(t)=2190 e^{-0.3 t}$ a. What is $B^{\prime}(3)$ ?

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Problem 2745

Graph the exponential function. $f(x)=2\left(\frac{2}{3}\right)^{x}$
Plot five points on the graph of the function, and also draw the asymptote. Then click on the graph-afunction button.

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Problem 2746

$\qquad$ 52. Find the derivative of the function. $f(x)=(\ln x)^{3}$ a. $f^{\prime}(x)=3(\ln x)^{2}$ b. $f^{\prime}(x)=\frac{(\ln x)^{6}}{x^{2}}$ c. $f^{\prime}(x)=2 x(\ln x)^{3}$ d. $f^{\prime}(x)=\frac{9(\ln x)^{8}}{x}$

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Problem 2747

The graph of an exponential function is shown in the figure below. The horizontal asymptote is shown as a dashed line. Find the range and the domain.
Write your answers as inequalities, using $x$ or $y$ as appropriate. Or, you may instead click on "Empty set" or "All reals" as the answer. (a) range: $\square$ (b) domain: $\square$
$\square$ $\square$ $>$ $\square$
$\square$ $\square$ $\square$

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Problem 2748

Evaluate the definite integral. $\int_{-3}^{3}\left(e^{x}-e^{-x}\right) d x$
STEP 1: Begin by finding the general integral. $\int_{-3}^{3}\left(e^{x}-e^{-x}\right) d x=[\square]_{-3}^{3}$
STEP 2: Substitute the limits of integration. $\square$ $-\left(e^{-3}+e^{3}\right)$
STEP 3: Simplify. $\square$ Need Help? Read it Watch it

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Problem 2749

You deposit $\$ 7000$ in an account earning $6 \%$ interest compounded continuously. The amount of money in account after $t$ years is given by $A(t)=7000 e^{0.06 t}$ .
How much will you have in the account in 7 years? \ $10647.00$ \square$ 0 Round your answer to decimal places.
How long will it be until you have $\$ 15700$ in the account? $\square$ years. Round your answer to 2 decimal places.

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Problem 2750

[-/0.76 Points] DETAILS MY NOTES LARAPCALC10 5.4.060.
Find the average value of the function on the interval. $f(x)=\frac{1}{(x-6)^{2}} ; \quad[0,5]$ $\square$ Find all $x$ -values in the interval for which the function is equal to its average value. (Enter your answers as a comma-separated list.) $x=$ $\square$ Need Help? Read It Watch it

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Problem 2751

The hands on a clock are of length 7 . inches (minute hand) and 6 , inches (hour hand). How fast is the distance between the tips of the hands changing at 9:00? (Assume this is a quality clock with continuous motion, not one of the cheap, jerky ones like they apparently install in $\$ 80$ million schools these days.)
Hint: Be sure to consider whether the distance between the hands is increasing or decreasing at this instant. (Give the exact answer OR round to three decimal places) Number Units

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Problem 2752

15. For each function, determine the equations of any asymptotes, and the locations of any missing points. a) $y=\frac{2 x}{5 x+3}$ b) $y=\frac{3 x+33}{-4 x^{2}-42 x+22}$ c) $y=\frac{3 x^{2}-3}{x-1}$ $H A=d \quad H A=a / c$

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Problem 2753

Which of the following statements is not true for the graph of $f(x)=e^{x}$ ?
Choose the statement that is not true below. A. The graph of $f(x)=e^{x}$ intersects the $y$ -axis at $(0,1)$ . B. The graph of $f(x)=e^{x}$ approaches 0 as $x$ approaches negative infinity. C. The line $y=0$ is a horizontal asymptote. D. The graph of $f(x)=e^{x}$ lies between the graphs of $y=3^{x}$ and $y=4^{x}$ .

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Problem 2754

Find the polynomial with a leading coefficient of either 1 or -1 and with smallest possible degree that matches the given graph.
The polynomial function is $f(x)=$ $\square$

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Problem 2755

Find a function of the form $y=A \sin (x)+D$ or $y=A \cos (x)+D$ whose graph matches the function shown below:
Leave your answer in exact form; if necessary, type pi for $\pi$ . $y=$ $\square$ Next Question

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Problem 2756

7. A hot air balloon is launched into the air with a human pilot. The twice-differentiable function $/ \mathrm{h}$ models the balloon's height, measured in feet, at time $t$ , measured in minutes. The table above gives values of the $h(t)$ and the vertical velocity $v(t)$ of the balloon at selected times $t$ . \begin{tabular}{|c|c|c|c|c|} \hline \begin{tabular}{c} $t$ \\ minutes \end{tabular} & 0 & 6 & 10 & 40 \\ \hline \begin{tabular}{c} $h(t)$ \\ feet \end{tabular} & 0 & 46 & 35 & 105 \\ \hline \begin{tabular}{c} $v(t)$ \\ feet per minute \end{tabular} & 0 & 6 & 20 & 1 \\ \hline \end{tabular} a. For $6 \leq t \leq 10$ , must there be a time $t$ when the balloon is 50 feet in the air? Justify your answer. b. For $10 \leq t \leq 40$ , must there be a time $t$ when the balloon's velocity is 3 feet per second? Justify your answer.
8. A particle moves along the $x$ -axis so that its position at any time $t \geq 0$ is given by $x(t)=t^{3}-3 t^{2}+t+$ 1. For what values of $t, 0 \leq t \leq 2$ , is the particle's instantaneous velocity the same as its average velocity on the closed interval $[0,2]$ ? Use a calculator for this problem.
9. Let $g$ be a continuous function. The graph of the piecewise-linear function $g^{\prime}$ , the derivative of $g$ , is shown above for $-4 \leq x \leq 4$ . a. Find the average rate of change of $g^{\prime}(x)$ on the interval $-4 \leq x \leq 4$ . b. Does the Mean Value Theorem applied on the interval $-4 \leq x \leq 4$ guarantee a value of $c$ , for $-4 \leq x \leq 4$ , such that $g^{\prime \prime}(\mathrm{c})$ is equal to this average rate of change? Why of why not?
Graph of $g^{\prime}$

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Problem 2757

Patt 1 of 2 Points: 0 of 1 Sive
The number of daily active users (in millions) on a social media site from 2017 to 2022 can be approximated by the function $f(x)=93.1677+64.5933 \ln x$ where $\mathrm{x}=1$ represents 2017, $\mathrm{x}=2$ represents 2018, and so on. (a) What does this function predict for the number of daily active users on the site in 2022? (b) According to this model, when did the number of daily active users on the site reach 160 million? (Hint: Substitute for $f(x)$ , and then write the equation in exponential form to solve it.) (a) What does this function predict for the number of daily active users in 2022? $\square$ million (Round to the neare tenth as needed.)

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Problem 2758

In December, the hours of daylight in San Francisco reach a low of approximately $9 \frac{1}{2}$ hours. The hours of daylight increases over the next six months, and in June, it reaches a high of approximately 15 hours before starting to decrease and reach its low again in December.
Which of the sinusoidal functions below could possibly model the number of daylight hours, $D$ , as a function of $t$ , the number of months since December? $D=2.75 \cos \left(\frac{\pi}{6} t\right)+12.25$ $D=-2.75 \cos \left(\frac{\pi}{6} t\right)+12.25$ $D=-2.75 \cos \left(\frac{\pi}{6}(t-3)\right)+12.25$ $D=2.75 \sin \left(\frac{\pi}{6} t\right)+12.25$ $D=2.75 \sin \left(\frac{\pi}{6}(t-3)\right)+12.25$

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Problem 2759

Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of $f(x)=2+3 e^{-0.2 x}$ . ine x-intercept(s) is/are at $x=$ $\square$ (Type an integer or a decimal. Use a comma to separate answers as needed.) B. There are no $x$ -intercepts.
Find the $y$ -intercepts of $f(x)$ . Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The $y$ -intercept(s) is/are at $y=5$ . (Type an integer or a decimal. Use a comma to separate answers as needed.) B. There are no $y$ -intercepts.
Find any horizontal asymptotes of $f(x)$ . Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one horizontal asymptote, $y=2$ . (Type an equation.) B. The function has two horizontal asymptotes. The top asymptote is $\square$ and the bottom asymptote is $\square$ (Type equations.) C. There are no horizontal asymptotes.
Find any vertical asymptotes of $f(x)$ . Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one vertical asymptote, $\square$ . (Type an equation.) B. The function has two vertical asymptotes. The leftmost asymptote is $\square$ and the rightmost asymptote is $\square$ . (Type equations.) C. There are no vertical asymptotes.

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Problem 2760

\begin{tabular}{|c|c|} \hline x & f(x) \\ \hline 3 & \square \\ \hline 5 & 4 \\ \hline 7 & 3 \\ \hline \end{tabular}
The user is trying to find the missing value for $x=3$ in the table. The given values are $f(5) = 4$ and $f(7) = 3$ . The $x$ values increase by 2 each time. The user is seeking to determine the missing value $f(3)$ , potentially by identifying a pattern or rule in the function $f(x)$ .

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Problem 2761

Calculate the linear approximation for $f(x): f(x) \approx f(a)+f^{\prime}(a)(x-a)$ . $f(x)=\frac{1}{2+x} \text { at } a=0$

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Problem 2762

QUESTION 3 Use logarithmic differentiation to find $\frac{d y}{d x}$ if $y=(\tanh (x+1))^{\sin ^{-1}\left(x^{2}\right)}$ .

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Problem 2763

Part 1 of 3
The graph below shows a rectangular sum of $n=8$ rectangles to approximate the area under the line from $x=0$ to $x=2$ .
Is this a right-hand or left-hand sum? right-hand sum $\checkmark^{\vee} \sigma^{\infty}$ $\square$ $\qquad$ What is the equation of the line? $y=2 x \quad$ Part 3 of 3
What is the value of the sum?

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Problem 2764

\begin{tabular}{|l|c|} \hline \multirow{2}{*}{ Function } & Is the function a polynomial? \\ \cline { 2 - 3 } (a) $u(x)=6 x^{-3}+4 x$ & Yes \\ (b) $v(x)=4(x-4)(x+2)$ & No \\ (c) $g(x)=2-\frac{4}{x}$ & \\ (d) $f(x)=-8 \sqrt{x}$ & \\ \hline \end{tabular}

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Problem 2765

For the polynomial below, 1 is a zero. $h(x)=x^{3}+3 x^{2}-2 x-2$
Express $h(x)$ as a product of linear factors. $h(x)=$

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Problem 2766

Using the remainder theorem to
Use the remainder theorem to find $P(1)$ for $P(x)=-x^{4}+3 x^{3}+4 x^{2}-4$ . Specifically, give the quotient and the remainder for the associated division and the value of $P(1)$ . $\square$

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Problem 2767

Suppose there are 20 bacteria in a Petri dish at start time. Seven hours later, there are 390 bacteria in the dish. 1.) Express the number of bacteria, $P$ , as a function of $t$ hours passed.
Note: Round the growth rate to 4 dec . places. $P(t)=$ $\square$ 2.) Use the model from part a to determine the number of bacteria, rounded to a whole number, in the dish after 17 hours: $\square$

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Problem 2768

Graph the integrand. $\begin{array}{c} \int_{-4}^{1} x^{3} d x \\ f(x)= \end{array}$ $\square$

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Problem 2769

What does the unit circle help to illustrate about trigonometric functions? a. Their maximum values only b. Their periodic nature c. Their irrelevance to geometry d. Their independence from angles

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Problem 2770

8. Find constants $a, b$ so that $f(z)=-x^{2}+2 x+y^{2}+i(a y+b x y)$ is analytic. A. $a=2, b=0$ (B) $a=2, b=-2$ C. $a=-2, b=2$ D. $a=-2, b=0$

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Problem 2771

Use the figure to evaluate the definite integrals $\int_{0}^{4} g(t) d t$ and $\int_{1}^{5} g(t) d t$ . (Express numbers in exact form. Use symbolic notation and fractions where needed.)

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Problem 2772

\#1: Find the domain, range, and asymptotes for $f(x)=\frac{x^{2}+7 x-5}{5 x^{2}-7 x-6}$ .

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Problem 2773

(1 point) Find an antiderivative $F(x)$ of $f(x)=5 x-\sqrt{x}$ . $F(x)=$

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Problem 2774

A farmer wants to fence in a rectangular plot of land adjacent to the north wall of his barn. No fencing is needed along the barn, and the fencing along the west side of the plot is shared with a neighbor who will split the cost of that portion of the fence. If the fencing costs $\$ 10$ per linear foot to install and the farmer is not willing to spend more than $\$ 7000$ , find the dimensions for the plot that would enclose the most area. (Enter the dimensions as a comma separated list.)

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Problem 2775

A farmer wants to fence in a rectangular plot of land adjacent to the north wall of his barn. No fencing is needed along the barn, and the fencing along the west side of the plot is shared with a neighbor who will split the cost of that portion of the fence. If the fencing costs $\$ 8$ per linear foot to install and the farmer is not willing to spend more than $\$ 4000$ , find the dimensions for the plot that would enclose the most area. (Enter the dimensions as a comma separated list.)

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Problem 2776

Fill in the blank to complete the fundamental trigonometric identity. $\csc (-u)=\square$

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Problem 2777

Find each of the following in rectangular form (a) $\sin (1-i)$ (c) $\ln \left(-e^{(1+i)}\right)$ (b) $i^{3 / 4}$ (d) $\ln (i+\sqrt{3})$

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Problem 2778

Find the second derivative of the function $f(x) = \frac{1}{4} \sqrt{x} + \frac{1}{x}$ , where $x > 0$ .

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Problem 2779

The problem: For the function $f(x)=\frac{12 x}{(x+3)^{2}}$ - Determine the domain of $f$ , any $x$ or $y$ intercepts, and any horizontal or vertical asymptotes. - Find the critical points of $f$ , and determine the intervals on which $f(x)$ is increasing/decreasing. - Determine the intervals on which the graph of $f$ is concave up/concave down, and the location of any points of inflection. - Sketch the graph of $f$ .
For your reference, note that $f^{\prime}(x)=\frac{12(3-x)}{(x+3)^{3}}$ , and $f^{\prime \prime}(x)=\frac{12(x-6)}{(x+3)^{4}}$ .You do not need compute these derivatives yourself.
Let $f(x)=x^{4}+e^{-3 x}$ . Which of the following is true at $x=0$ ?

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Problem 2780

2. Determine the domain of $y=\log (4 x-7)$ . A. $x>\frac{7}{4}$ B. $x>\frac{4}{7}$ C. $x>-\frac{4}{7}$ D. $x>-\frac{7}{4}$

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Problem 2781

The extent to which species compete for resources is often measured by the niche overlap. If the horizontal axis represents a continuum of different resource types (for example, seed sizes for certain bird species), then a plot of the degree of preference for these resources is called a species' niche. The degree of overlap of two species' niches is then a measure of the extent to which they compete for resources. The niche overlap for a species is the fraction of the area under its preference curve that is also under the other species' curve. The niches displayed in the figure are given by $\begin{array}{ll} n_{1}(x)=(x-4)(6-x) & 4 \leq x \leq 6 \\ n_{2}(x)=(x-5)(7-x) & 5 \leq x \leq 7 \end{array}$
Estimate the niche overlap using midpoints. (Use 10 subintervals. Round your answer to the nearest whole number.) $\square$ \%

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Problem 2782

4. The imaginary part of: $\int_{z=-i}^{z=i-1} \frac{d z}{z^{3}}$ \begin{tabular}{|l|l|l|l|l|} \hline a. -0.25 & b. 0.25 & c. 0 & d. -3.14 & e. 3.14 \\ \hline \end{tabular}

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Problem 2783

2. A ferris wheel has a diameter of 40 m and rotates once every 24 s . a) Draw a graph to show a person's height above or below the centre of rotation starting at the lowest position. (ie. make the middle of the ferris wheel be your 0 value on the $y$ -axis). $\begin{array}{l} a x=20 \quad a=\frac{|\max x-\min |}{2} \\ \text { in }=-20 \quad=\frac{(20-(-20) \mid}{2} \\ =\frac{\operatorname{max+min}}{2} \\ =\frac{20+(-20)}{20^{2}}=\frac{(20+20 \mid}{2} \\ =\frac{120 / 2}{2} \\ =0 / 2 \\ =\sigma \end{array}$ b) Find an equation of the graph in a)

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Problem 2784

2. A ferris wheel has a diameter of 40 m and rotates once every 24 s . a) Draw a graph to show a person's height above or below the centre of rotation starting at the lowest position. (ie. make the middle of the ferris wheel be your 0 value on the $y$ -axis). $\begin{array}{l} \text { sin } \rightarrow \text { no pha } \\ 2=0\end{array}$ $\begin{array}{l} \begin{array}{rl} \max =20 & a \end{array}=\frac{|\max -\min |}{2} \quad+=0 \\ C=\frac{\text { maxrmin }}{2}=\frac{(20-(-20) \mid}{2} \\ =\frac{20+(-20)}{20}=\frac{(20+20)}{2} \\ \begin{array}{l} =\frac{20+(-20)}{20^{2}-20} \\ =\frac{-h e c o i}{2} \end{array} \\ \begin{array}{l} =0 / 2 \quad=20 \\ =\sigma \quad \text { peribs }=24 \end{array} \\ \begin{array}{r} =\sigma / 2 \text { perios }=2 u \\ =\sigma \quad \therefore k=\frac{2}{2} \end{array} \\ \therefore k=\frac{2 \pi}{24}=\frac{\pi}{12} \end{array}$ b) Find an equation of the graph in a) $\qquad$

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Problem 2785

Find the derivative of the function $y = e^{x} \sinh^{-1} x^{2}$ .

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Problem 2786

Leila is walking. $D(t)$ , given below, is her distance in kilometers from Glen City after $t$ hours of walking. $D(t)=12.8-4 t$
Complete the following statements.
Let $D^{-1}$ be the inverse function of $D$ . Take $x$ to be an output of the function $D$ . That is, $x=D(t)$ and $t=D^{-1}(x)$ . (a) Which statement best describes $D^{-1}(x)$ ? The reciprocal of her distance from Glen City (in kilometers) after walking $x$ hours. The amount of time she has walked (in hours) when she is $x$ kilometers from Glen City. Her distance from Glen City (in kilometers) after she has walked $x$ hours. The ratio of the amount of time she has walked (in hours) to her distance from Glen City (in kilometers), $x$ . (b) $D^{-1}(x)=$ $\square$ $\square$ (c) $D^{-1}(8.4)=$ $\square$

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Problem 2787

For each graph, choose the function that best describes it. (a) (Choose one) (b) (Choose one)

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Problem 2788

(b) $f(x)=3 x$ $\begin{array}{l} g(x)=3 x \\ f(g(x))= \\ g(f(x))= \end{array}$ $f$ and $g$ are inverses of each other $f$ and $g$ are not inverses of each other

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Problem 2789

$f(x)=x+3-1 / x$ 1) calculer les limites aux bormnes de Df 2) Deteminer le sems de variation 3) Determinar l'é de l'asymptecte ablique 4) Representation graphique

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Problem 2790

9.1 For each polynomial given below, find the degree, the leading term, the leading coemcient, and the constant term. (a) $f(x)=-\frac{1}{2} x+3 x^{4}$ (b) $g(x)=\left(\frac{1}{3} x^{2}-3\right)(5+2 x)^{2}$ (c) $f(x)+g(x)$ (d) $f(x) \cdot g(x)$ 9.2 (a) Set up the sign diagrams and determine the end behaviour for the polynomials from exercise 9.1 .

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Problem 2791

9. $\begin{array}{l} f: x \mapsto 2 x-1 \\ g: x \mapsto \frac{3}{x}, x \neq 0 \end{array}$ May O4 3H (a) Find the value of (i) $\mathrm{f}(\mathrm{3})$ , (ii) $\mathrm{fg}(6)$ . (b) Express the inverse function $\mathrm{f}^{-1}$ in the form $\mathrm{f}^{-1}: x \mapsto \ldots$ (c) (i) Express the composite function gf in the form $\mathrm{gf}: x \mapsto \ldots$ . (ii) Which value of $x$ must be excluded from the domain of $g$ ?

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Problem 2792

8) Find the values of $(x)$ which $f(x)=\frac{x-3}{|x|-3}$ that not continuous termine whether scontinuity is removable? a) 3 b) -3 c) 0 d) $\{3,-3\}$

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Problem 2793

6) Find the horizontal asymptote of $f(x)=\frac{e^{x}-e^{-x}}{e^{-x}+e^{x}}$ ? a) $y=1$ b) $y=-1$ c) $y=1,-1$ d) None

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Problem 2794

Given $f(x)=(x+8)(x+7)(x+2)$ find the roots give your answers $x_{1} \leq x_{2} \leq x_{3}$ $\begin{array}{l} x_{1}=\square \\ x_{2}=\square \\ x_{3}=\square \end{array}$
Find the $y$ -intercept $=(0$ , $\square$

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Problem 2795

The hands on a clock are of length 6.5 inches (minute hand) and 5.5 inches (hour hand). How fast is the distance between the tips of the hands changing at 9:00? (Assume this is a quality clock with continuous motion, not one of the cheap, jerky ones like they apparently install in $\$ 80$ million schools these days.)
Hint: Be sure to consider whether the distance between the hands is increasing or decreasing at this instant. (Give the exact answer OR round to three decimal places) Number Units

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Problem 2796

```latex f(x) \text{ est une fonction périodique telle que } f(x) = 2x - 1 \text{ pour } x \in \left(0, \frac{3}{2}\right).
\text{Nous devons dessiner la courbe périodique sur l'intervalle } [-1, 2] \text{ avec une période } T = \frac{3}{2}. ```

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Problem 2797

(2) $x+y=x+2 y-x y$ closed, Associative, not commutabive $\left.\begin{array}{rl} x a=x+2 a-a x=x & a(2-x)=0 \\ a=0 \end{array}\right]$ 26/10

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Problem 2798

COPY - Let $f(x)=2 x-1, g(x)=3 x$ , and $h(x)=x^{2}+1$ . Find $h(g(-4))$

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Problem 2799

Let $f(x)=2 x-1, g(x)=3 x$ , and $h(x)=x^{2}+1$ . Find $f(h(7))$

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Problem 2800

Composition of functions
COPY - Let $f(x)=2 x-1, g(x)=3 x$ , and $h(x)=x^{2}+1$ . Find $g(f(0))$

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Function

Problem 2701

Problem 2702

Problem 2703

Problem 2704

Problem 2705

Problem 2706

List the value of all three trigonometric functions for the given angle π/3\pi / 3π/3.

Problem 2707

fr(x)=(ex2x3)′=(ex2⋅x1/5)′\begin{aligned} f^{r}(x) & =\left(\frac{e^{x^{2}}}{\sqrt[3]{x}}\right)^{\prime} \\ & =\left(e^{x^{2}} \cdot x^{1 / 5}\right)^{\prime}\end{aligned}fr(x)​=(3x​ex2​)′=(ex2⋅x1/5)′​

Problem 2708

Problem 2709

Problem 2710

Problem 2711

Problem 2712

Problem 2713

Use the graph to find (a) the xxx-intercept(s) and (b) the zero(s) of the function. (a) The xxx-intercept(s) is/are □\square□ . (Type an ordered pair. Use a comma to separate answers as needed.)

Problem 2714

Find the value of (f∘g)(5)(f \circ g)(5)(f∘g)(5) given that f(x)=2xf(x) = 2\sqrt{x}f(x)=2x​ and g(x)=2g(x) = 2g(x)=2.

Problem 2715

Problem 2716

Problem 2717

Problem 2718

Find the limit. Use l'Hospital's rule where appropriate. If there is a more elementary method, consider using it. lim⁡x→(π/2)+cos⁡(x)1−sin⁡(x)\lim _{x \rightarrow(\pi / 2)^{+}} \frac{\cos (x)}{1-\sin (x)}x→(π/2)+lim​1−sin(x)cos(x)​

Problem 2719

Points: 0 of 1If f(x)=xf(x)=\sqrt{x}f(x)=x​ and g(x)=−2x−2g(x)=-2 x-2g(x)=−2x−2, find (f∘g)(x)(f \circ g)(x)(f∘g)(x) and (g∘f)(x)(g \circ f)(x)(g∘f)(x). (f∘g)(x)=(f \circ g)(x)=(f∘g)(x)= □\square□ (Simplify your answer. Type an exact answer, using radicals as needed.)

Problem 2720

Points: 0Find (f∘g)(x)(f \circ g)(x)(f∘g)(x) and (g∘f)(x)(g \circ f)(x)(g∘f)(x) and the domain of each. f(x)=7x−5,g(x)=x+57f(x)=7 x-5, g(x)=\frac{x+5}{7}f(x)=7x−5,g(x)=7x+5​ (f∘g)(x)=(f \circ g)(x)=(f∘g)(x)= □\square□ (Simplify your answer.)

Problem 2721

If f(x)=x2−6x+2f(x)=x^{2}-6 x+2f(x)=x2−6x+2 and g(x)=−2xg(x)=-2 xg(x)=−2x, find the following composition. (g∘f)(2)(g∘f)(2)=□\begin{array}{r} (g \circ f)(2) \\ (g \circ f)(2)=\square \end{array}(g∘f)(2)(g∘f)(2)=□​

Problem 2722

Find (f∘g)(x)(f \circ g)(x)(f∘g)(x) and (g∘f)(x)(g \circ f)(x)(g∘f)(x). f(x)=−3x2−1,g(x)=2xf(x)=-3 x^{2}-1, g(x)=2 xf(x)=−3x2−1,g(x)=2x (f∘g)(x)=(f \circ g)(x)=(f∘g)(x)= □\square□ (Simplify your answer.)

Problem 2723

Solve by factoring: f(x)=p2−5p−14f(x)=p^{2}-5 p-14f(x)=p2−5p−14

Problem 2724

ChallenGE YOURSELF!!!Which of the following will be the xxx-intercept of y=log⁡b(x−k)y=\log _{b}(x-k)y=logb​(x−k) based on the constants bbb and kkk ? (1) x=k+1x=k+1x=k+1 (2) x=b+1x=b+1x=b+1 (3) x=bk+1x=b^{k}+1x=bk+1 (4) x=kb+1x=k^{b}+1x=kb+1

Problem 2725

Problem 2726

Problem 2727

The yyy-intercept of the function y=0.5x3−2.5x2+0.1x+10y=0.5 x^{3}-2.5 x^{2}+0.1 x+10y=0.5x3−2.5x2+0.1x+10 is Select one: a. -2.5 b. 0.5 c. 10 d. 0.1

Problem 2728

Problem 2729

Find the average value of the following function over the given interval. Draw a graph of the function and indicate the average value. f(x)=8x on [3,3e]f(x)=\frac{8}{x} \text { on }[3,3 e]f(x)=x8​ on [3,3e]

Problem 2730

3. A company estimates that its sales will grow continuously at a rate given by the function S′(t)=S^{\prime}(t)=S′(t)= 20et20 e^{t}20et where S′(t)S^{\prime}(t)S′(t) is the rate at which sales are increasing in dollars per day, on day ttt. Find the accumulated sales for the first 5 days.

Problem 2731

Find fff. (Use CCC for the constant of the first antiderivative and DDD for the constant of the second antiderivative.) f(x)=24x3−18x2+8xf(x)=□\begin{array}{l} f(x)=24 x^{3}-18 x^{2}+8 x \\ f(x)=\square \end{array}f(x)=24x3−18x2+8xf(x)=□​ Need Help? Read It Watch it

Problem 2732

Problem 2733

option. Which of the following will give the most accurate approximation of the definite integral ∫01g(x)dx\int_{0}^{1} g(x) d x∫01​g(x)dx by Simpson's rule? intervals of width 0.1 unit each intervals of width 0.25 unit each intervals of width 0.2 unit each intervals of width 0.5 unit each

Problem 2734

Graph the exponential function. f(x)=2xf(x)=2^{x}f(x)=2xPlot five points on the graph of the function, and also draw the asymptote. Then click on the graph-afunction button.

Problem 2735

Problem 2736

Using the rational root theorem, list out all possible/candidate rational roots of f(x)=−6x4+23x3−4x2−12x+9f(x)=-6 x^{4}+23 x^{3}-4 x^{2}-12 x+9f(x)=−6x4+23x3−4x2−12x+9. Express your answer as integers or as fractions in simplest form. Use commas to separate.

Problem 2737

Problem 2738

Problem 2739

Problem 2740

Graph the exponential function. f(x)=(32)xf(x)=\left(\frac{3}{2}\right)^{x}f(x)=(23​)xPlot five points on the graph of the function, and also draw the asymptote. Then click on the graph-afunction button.

Problem 2741

Explain why the vertex for f(x)=a∣x−h∣+kf(x)=a|x-h|+kf(x)=a∣x−h∣+k is unaffected by aaa.

Problem 2742

Problem 2743

Graph the exponential function. f(x)=−14(4)xf(x)=-\frac{1}{4}(4)^{x}f(x)=−41​(4)xPlot five points on the graph of the function, and also draw the asymptote. Then click on the graph-afunction button.

Problem 2744

1. Alvaro is studying the extinction of the bear population of Siberia over time. The following function gives the number of bears ttt years since Alvaro started tracking it. B(t)=2190e−0.3tB(t)=2190 e^{-0.3 t}B(t)=2190e−0.3t a. What is B′(3)B^{\prime}(3)B′(3) ?

Problem 2745

Graph the exponential function. f(x)=2(23)xf(x)=2\left(\frac{2}{3}\right)^{x}f(x)=2(32​)xPlot five points on the graph of the function, and also draw the asymptote. Then click on the graph-afunction button.

Problem 2746

Problem 2747

Problem 2748

Problem 2749

Problem 2750

Problem 2751

Problem 2752

Problem 2753

Problem 2754

Find the polynomial with a leading coefficient of either 1 or -1 and with smallest possible degree that matches the given graph.The polynomial function is f(x)=f(x)=f(x)= □\square□

Problem 2755

List the value of all three trigonometric functions for the given angle $\pi / 3$ .

$\begin{aligned} f^{r}(x) & =\left(\frac{e^{x^{2}}}{\sqrt[3]{x}}\right)^{\prime} \\ & =\left(e^{x^{2}} \cdot x^{1 / 5}\right)^{\prime}\end{aligned}$

Use the graph to find (a) the $x$ -intercept(s) and (b) the zero(s) of the function. (a) The $x$ -intercept(s) is/are $\square$ . (Type an ordered pair. Use a comma to separate answers as needed.)

Find the value of $(f \circ g)(5)$ given that $f(x) = 2\sqrt{x}$ and $g(x) = 2$ .

Find the limit. Use l'Hospital's rule where appropriate. If there is a more elementary method, consider using it. $\lim _{x \rightarrow(\pi / 2)^{+}} \frac{\cos (x)}{1-\sin (x)}$

Points: 0 of 1
If $f(x)=\sqrt{x}$ and $g(x)=-2 x-2$ , find $(f \circ g)(x)$ and $(g \circ f)(x)$ . $(f \circ g)(x)=$ $\square$ (Simplify your answer. Type an exact answer, using radicals as needed.)

Points: 0
Find $(f \circ g)(x)$ and $(g \circ f)(x)$ and the domain of each. $f(x)=7 x-5, g(x)=\frac{x+5}{7}$ $(f \circ g)(x)=$ $\square$ (Simplify your answer.)

If $f(x)=x^{2}-6 x+2$ and $g(x)=-2 x$ , find the following composition. $\begin{array}{r} (g \circ f)(2) \\ (g \circ f)(2)=\square \end{array}$

Find $(f \circ g)(x)$ and $(g \circ f)(x)$ . $f(x)=-3 x^{2}-1, g(x)=2 x$ $(f \circ g)(x)=$ $\square$ (Simplify your answer.)

Solve by factoring: $f(x)=p^{2}-5 p-14$

ChallenGE YOURSELF!!!
Which of the following will be the $x$ -intercept of $y=\log _{b}(x-k)$ based on the constants $b$ and $k$ ? (1) $x=k+1$ (2) $x=b+1$ (3) $x=b^{k}+1$ (4) $x=k^{b}+1$

The $y$ -intercept of the function $y=0.5 x^{3}-2.5 x^{2}+0.1 x+10$ is Select one: a. -2.5 b. 0.5 c. 10 d. 0.1

Find the average value of the following function over the given interval. Draw a graph of the function and indicate the average value. $f(x)=\frac{8}{x} \text { on }[3,3 e]$

3. A company estimates that its sales will grow continuously at a rate given by the function $S^{\prime}(t)=$ $20 e^{t}$ where $S^{\prime}(t)$ is the rate at which sales are increasing in dollars per day, on day $t$ . Find the accumulated sales for the first 5 days.

Find $f$ . (Use $C$ for the constant of the first antiderivative and $D$ for the constant of the second antiderivative.) $\begin{array}{l} f(x)=24 x^{3}-18 x^{2}+8 x \\ f(x)=\square \end{array}$ Need Help? Read It Watch it

option. Which of the following will give the most accurate approximation of the definite integral $\int_{0}^{1} g(x) d x$ by Simpson's rule? intervals of width 0.1 unit each intervals of width 0.25 unit each intervals of width 0.2 unit each intervals of width 0.5 unit each

Graph the exponential function. $f(x)=2^{x}$
Plot five points on the graph of the function, and also draw the asymptote. Then click on the graph-afunction button.

Using the rational root theorem, list out all possible/candidate rational roots of $f(x)=-6 x^{4}+23 x^{3}-4 x^{2}-12 x+9$ . Express your answer as integers or as fractions in simplest form. Use commas to separate.

Graph the exponential function. $f(x)=\left(\frac{3}{2}\right)^{x}$
Plot five points on the graph of the function, and also draw the asymptote. Then click on the graph-afunction button.

Explain why the vertex for $f(x)=a|x-h|+k$ is unaffected by $a$ .

Graph the exponential function. $f(x)=-\frac{1}{4}(4)^{x}$
Plot five points on the graph of the function, and also draw the asymptote. Then click on the graph-afunction button.

1. Alvaro is studying the extinction of the bear population of Siberia over time. The following function gives the number of bears $t$ years since Alvaro started tracking it. $B(t)=2190 e^{-0.3 t}$ a. What is $B^{\prime}(3)$ ?

Graph the exponential function. $f(x)=2\left(\frac{2}{3}\right)^{x}$
Plot five points on the graph of the function, and also draw the asymptote. Then click on the graph-afunction button.

Find the polynomial with a leading coefficient of either 1 or -1 and with smallest possible degree that matches the given graph.
The polynomial function is $f(x)=$ $\square$

Find a function of the form $y=A \sin (x)+D$ or $y=A \cos (x)+D$ whose graph matches the function shown below:
Leave your answer in exact form; if necessary, type pi for $\pi$ . $y=$ $\square$ Next Question

Calculate the linear approximation for $f(x): f(x) \approx f(a)+f^{\prime}(a)(x-a)$ . $f(x)=\frac{1}{2+x} \text { at } a=0$

QUESTION 3 Use logarithmic differentiation to find $\frac{d y}{d x}$ if $y=(\tanh (x+1))^{\sin ^{-1}\left(x^{2}\right)}$ .

For the polynomial below, 1 is a zero. $h(x)=x^{3}+3 x^{2}-2 x-2$
Express $h(x)$ as a product of linear factors. $h(x)=$

Using the remainder theorem to
Use the remainder theorem to find $P(1)$ for $P(x)=-x^{4}+3 x^{3}+4 x^{2}-4$ . Specifically, give the quotient and the remainder for the associated division and the value of $P(1)$ . $\square$

Graph the integrand. $\begin{array}{c} \int_{-4}^{1} x^{3} d x \\ f(x)= \end{array}$ $\square$

8. Find constants $a, b$ so that $f(z)=-x^{2}+2 x+y^{2}+i(a y+b x y)$ is analytic. A. $a=2, b=0$ (B) $a=2, b=-2$ C. $a=-2, b=2$ D. $a=-2, b=0$

Use the figure to evaluate the definite integrals $\int_{0}^{4} g(t) d t$ and $\int_{1}^{5} g(t) d t$ . (Express numbers in exact form. Use symbolic notation and fractions where needed.)

\#1: Find the domain, range, and asymptotes for $f(x)=\frac{x^{2}+7 x-5}{5 x^{2}-7 x-6}$ .

(1 point) Find an antiderivative $F(x)$ of $f(x)=5 x-\sqrt{x}$ . $F(x)=$