Browse Derivatives problems Studdy Solved!

Problem 501

3. Determine la derivada direccional de $f(x, y)=2 x^{2} y^{3}+6 x y$ en $(1,1)$ en la dirección del vector unitario cuyo ángulo con el eje x positivo es $\frac{\pi}{6}$ . Además, determine el gradiente de fen ( 1,1 ) Recuerde: $u=\cos \theta i+\operatorname{sen} \theta j$

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Problem 502

Question Given the function $f(x)=\frac{x}{1-4 x^{3}}$ , find $f^{\prime}(x)$ in simplified form.
Answer Attempt 1 out of 3

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Problem 503

Question Given the function $y=\frac{3 x^{2}-2}{1-x^{2}}$ , find $\frac{d y}{d x}$ in simplified form.
Answer Attempt 1 out of 3

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Problem 504

(1 point) Suppose that $f(x)=x-7 x^{1 n}$ (A) Find all critical values of $f$ , If there are no critical values, onter -1000 . If there are more than one, enter them separated by commas.
Critical value $(\mathrm{s})=0,1$ (B) Use interval notation to indicate where $f(x)$ is increasing.
Note: When using interval notation in WeBWork, you use I for $\infty,-1$ for $-\infty$ , and $\mathbf{U}$ for the union symbol, If there are no values that satisfy the required condition, then enter " $00^{\circ}$ without the quotation marks. Increasing: $(-\infty, 0) \cup(1, \infty)$ (C) Use interval notation to indicate where $f(x)$ is decreasing.
Decreasing: $(0,1)$ (D) Find the $x$ -coordinates of all local maxima of $f$ . If there are no local maxima, enter -1000. If there are more than one, enter them separated by commas.
Local maxima at $x=$ $\square$ (E) Find the $x$ -coordinates of all local minima of $f$ . If there are no local minima, enter -1000 . If there are more than one, enter them separated by commas.
Local minima at $x=1$ (A) Use interval notation to indicate where $f(x)$ is concave up.
Concave up: $(0, \infty)$ (G) Use interval notation to indicate where $f(x)$ is concave down.
Concave down: $(-\infty, 0)$ (H) Find all inflection points of $f$ . If there are no inflection points, enter -1000 . If there are more than one, enter them separated by commas.
Inflection point(s) at $x=$ $\square$ (1) Use all of the preceding information to sketch a graph of $f$ . When you're finished, enter a " 1 " in the box below.
Graph Complete: $\square$

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Problem 505

$f(x)=x-7 x^{1 / 7}$ (A) Find all critical values of $f$ . If there are no critical values, enter -1000 . If there are more than one, enter them separated by commas.
Critical value $(s)=0,1$ . (B) Use interval notation to indicate where $f(x)$ is increasing.
Note: When using interval notation in WeBWorK, you use I for $\infty,-\mathrm{I}$ for $-\infty$ , and $\mathbf{U}$ for the union symbol. If there are no values that satisfy the required condition, then enter the quotation marks.
Increasing: $(-\infty, 0) \cup(1, \infty)$ (C) Use interval notation to indicate where $f(x)$ is decreasing.
Decreasing: $(0,1)$ (D) Find the $x$ -coordinates of all local maxima of $f$ . If there are no local maxima, enter -1000 . If there are more than one, enter them separated by commas.
Local maxima at $x=$ $\square$ 0 (E) Find the $x$ -coordinates of all local minima of $f$ . If there are no local minima, enter -1000 . If there are more than one, enter them separated by commas.
Local minima at $x=$ $\square$ (F) Use interval notation to indicate where $f(x)$ is concave up.
Concave up: $(0, \infty)$ (G) Use interval notation to indicate where $f(x)$ is concave down.
Concave down: $(-\infty, 0)$ (H) Find all inflection points of $f$ . If there are no inflection points, enter -1000 . If there are more than one, enter them separated by commas. Inflection point(s) at $x=$ $\square$

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Problem 506

(1 point) Suppose that $f(x)=x^{1 / 3}(x+3)^{2 / 3}$ (A) Find all critical values of $f$ if there are no critical values, enter None. If there are more than one, enter them separated by commas Critical value(s) $=-3,-1,0$ (B) Use interval notation to indicate where $f(x)$ is increasing
Note: When using interval notation in WeBWork, you use Ifor $\infty,-1$ for $-\infty$ , and $\mathbf{U}$ for the union symbol. If there are no values that satisfy the required condition, then enter " 0 " without the quotation marks Increasing $(-3,-1) U(0, I)$ (C) Use interval notation to indicate where $f(x)$ is decreasing
Decreasing $\square$ (D) Find the $x$ -coordinates of all local maxima of $f$ If there are no local maxima, enter None If there are more than one, enter them separated by commas
Local maxima at $x=-1$ $\square$ (E) Find the $x$ -coordinates of all local minima of $f$ If there are no local minima, enter None If there are more than one, enter them separated by commas
Local minima at $x=0$ $\square$ (F) Use interval notation to indicate where $f(x)$ is concave up
Concave up $\square$ (G) Use interval notation to indicate where $f(x)$ is concave down.
Concave down: $\square$ (H) Find all inflection points of $f$ . If there are no inflection points, enter None. If there are more than one, enter them separated by commas Inflection point(s) at $x=$ $\square$ (I) Use all of the preceding information to sketch a graph of $f$ When you're finished, enter a 1 in the box below
Graph Complete $\square$

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Problem 507

A balloon rises vertically from the ground 200 m away from an observer. It rises with a position function $h=50 t^{2}$ where $h$ is the height of the balloon in metres and $t$ is in seconds. How fast is the angle of elevation changing 2 seconds after the balloon leaves the ground?

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Problem 508

8. The function $f(x)=-x e^{x}+2$ is concave down at $x=0$ . a. Find the tangent line of $f$ at $x=0$ . b. What is the estimate for $f(-0.1)$ using the local linear approximation for $f$ at $x=0$ ? c. Is it an underestimate or overestimate? Explain.

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Problem 509

13. The function $f$ given by $f(x)=x^{3}+12 x-24$ is A. increasing for $x<-2$ , decreasing for $-2<x<2$ , increasing for $x>2$ B. decreasing for $x<0$ , increasing for $x>0$ C. increasing for all $x$ D. decreasing for all $x$ E. decreasing for $x<-2$ , increasing for $-2<x<2$ , decreasing for $x>2$

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Problem 510

If $y=\sin \left(x^{2}\right)$ , what is $\frac{d y}{d x}$ using the chain rule? - I. $2 x \cos \left(x^{2}\right)$ - II. $2 x \sin \left(x^{2}\right)$ - III. $\cos \left(x^{2}\right)$ - IV. $2 x \sin (x)$ A. IV B. I C. III D. II

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Problem 511

2. The total stopping distance $T$ of a vehicle is $T=2.5 x+0.5 x^{2}$ where $T$ is in feet and $x$ is the speed in miles-per-hour. Approximate the change and percent change in the total stopping distance as the speed changes from $x=25$ to $x=26 \mathrm{mph}$ .

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Problem 512

Die Eìnführung der Rippenqualle Beroe ovata, einem Fressfeind, konnte die Population im Schwarzen Meer zurückdrängen. Die Funktion g mit $g(x)=30+x^{2} \cdot e^{-0,1 \cdot x}, 0 \leq x \leq 80$ modelliert die Populationsdichte mit Fressfeind. Die folgenden Aufgaben beziehen sich alle auf die Funktion g. e) Berechnen Sie die maximale Anzahl an Mnemiopsis leidyi pro Kubikmeter. f) Berechnen Sie den Zeitpunkt, zu dem die Population an stärksten abnimmt. 3 g) Bestimmen Sie die durchschnittliche Änderungsrate im Zeitintervall [20; 80]. Untersuchen Sie, ob es einen Zeitpunkt in diesem Zeitintervall gibt, an dem die momentane Änderungsrate so groß ist wie die durchschnittliche Änderungsrate des Zeitintervalls. 4 5
Eine vereinfachte Modellierung geht davon aus, dass die Populationsdichte ab einem bestimmten Zeitpunkt $z$ durch die Tangente an den Graphen von $g$ im Punkt $P(z \mid g(z))$ beschrieben werden kann. h) Bestimmen Sie für $z=70$ die Gleichung der Tangente und den Zeitpunkt, zu dem die Populationsdichte nach diesem vereinfachten Modell null ist. 5

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Problem 513

The derivative of $\sqrt{\sin 2 x}$ is
Select one: $\frac{\cos x}{\sqrt{\sin x}}$ $-\frac{\cos 2 x}{\sqrt{\sin 2 x}}$ $\frac{\cos 2 x}{\sqrt{\sin 2 x}}$ $\frac{\cos 2 x}{2 \sqrt{\sin 2 x}}$ Check

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Problem 514

3. Soit g la fonction définie sur $\left[1 ;+\infty\left[\right.\right.$ par: $\quad g(x)=x^{3}-3 x-3$ . a) Montrer que la fonction $g$ admet une fonction réciproque définie sur un intervalle $J$ à déterminer. b) Montrer que l'équation $g(x)=0$ admet une unique solution $\alpha$ dans $[1 ;+\infty[$ . c) Montrer que $\left(g^{-1}\right)^{\prime}(0)=\frac{1}{3\left(\alpha^{2}-1\right)}$ .

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Problem 515

2. (1,0 ponto cada item) Usando as regras de diferenciaçāo, encontre as derivadas das funçöes a seguir, simplificando ao máximo se possível. (a) $y=\frac{\operatorname{sen} x}{\cos x-1}$ (c) $y=\cos \left(2^{x}\right)$ (b) $f(x)=x \operatorname{arctg} x-\frac{1}{2} \ln \left(1+x^{2}\right)$ (d) $y=(x)^{(\cosh x)}$

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Problem 516

1. Kurvendiskussion
Gegeben ist die Funktion $f(x)=(x-1) \cdot e^{x}$ . a) Bestimmen Sie die Ableitungen $\mathrm{f}^{\prime}, \mathrm{f}^{\prime \prime}$ und $\mathrm{f}^{\prime \prime \prime}$ . b) Untersuchen Sie die Funktion f auf Nullstellen. c) Die Funktion f besitzt ein Extremum und einen Wendepunkt. Wo liegen diese Punkte? d) Untersuchen Sie das Verhalten von f für $\mathrm{x} \rightarrow-\infty$ bzw. $\mathrm{x} \rightarrow \infty$ mit einer Tabelle. e) Skizzieren Sie den Graphen von $\mathrm{f}(-3 \leq \mathrm{x} \leq 2)$ .

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Problem 517

7-18 Use la parte 1 del teorema fundamental del cálculo para encontrar la derivada de cada una de las siguientes funciones.
7. $g(x)=\int_{1}^{x} \frac{1}{t^{3}+1} d t$
8. $g(x)=\int_{3}^{x} e^{t^{2}-t} d t$

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Problem 518

Find $y^{\prime}$ . $\begin{array}{l} y=\left(\frac{x+1}{x-7}\right)^{7} \\ y^{\prime}=\square \end{array}$

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Problem 519

What is the first derivative of $f(x)=\log _{2}\left(6 x^{3}+2 x+6\right) ?$
Select the correct answer below: $\log _{2}\left(18 x^{2}+2\right)$ $\frac{1}{(\ln 2)\left(6 x^{3}+2 x+6\right)}$ $\frac{18 x^{2}+2}{(\ln 2)\left(6 x^{3}+2 x+6\right)}$ $\frac{(\ln 2)\left(18 x^{2}+2\right)}{6 x^{3}+2 x+6}$ $\frac{18 x^{2}+2}{6 x^{3}+2 x+6}$

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Problem 520

If $r(x)=\log _{5}\left(\frac{x^{6} 5^{x}}{5 x^{2}+5}\right)$ , find $r^{\prime}(x)$
Select the correct answer below: $r^{\prime}(x)=\frac{6}{x \ln 5}+1-\frac{10 x}{\left(5 x^{2}+5\right) \ln 5}$ $r^{\prime}(x)=\frac{5 x^{2}+5}{x^{5}(5) \ln 5}$ $r^{\prime}(x)=\frac{6}{x \ln 5}+\frac{x 5^{x-1}}{5^{2} \ln 5}-\frac{10 x}{\left(5 x^{2}+5\right) \ln 5}$ $r^{\prime}(x)=\frac{6}{x}+1-\frac{10 x}{5 x^{2}+5}$

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Problem 521

What is $g^{\prime}(x)$ when $g(x)=\ln \left(3 x^{3}+2\right)$ ?
Select the correct answer below: $\ln \left(9 x^{2}\right)$ $\frac{1}{3 x^{3}+2}$ $\frac{3 x^{3}+2}{9 x^{2}}$ $\frac{9 x^{2}}{\ln \left(3 x^{3}+2\right)}$ $\frac{9 x^{2}}{3 x^{3}+2}$

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Problem 522

What is the first derivative of $t(x)=\ln \left(4 x^{3}+4 x^{2}+6 x+4\right)$ ?
Select the correct answer below: $\frac{1}{4 x^{3}+4 x^{2}+6 x+4}$ $\frac{12 x^{2}+8 x+6}{4 x^{3}+4 x^{2}+6 x+4}$ $\frac{12 x^{2}+8 x+6}{\ln \left(4 x^{3}+4 x^{2}+6 x+4\right)}$ $\ln \left(12 x^{2}+8 x+6\right)$ $\frac{4 x^{3}+4 x^{2}+6 x+4}{12 x^{2}+8 x+6}$

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Problem 523

Let $f(x)=-5 \ln (\sin x)$
Then $f^{\prime}(x)=$ $\square$ and $f^{\prime \prime}(x)=$ $\square$

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Problem 524

Question 3 (1 point) If $f(x)=2 \cos 3 x$ , find $f^{\prime}\left(\frac{\pi}{3}\right)$ . a) 3 b) 0 c) -6 d) 6

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Problem 525

For Problems 12 and 13, consider the table below \begin{tabular}{|c|c|c|c|c|} \hline $x$ & $f(x)$ & $f^{\prime}(x)$ & $g(x)$ & $g^{\prime}(x)$ \\ \hline 1 & 2 & $\frac{1}{2}$ & -3 & 5 \\ \hline 2 & 3 & 1 & 0 & 4 \\ \hline 3 & 4 & 2 & 2 & 3 \\ \hline 4 & 6 & 4 & 3 & $\frac{1}{2}$ \\ \hline \end{tabular}
12. If $h(x)=f^{-1}(x)$ find $h^{\prime}(3)$
13. If $h(x)=g^{-1}(x)$ find $h^{\prime}(-3)$

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Problem 526

2. The graph of $g^{\prime}$ , the derivative of $g$ , is shown above for $[-2,11]$ and consists of three line segments and a semicircle. Use the graph of $g^{\prime}$ to answer the following. A) Find all critical values of $g(x)$ in the open interval $(-2,11)$ . B) At which value(s) of $x$ does $g(x)$ have a local minimum? Explain your reasoning. C) On what interval(s) in $[-2,11]$ is $g(x)$ increasing? Give a reason for your answer D) Does $g(x)$ have a local minimum, local maximum, or neither at $x=4$ ? Give a reason for your answer. E) Let $h(x)=x g(x)$ . If $h(x)$ has a relative maximum at $x=4$ , find $h(4)$ . Show the work that leads to your answer.

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Problem 527

Use implicit differentiation to find an equation of the tangent line to the curve $\sin (x+y)=8 x-8 y$ at the point $(\pi, \pi)$ . Tangent Line Equation: $\square$

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Problem 528

What is the first derivative of $q(x)=\left(4^{x}\right)^{5} ?$
Select the correct answer below: $q^{\prime}(x)=4^{5 x}(5 \ln 4)$ $q^{\prime}(x)=5\left(4^{x}\right)^{4}$ $q^{\prime}(x)=5\left(4^{5 x}\right)$ 人to $q^{\prime}(x)=(5 x) 4^{5 x-1}$ $q^{\prime}(x)=4^{5 x}(4 \ln 5)$

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Problem 529

Name MP2T2B Mathematics Department Jenlsha Thempson, AP(S) $11 / 20 / 2024$ Score $\qquad$ Questions 1-13 are 6 points each. Questions 14 is 8 points and question 15 is 14 points. MCs21X - AP Calculus BC Ms. M. E. Dela Cruz
1. Using a GC, find, correct to 3 decimal places, an estimate for the value of $x$ at which the graph of $y=x^{4}-2 x^{3}+5 x-8$ has a horizontal tangent. $y^{\prime}=4 x^{3}-6 x^{2}+5$
2. Which is true about the function $f(x)=\sqrt[3]{x^{2}}$ ? (A) It has a vertical tangent at $\mathrm{x}=0$ . (B) It has a stationary point at $x=0$ . (D) It has a cusp at $x=0$ . (C) It has a relative maximum at $x=0$ . (E) It is discontinuous at $x=0$ .
3. If $y=\frac{x}{x^{2}+4}$ , then $\frac{d y}{d x}=$ (A) $\frac{x^{2}-4}{\left(x^{2}+4\right)^{2}}$ (B) $\frac{4-x^{2}}{\left(x^{2}+4\right)^{2}}$ $(1)\left(x^{2}+4\right)+(x)(2 x)$ $(x-4)+(x-1)$
4. Which of the following functiche (D) $\frac{4-x^{2}}{x^{2}+4}$ (E) $\frac{x^{2}-4}{x^{2}+4}$ (A) $y=x^{3}$ (B) $y=\frac{1}{x+1}$ (C) $y=|x|$ (D) $y=\sqrt{x}$ (E) $y=(x-2)^{\frac{2}{3}}$
5. If $f$ is continuous on $[a, b]$ which of the following is always true? (A) $\quad f$ is differentiable on $(a, b)$ . (B) $\quad f$ is either increasing or decreasing on [a, b] (C) $f$ has both a maximum and a minimum value on [a, b]. (D) The maximum value of $f$ is greater than the minimum value of $f$ .
6. Find all the intervals over which the function $f(x)=x^{3}-3 x^{2}$ is (a) increasing $(6, \infty)$ $3 x^{2}-6 x$ 6 $3 x\left(x^{2}-6\right) \quad x=6$ (b) decreasing $(-\infty, 0) \cup(0,6)$
7. Write, in point-slope form, the equation of the tangent line to the graph of $y=x-x^{2}$ at $(1,0)$ .
8. Find the slope-intercept form of the equation of the normal line to the curve $y=x^{3}$ at the point at which $x=\frac{1}{3}$ .
9. Find the exact values of the absolute extrema of the function $f(x)=\frac{x^{3}}{x+2}$ on $[-1,1]$

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Problem 530

If $t$ is in minutes since the drug was administered, the concentration, $C(t) \mathrm{in} \mathrm{ng} / \mathrm{ml}$ , of a drug in a patient's bloodstream is given by $C(t)=20 t e^{-0.04 t}$ . (a) How long does it take for the drug to reach peak concentration? What is the peak concentration?
Round your answers to one decimal place.
The drug reaches its peak concentration at i $\square$ minutes.
The peak concentration is i $\square$ $\mathrm{ng} / \mathrm{ml}$ . (b) What is the concentration of the drug in the body after 15 minutes? After an hour?
Round your answers to one decimal place. The concentration after 15 minutes is $\square$ $\mathrm{ng} / \mathrm{ml}$ .
The concentration after 60 minutes is $\square$ i $\mathrm{ng} / \mathrm{ml}$ . (c) If the minimum effective concentration is $10 \mathrm{ng} / \mathrm{ml}$ , when should the next dose be administered? cytosolic proteins cytoso

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Problem 531

7. $y=\frac{x^{2}+1}{x}$ a. Find intervals increasing or decreasing
6. find intervaler concave up or concave down c. find all arymptoter d. points at local max, local min, paI e.sketch the function

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Problem 532

7. [0.6/1 Points] DETAILS MY NOTES SCALCET9 4.7.065. PREVIOUS ANSWERS ASK YOUR TEACHE
If $C(x)$ is the cost of producing $x$ units of a commodity, then the average cost per unit is $c(x)=\frac{C(x)}{x}$ . Consider the cost function $C(x)$ given below. (Round your answers to the nearest cent.) $C(x)=54,000+140 x+4 x^{3 / 2}$ (a) Find the total cost (in dollars) at a production level of 1,000 units. \ $320,491.11$ \square $(b) Find the average cost (in dollars per unit) at a production level of 1,000 units. \$$ \square $320.49$ \checkmark $per unit (c) Find the marginal cost (in dollars per unit) at a production level of 1,000 units. \$$ \square $329.74 - per unit (d) Find the production level (in units) that will minimize the average cost.$ \square $units (e) What is the minimum average cost (in dollars per unit)? \$$ \square$ per unit

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Problem 533

Find the intervals where the function $f(x) = \frac{1}{2} x^4 - 4x^2 + 7$ is concave upward.

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Problem 534

Given the values of the derivative $f^{\prime}(x)$ in the table and that $f(0)=150$ , find or estimate $f(x)$ for $x=0,2,4,6$ . \begin{tabular}{|c|c|c|c|c|} \hline $x$ & 0 & 2 & 4 & 6 \\ \hline $f^{\prime}(x)$ & 13 & 28 & 43 & 55 \\ \hline $f(x)$ & $\square$ & & & $\square$ \\ \hline \end{tabular}

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Problem 535

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Problem 536

Find the following derivative: ( 1 Mark) $f(x)=8 x^{3}-\frac{4}{x^{4}} x-12 x-0.1$

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Problem 537

6) An inverted conical tank with radius 3 meter and height 4 meter is fully filled with water. The water is leaking from the vertex of the tank at a rate of $0.2 \mathrm{~m}^{3} \mathrm{~s}^{-1}$ . At the instant when the height of water is 2 meters, calculate the rate of change of (a) the water level. (7 marks) (b) the radius of the water surface. (2 marks)

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Problem 538

tion de la tangente à la courbe représentative de $f$ au point d'abscisse a. $f(x)=-x^{2}+2 x-8 ; a=-2$

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Problem 539

Determination de tangentes Pour les exercices 665 à G2\ $, déterminez une équation de la tangente à la courbe représentative de$ f $au point d'abscisse$ a$.
67 数 $f(x)=\frac{x+3}{1-2 x} ; a=-1$ .

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Problem 540

Set $x^{}=1$ . For each of the implicit curves below, do the following: - Find $y^{}$ (there may be more than one possible $y^{}$ value) where $\left(x^{}, y^{}\right)$ is a point on the curve and - calculate $\left.\frac{d y}{d x}\right|_{\left(x^{}, y^{*}\right)}$ .
The implicit curves are (a) $x^{2} y+6 x^{4} y^{2}=\sin (\pi x)$ (b) $\frac{(x-1) y}{\sqrt{1+x^{2}}}+(1+x y)=\tan ((x-1) y)$ (c) $x^{2}-y^{2}=\cos \left(\frac{\pi}{2} x+(x-1) y\right)$

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Problem 541

Set $x^{}=1$ . For each of the implicit curves below, do the following: Find $y^{}$ (there may be more than one possible $y^{}$ value) where $\left(x^{}, y^{}\right)$ is a point the curve and - calculate $\left.\frac{d y}{d x}\right|_{\left(x^{}, y^{*}\right)}$ .
The implicit curves are

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Problem 542

1 - 2

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Problem 543

The function $f(x)=3 x+5 x^{-1}$ has one local minimum and one local maximum. This function has a local maximum at $x=$ with value and a local minimum at $x=$ $\square$ with value $\square$

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Problem 544

Delow is the function $f(x)$ .
Over which interval of $x$ values is $f^{\prime}>0$ ? $(3, \infty)$ $[3, \infty)$ $(-\infty, 3)$ $(-\infty, 3]$ $(-\infty, \infty]$
Over which interval of $x$ values is $f^{\prime}<0$ ? $(3, \infty)$ $[3, \infty)$ $(-\infty, 3)$ $(-\infty, 3]$ $(-\infty, \infty]$
Over the interval $(-\infty, \infty)$ , this function is concave up ( $f^{\prime \prime}>0$ ) concave down ( $f^{\prime \prime \prime}<\mathbf{0}$ )

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Problem 545

Consider the function $f(x)=x^{2} e^{6 x}$ . $f(x)$ has two inflection points at $x=C$ and $x=D$ with $C<D$ where $C$ is $\square$ and $D$ is $\square$ Finally for each of the following intervals, tell whether $f(x)$ is concave up or concave down. $(-\infty, C)$ : Select an answer $\checkmark$ $(C, D)$ : Select an answer $\checkmark$ $(D, \infty)$ Select an answer $\checkmark$ Question Help: Video

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Problem 546

pondus Monitor Question 7, 4.4.56 Test Score: $75.2 \%, 75.2$ of 100 points Points: 7.2 of 12 Close
Use the derivative $f^{\prime}(x)=(x-2)(x+2)(x+4)$ to determine the local maxima and minima of $f$ and the intervals of increase and decrease. Sketch a possible graph of $f(f$ is not unique).
The local maximum/maxima is/are at $x=-2$ . (Use a comma to separate answers as needed.) The local minimum/minima is/are at $x=-4,2$ . (Use a comma to separate answers as needed.) The interval(s) of increase is(are) $(-4,-2),(2, \infty)$ . (Type your answer in interval notation. Use a comma to separate answers as needed.) The interval(s) of decrease is(are) $(-\infty,-4),(-2,2)$ . (Type your answer in interval notation. Use a comma to separate answers as needed.) Which is a possible graph of $f$ ? A. B. c. D.

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Problem 547

J Test Booklet
FRQ 4.1 and 4.2 hegafice (gany buck herve) Caren rides her bicycle along a straight road from home to school, starting at home at time $t=0$ minutes and arriving at school at time $t=12$ minutes. During the time interval $0 \leq t \leq 12$ minutes, her velocity $v(t)$ , in miles per minute, is modeled by the piecewise-linear function whose graph is shown above.
2. Shortly after leaving home, Caren realizes she left her calculus homework at home, and she returns to get it. At what time does she turn around to go back home? Give a reason for your answer. Canen turns aroun I to go hame at $t=2$ minutes beracde at theat tiphe we can see that her nelocity deereazes arich cun mean she went back hone. A particle moves along a straight line. For $0 \leq t \leq 5$ , the velocity of the particle is given by $v(t)=-2+\left(t^{2}+3 t\right)^{6 / 5}-t^{3}$ , and the $\begin{array}{l} \text { position of the particle is given by } s(t) \text {. It is known that } s(0)=10 . \quad-2+\left(t^{2}+3 t\right)^{6 / 5}-1^{3}=2 \\ v(t)=-2 \\ -2+\left(t^{2}+3 t\right)^{6 / 5}-t^{3}=-2 \\ \left.\left(t^{2}+3 t\right)^{6 / 5}-t^{3}=0 \quad\left(t^{2}+3 t\right)^{6 / s}=t^{3} \quad-3 t\right)^{6 / 5}-t^{3}=4 \end{array}$
3. 国 Find all values of $t$ in the interval $2 \leq t \leq 4$ for which the speed of the particle is 2 . $|V(t)|=2$ in the interval $2 \leq t \leq y$ are $t \approx 2.80$ and $t \approx 3.292$ . $a(t)=v(t)=-\sin \left(\frac{t^{2}}{2}\right)-(t+1) \cos \left(\frac{t^{2}}{2}\right) \cdot t$
A particle moves along the $x$ -axis so that its velocity at time $t$ is given by $a(t)=-\sin \left(\frac{t^{2}}{2}\right)-t(t+1) \cos \left(\frac{t^{2}}{2}\right)$ $\begin{array}{l} v(t)=-(t+1) \sin \left(\frac{t^{2}}{2}\right), a(2)=-\sin \left(\frac{2^{2}}{2}\right)-2(2-1) \cos \left(\frac{z^{2}}{2}\right) \\ v(2)=-(2+1) \sin \left(\frac{2^{2}}{2}\right)=-3 \sin (2)=-\sin (2)-6 \cos (2) \end{array}$
At time $t=0$ , the particle is at position $x=1$ .
4. 园 Find the acceleration of the particle at time $t=2$ . Is the speed of the particle increasing at $t=2$ ? Why or why not? The ucueteration at $y=2$ is Page 2 of 2 AP Calculus AB

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Problem 548

9. Determine the derivative of the following functions: $\begin{array}{ll} f(x)=\left(x^{2}+x+1\right)^{3} ; & f(x)=\sqrt{x}+\frac{1}{x} \\ f(x)=\frac{\sin \left(x^{3}+1\right)}{\cos (2 x+2)} ; & f(x)=\sin \left(\sqrt{x^{2}+1}\right) \\ f(x)=\operatorname{tg}\left(x^{2}\right) ; & f(x)=e^{x^{2} \sin x} \end{array}$

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Problem 549

9. Determine the derivative of the following functions: $\begin{array}{ll} f(x)=\left(x^{2}+x+1\right)^{3} ; & f(x)=\sqrt{x}+\frac{1}{x} \\ f(x)=\frac{\sin \left(x^{3}+1\right)}{\cos (2 x+2)} ; & f(x)=\sin \left(\sqrt{x^{2}+1}\right) \\ f(x)=\operatorname{tg}\left(x^{2}\right) ; & f(x)=e^{x^{2} \sin x} \end{array}$

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Problem 550

9. Determine the derivative of the following functions: $\begin{array}{ll} f(x)=\left(x^{2}+x+1\right)^{3} ; & f(x)=\sqrt{x}+\frac{1}{x} \\ f(x)=\frac{\sin \left(x^{3}+1\right)}{\cos (2 x+2)} ; & f(x)=\sin \left(\sqrt{x^{2}+1}\right) \\ f(x)=\operatorname{tg}\left(x^{2}\right) ; & f(x)=e^{x^{2} \sin x} \end{array}$

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Problem 551

Consider the following. (If an answer does not exist, enter DNE.) $f(x)=7 \sin (x)+7 \cos (x), \quad 0 \leq x \leq 2 \pi$ (a) Find the interval(s) on which $f$ is increasing. (Enter your answer using interval notation.) $\square$ (b) Find the interval(s) on which $f$ is decreasing. (Enter your answer using interval notation.) $\square$ (c) Find the local minimum and maximum values of $f$ . local minimum value $\square$ local maximum value $\square$

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Problem 552

WeBWorK 5 - Topics 10 - 12: Problem 3 (1 point)
Consider the function $f(x)$ whose second derivative is $f^{\prime \prime}(x)=10 x+5 \sin (x)$ . If $f(0)=4$ and $f^{\prime}(0)=2$ , what is $f(x)$ ? Answer: $\square$

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Problem 553

What is the first derivative of $p(x)=\ln \left(4 x^{3}+x^{2}+1\right)$ ?
Select the correct answer below: $\frac{12 x^{2}+2 x}{4 x^{3}+x^{2}+1}$ $\frac{4 x^{3}+x^{2}+1}{12 x^{2}+2 x}$ $\frac{1}{4 x^{3}+x^{2}+1}$ $\ln \left(12 x^{2}+2 x\right)$ $\frac{12 x^{2}+2 x}{\ln \left(4 x^{3}+x^{2}+1\right)}$

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Problem 554

What is the first derivative of $q(x)=\log _{5}\left(\frac{2 x^{2}+8}{x^{4} 5^{x}}\right) ?$
Select the correct answer below: $q^{\prime}(x)=\frac{4 x}{2 x^{2}+8}-\frac{4}{x}-1$ $q^{\prime}(x)=\frac{1}{\left(2 x^{2}+8\right) \ln 5}-\frac{4}{x \ln 5}-1$ $q^{\prime}(x)=\frac{4 x}{\left(2 x^{2}+8\right) \ln 5}-\frac{4}{x \ln 5}-1$ $q^{\prime}(x)=\frac{x^{4}(5) \ln 5}{2 x^{2}+8}$

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Problem 555

Consider the equation below. (If an answer does not exist, enter DNE.) $f(x)=6 \cos ^{2}(x)-12 \sin (x), \quad 0 \leq x \leq 2 \pi$ (a) Find the interval on which $f$ is increasing. (Enter your answer using interval notation.) $\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$
Find the interval on which $f$ is decreasing. (Enter your answer using interval notation.) $\left(0, \frac{\pi}{2}\right) \cup\left(\frac{3 \pi}{2}, 2 \pi\right)$ (b) Find the local minimum and maximum values of $f$ . local minimum value $-12$ local maximum value 12 (c) Find the inflection points. (Order your answers from smallest to largest $x$ , then from smallest to largest $y$ .) $\begin{array}{l} (x, \sqrt{x})=(\square) \\ (x, y)=(\square) \end{array}$
Find the interval on which $f$ is concave up. (Enter your answer using interval notation.) $\square$ Find the interval on which $f$ is concave down. (Enter your answer using interval notation.) $\square$

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Problem 556

Consider the equation below. (If an answer does not exist, enter DNE.) $f(x)=6 \cos ^{2}(x)-12 \sin (x), \quad 0 \leq x \leq 2 \pi$ (a) Find the interval on which $f$ is increasing. (Enter your answer using interval notation.) $\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$
Find the interval on which $f$ is decreasing. (Enter your answer using interval notation.) $\left(0, \frac{\pi}{2}\right) \cup\left(\frac{3 \pi}{2}, 2 \pi\right)$ (b) Find the local minimum and maximum values of $f$ . local minimum value $\quad-12$ local maximum value 12 (c) Find the inflection points. (Order your answers from smallest to largest $x$ , then from smallest to largest $y$ .) $\begin{array}{l} (x, y)=\left(\frac{\pi}{6}, \frac{3}{2}\right) \\ (x, y)=\left(\frac{5 \pi}{6}, \frac{3}{2}\right) \end{array}$
Find the interval on which $f$ is concave up. (Enter your answer using interval notation.) $\left(0, \frac{\pi}{6}\right) \cup\left(\frac{5 \pi}{6}, \frac{3 \pi}{2}\right)$
Find the interval on which $f$ is concave down. (Enter your answer using interval notation.)

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Problem 557

If $f(x)=\sin \left(4 x+\frac{\pi}{2}\right)$ , find $f^{\prime}\left(\frac{\pi}{4}\right)$ . 0 4 $-4$ 1

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Problem 558

The position of an object at t seconds is given in feet by $f(t)=3 t^{2}+4 t^{-1}+\sqrt{t}$
What is the object's velocity at 4 seconds?

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Problem 559

Find the derivative $f^{\prime}(x)$ of $f(x)=|x|$ for $x>0$ , $x<0$ , and at $x=0$ . Explain why it's undefined at $0$ .

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Problem 560

Find the $x$ values where the function $f$ has a relative maximum given $f^{\prime}(x)=x^{2}(x+1)^{3}(x-4)^{2}$ .

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Problem 561

Given values of $f^{\prime \prime}(x)$ : at $x=-1$ is -4, $x=0$ is -1, $x=1$ is 2, $x=2$ is 5, $x=3$ is 8. What type of function is $f$ ? (A) linear (B) quadratic (C) cubic (D) fourth-degree (E) exponential

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Problem 562

Find the value of $c$ such that $f(x)=x+\frac{c}{x}$ has a local minimum at $x=3$ .

See Solution

Problem 563

Find the time $t$ (from 0 to 10) when the object with velocity $v(t)=t \cos t-\ln (t+2)$ reaches maximum speed.

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Problem 564

Find the time $t$ in $[0, 10]$ when the velocity $v(t)=t \cos t - \ln(t+2)$ is maximized. Options: A. 9.5 B. 5.1 C. 6.4 D. 7.6

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Problem 565

Estimate $y$ at $x=3.1$ using the tangent line of the curve $x^{3}+x \tan y=27$ at $(3,0)$ . Choose from (A) -2.7 (B) -0.9 (C) 0 (D) 0.1 (E) 3.0.

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Problem 566

Find the rate of change of sales $S(t)=10,000+2000 t-200 t^{2}$ and interpret $S^{\prime}(t)$ .

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Problem 567

Find the derivative of the function $f(x)=6 \cdot \sqrt{x} \cdot(x-5)$ . What is $f^{\prime}(x)$ ?

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Problem 568

Find the derivative $f^{\prime}(x)$ of the function $f(x)=-3 \cdot \ln (4 \cdot x)$ and calculate $f^{\prime}(2)$ .

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Problem 569

Find the derivative of $f(x)=\frac{\sqrt{x}-7}{\sqrt{x}+7}$ and calculate $f^{\prime}(5)$ .

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Problem 570

Find the derivative $f'(x)$ of the function $f(x)=\sqrt{4x^2 + 4x + 4}$ and evaluate it at $x=4$ .

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Problem 571

Calculate the derivative of $g(x)=(4x^{2}+5x) \cdot e^{x}$ .

See Solution

Problem 572

Find $y^{\prime}(4)$ using implicit differentiation for $4 \cdot x^{2}+3 \cdot x+x \cdot y=4$ and $y(4)=-18$ .

See Solution

Problem 573

Find the derivative $f^{\prime}(x)$ of the function $f(x)=2+\frac{7}{x}+\frac{7}{x^{2}}$ .

See Solution

Problem 574

Find the derivative of $R(j)=\left(\ln \left(j^{2}\right)\right)^{2}$ at $j=e$ .

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Problem 575

Find the derivative $f^{\prime}(t)$ of the function $f(t)=(t^{2}+5t+3)(2t^{-2}+6t^{-5})$ .

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Problem 576

Find the slope of the tangent line to the curve $4xy^{3}+3xy=7$ at the point $(1,1)$ using implicit differentiation.

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Problem 577

Find the marginal cost for both businesses given their cost functions: $C_A(x)=200+25x+0.1x^2$ and $C_B(x)=400+80x+0.06x^2$ . For $x=500$ , which business has the lowest cost for the next tire?

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Problem 578

Find the critical number of the function $f(x)=(8 \cdot x-7) \cdot e^{5 \cdot x}$ .

See Solution

Problem 579

The demand function is $D(x)=967-25-x$ .
(a) Find the elasticity of demand. (b) Find the price where elasticity equals 1. (c) Determine the price for maximum revenue.

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Problem 580

Find the value of $x$ that satisfies the equation $\frac{d}{d x}(x \cdot(967-25-x))=0$ .

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Problem 581

Projectile height is given by $h(t)=-16 \cdot t^{2}+256-t$ . Find: (a) average velocity for 3s, (b) speed & height at 6s, (c) max height & time, (d) acceleration at $t=5$ .

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Problem 582

Helium fills a balloon at 2 ft³/s. Find the radius increase rate (in ft/s) after 7 minutes. Use $V=\frac{4}{3} \cdot \pi \cdot r^{3}$ .

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Problem 583

Find the average rate of change of profit $P(x)=-400x^2+6800x-12000$ over $[6,6+h]$ for $h=1, 0.1, 0.01, 0.001, 0.0001$ . What do the results indicate?

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Problem 584

A particle's position is given by $s=2 \cdot t^{3}-7 \cdot t^{2}-9 \cdot t+12$ . Find velocity, acceleration, and specific values.

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Problem 585

Projectile height is given by $h(t)=-16 \cdot t^{2}+256 \cdot t$ . Find average velocity, speed, max height, and acceleration at $t=5$ .

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Problem 586

Differentiate $y=\sqrt{a+x}$ .

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Problem 587

Differentiate $y=\frac{1}{\sqrt{a+x^{2}}}$ with respect to $x$ .

See Solution

Problem 588

Find the derivative of $y=x^{3} \ln (x)$ with respect to $x$ : $\frac{d y}{d x}$ .

See Solution

Problem 589

Find the tangent line to $f(x)=-x^{2}+5$ at $x=5$ and use it to estimate $f(5.1)$ .

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Problem 590

Find points $(x, y)$ where $\nabla f(x, y)=(0,0)$ for $f(x, y)=\frac{x^{3}}{3}-\frac{y^{3}}{3}+3xy$ and classify them.

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Problem 591

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the equation $\sqrt{y}+x y=5$ .

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Problem 592

Find $\frac{d y}{d x}$ in terms of $x$ given that $x=\sec 3 y$ .

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Problem 593

Find $\frac{d y}{d x}$ in terms of $t$ for $x=\sin(t)$ and $y=\cos^2(t)$ . Simplify your expression.

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Problem 594

Find $\frac{d y}{d x}$ in terms of $t$ for the equations $x=\sin^3 t$ and $y=\cos^4 t$ .

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Problem 595

Find and simplify $\frac{d y}{d x}$ using the parametric equations $x=\sin(t)$ and $y=\cos^2(t)$ .

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Problem 596

Find $\frac{d y}{d x}$ in terms of $t$ for the equations $x=\sin(t)$ , $y=\cos^2(t)$ .

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Problem 597

Find the rate of change of sales $S(t) = 10,000 + 2000t - 200t^2$ at $t=0$ , $t=4$ , and $t=8$ weeks.

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Problem 598

Monthly sales (in thousands) for a music store are given by $S(t)=\frac{200}{t^{2}+36}$ . Find $S(2)$ , $S'(2)$ , and estimate sales for month 3.

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Problem 599

Monthly sales for a record album are given by $S(t)=\frac{200 t}{t^{2}+36}$ . Find: a) $S(2)$ , b) $S^{\prime}(2)$ , c) estimate sales in month 3 using (a) and (b).

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Problem 600

Differentiate $x^{3} y^{2}$ with respect to $y$ .

See Solution

Derivatives

Problem 501

Problem 502

Question Given the function f(x)=x1−4x3f(x)=\frac{x}{1-4 x^{3}}f(x)=1−4x3x​, find f′(x)f^{\prime}(x)f′(x) in simplified form.Answer Attempt 1 out of 3

Problem 503

Question Given the function y=3x2−21−x2y=\frac{3 x^{2}-2}{1-x^{2}}y=1−x23x2−2​, find dydx\frac{d y}{d x}dxdy​ in simplified form.Answer Attempt 1 out of 3

Problem 504

Problem 505

Problem 506

Problem 507

A balloon rises vertically from the ground 200 m away from an observer. It rises with a position function h=50t2h=50 t^{2}h=50t2 where hhh is the height of the balloon in metres and ttt is in seconds. How fast is the angle of elevation changing 2 seconds after the balloon leaves the ground?

Problem 508

8. The function f(x)=−xex+2f(x)=-x e^{x}+2f(x)=−xex+2 is concave down at x=0x=0x=0. a. Find the tangent line of fff at x=0x=0x=0. b. What is the estimate for f(−0.1)f(-0.1)f(−0.1) using the local linear approximation for fff at x=0x=0x=0 ? c. Is it an underestimate or overestimate? Explain.

Problem 509

Problem 510

Problem 511

Problem 512

Problem 513

Problem 514

Problem 515

Problem 516

Problem 517

7-18 Use la parte 1 del teorema fundamental del cálculo para encontrar la derivada de cada una de las siguientes funciones.7. g(x)=∫1x1t3+1dtg(x)=\int_{1}^{x} \frac{1}{t^{3}+1} d tg(x)=∫1x​t3+11​dt8. g(x)=∫3xet2−tdtg(x)=\int_{3}^{x} e^{t^{2}-t} d tg(x)=∫3x​et2−tdt

Problem 518

Find y′y^{\prime}y′. y=(x+1x−7)7y′=□\begin{array}{l} y=\left(\frac{x+1}{x-7}\right)^{7} \\ y^{\prime}=\square \end{array}y=(x−7x+1​)7y′=□​

Problem 519

Problem 520

Problem 521

Problem 522

Problem 523

Let f(x)=−5ln⁡(sin⁡x)f(x)=-5 \ln (\sin x)f(x)=−5ln(sinx)Then f′(x)=f^{\prime}(x)=f′(x)= □\square□ and f′′(x)=f^{\prime \prime}(x)=f′′(x)= □\square□

Problem 524

Question 3 (1 point) If f(x)=2cos⁡3xf(x)=2 \cos 3 xf(x)=2cos3x, find f′(π3)f^{\prime}\left(\frac{\pi}{3}\right)f′(3π​). a) 3 b) 0 c) -6 d) 6

Problem 525

Problem 526

Problem 527

Use implicit differentiation to find an equation of the tangent line to the curve sin⁡(x+y)=8x−8y\sin (x+y)=8 x-8 ysin(x+y)=8x−8y at the point (π,π)(\pi, \pi)(π,π). Tangent Line Equation: □\square□

Problem 528

Problem 529

Problem 530

Problem 531

7. y=x2+1xy=\frac{x^{2}+1}{x}y=xx2+1​ a. Find intervals increasing or decreasing6. find intervaler concave up or concave down c. find all arymptoter d. points at local max, local min, paI e.sketch the function

Problem 532

Problem 533

Find the intervals where the function f(x)=12x4−4x2+7 f(x) = \frac{1}{2} x^4 - 4x^2 + 7 f(x)=21​x4−4x2+7 is concave upward.

Problem 534

Problem 535

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Problem 536

Find the following derivative: ( 1 Mark) f(x)=8x3−4x4x−12x−0.1f(x)=8 x^{3}-\frac{4}{x^{4}} x-12 x-0.1f(x)=8x3−x44​x−12x−0.1

Problem 537

Problem 538

tion de la tangente à la courbe représentative de fff au point d'abscisse a. f(x)=−x2+2x−8;a=−2f(x)=-x^{2}+2 x-8 ; a=-2f(x)=−x2+2x−8;a=−2

Problem 539

Problem 540

Problem 541

Problem 542

1 - 2

Problem 543

The function f(x)=3x+5x−1f(x)=3 x+5 x^{-1}f(x)=3x+5x−1 has one local minimum and one local maximum. This function has a local maximum at x=x=x= with value and a local minimum at x=x=x= □\square□ with value □\square□

Problem 544

Problem 545

Problem 546

Problem 547

Problem 548

Problem 549

Problem 550

Problem 551

Problem 552

WeBWorK 5 - Topics 10 - 12: Problem 3 (1 point)Consider the function f(x)f(x)f(x) whose second derivative is f′′(x)=10x+5sin⁡(x)f^{\prime \prime}(x)=10 x+5 \sin (x)f′′(x)=10x+5sin(x). If f(0)=4f(0)=4f(0)=4 and f′(0)=2f^{\prime}(0)=2f′(0)=2, what is f(x)f(x)f(x) ? Answer: □\square□

Problem 553

Problem 554

Problem 555

Problem 556

Problem 557

If f(x)=sin⁡(4x+π2)f(x)=\sin \left(4 x+\frac{\pi}{2}\right)f(x)=sin(4x+2π​), find f′(π4)f^{\prime}\left(\frac{\pi}{4}\right)f′(4π​). 0 4 −4-4−4 1

Problem 558

The position of an object at t seconds is given in feet by f(t)=3t2+4t−1+tf(t)=3 t^{2}+4 t^{-1}+\sqrt{t}f(t)=3t2+4t−1+t​What is the object's velocity at 4 seconds?

Problem 559

Find the derivative f′(x)f^{\prime}(x)f′(x) of f(x)=∣x∣f(x)=|x|f(x)=∣x∣ for x>0x>0x>0, x<0x<0x<0, and at x=0x=0x=0. Explain why it's undefined at 000.

Question Given the function $f(x)=\frac{x}{1-4 x^{3}}$ , find $f^{\prime}(x)$ in simplified form.
Answer Attempt 1 out of 3

Question Given the function $y=\frac{3 x^{2}-2}{1-x^{2}}$ , find $\frac{d y}{d x}$ in simplified form.
Answer Attempt 1 out of 3

A balloon rises vertically from the ground 200 m away from an observer. It rises with a position function $h=50 t^{2}$ where $h$ is the height of the balloon in metres and $t$ is in seconds. How fast is the angle of elevation changing 2 seconds after the balloon leaves the ground?

8. The function $f(x)=-x e^{x}+2$ is concave down at $x=0$ . a. Find the tangent line of $f$ at $x=0$ . b. What is the estimate for $f(-0.1)$ using the local linear approximation for $f$ at $x=0$ ? c. Is it an underestimate or overestimate? Explain.

7-18 Use la parte 1 del teorema fundamental del cálculo para encontrar la derivada de cada una de las siguientes funciones.
7. $g(x)=\int_{1}^{x} \frac{1}{t^{3}+1} d t$
8. $g(x)=\int_{3}^{x} e^{t^{2}-t} d t$

Find $y^{\prime}$ . $\begin{array}{l} y=\left(\frac{x+1}{x-7}\right)^{7} \\ y^{\prime}=\square \end{array}$

Let $f(x)=-5 \ln (\sin x)$
Then $f^{\prime}(x)=$ $\square$ and $f^{\prime \prime}(x)=$ $\square$

Question 3 (1 point) If $f(x)=2 \cos 3 x$ , find $f^{\prime}\left(\frac{\pi}{3}\right)$ . a) 3 b) 0 c) -6 d) 6

Use implicit differentiation to find an equation of the tangent line to the curve $\sin (x+y)=8 x-8 y$ at the point $(\pi, \pi)$ . Tangent Line Equation: $\square$

7. $y=\frac{x^{2}+1}{x}$ a. Find intervals increasing or decreasing
6. find intervaler concave up or concave down c. find all arymptoter d. points at local max, local min, paI e.sketch the function

Find the intervals where the function $f(x) = \frac{1}{2} x^4 - 4x^2 + 7$ is concave upward.

Find the following derivative: ( 1 Mark) $f(x)=8 x^{3}-\frac{4}{x^{4}} x-12 x-0.1$

tion de la tangente à la courbe représentative de $f$ au point d'abscisse a. $f(x)=-x^{2}+2 x-8 ; a=-2$

The function $f(x)=3 x+5 x^{-1}$ has one local minimum and one local maximum. This function has a local maximum at $x=$ with value and a local minimum at $x=$ $\square$ with value $\square$

WeBWorK 5 - Topics 10 - 12: Problem 3 (1 point)
Consider the function $f(x)$ whose second derivative is $f^{\prime \prime}(x)=10 x+5 \sin (x)$ . If $f(0)=4$ and $f^{\prime}(0)=2$ , what is $f(x)$ ? Answer: $\square$

If $f(x)=\sin \left(4 x+\frac{\pi}{2}\right)$ , find $f^{\prime}\left(\frac{\pi}{4}\right)$ . 0 4 $-4$ 1

The position of an object at t seconds is given in feet by $f(t)=3 t^{2}+4 t^{-1}+\sqrt{t}$
What is the object's velocity at 4 seconds?

Find the derivative $f^{\prime}(x)$ of $f(x)=|x|$ for $x>0$ , $x<0$ , and at $x=0$ . Explain why it's undefined at $0$ .

Find the $x$ values where the function $f$ has a relative maximum given $f^{\prime}(x)=x^{2}(x+1)^{3}(x-4)^{2}$ .

Given values of $f^{\prime \prime}(x)$ : at $x=-1$ is -4, $x=0$ is -1, $x=1$ is 2, $x=2$ is 5, $x=3$ is 8. What type of function is $f$ ? (A) linear (B) quadratic (C) cubic (D) fourth-degree (E) exponential

Find the value of $c$ such that $f(x)=x+\frac{c}{x}$ has a local minimum at $x=3$ .

Find the time $t$ (from 0 to 10) when the object with velocity $v(t)=t \cos t-\ln (t+2)$ reaches maximum speed.

Find the time $t$ in $[0, 10]$ when the velocity $v(t)=t \cos t - \ln(t+2)$ is maximized. Options: A. 9.5 B. 5.1 C. 6.4 D. 7.6

Estimate $y$ at $x=3.1$ using the tangent line of the curve $x^{3}+x \tan y=27$ at $(3,0)$ . Choose from (A) -2.7 (B) -0.9 (C) 0 (D) 0.1 (E) 3.0.

Find the rate of change of sales $S(t)=10,000+2000 t-200 t^{2}$ and interpret $S^{\prime}(t)$ .

Find the derivative of the function $f(x)=6 \cdot \sqrt{x} \cdot(x-5)$ . What is $f^{\prime}(x)$ ?

Find the derivative $f^{\prime}(x)$ of the function $f(x)=-3 \cdot \ln (4 \cdot x)$ and calculate $f^{\prime}(2)$ .

Find the derivative of $f(x)=\frac{\sqrt{x}-7}{\sqrt{x}+7}$ and calculate $f^{\prime}(5)$ .

Find the derivative $f'(x)$ of the function $f(x)=\sqrt{4x^2 + 4x + 4}$ and evaluate it at $x=4$ .

Calculate the derivative of $g(x)=(4x^{2}+5x) \cdot e^{x}$ .

Find $y^{\prime}(4)$ using implicit differentiation for $4 \cdot x^{2}+3 \cdot x+x \cdot y=4$ and $y(4)=-18$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=2+\frac{7}{x}+\frac{7}{x^{2}}$ .

Find the derivative of $R(j)=\left(\ln \left(j^{2}\right)\right)^{2}$ at $j=e$ .