Browse Derivatives problems Studdy Solved!

Problem 201

AD1 \& AD2: Product and Quotient Rules Differentiate the following functions. Show the $[\cdot]^{\prime}$ step in your work.
1. $f(x)=\left(x^{4}+5 x-1\right) \cdot(3 x+2)$
2. $g(x)=\frac{2 x^{4}+7 x}{5 x+1}$

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Problem 202

$-5 \cdot x \cdot y+e^{x+y}=4$ a. Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ in terms of $x$ and $y$ . $\frac{\mathrm{d} y}{\mathrm{~d} x}=$ $a^{b}$ $\sin (a)$ $\frac{\partial}{\partial x} f$ : $\infty$ $\alpha$ $\Omega$ $\frac{5 y-e^{x+y}}{e^{x+y}-5 x}$ b. Find the value of $\frac{d y}{d x}$ at the point $P(\sqrt{5},-\sqrt{5})$ . $\left.\frac{\mathrm{d} y}{\mathrm{~d} x}\right|_{(\sqrt{5},-\sqrt{5})}=$

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Problem 203

For the given cost function $C(x)=48400+200 x+x^{2}$ , which gives the total cost (\ $) for$ x $items. Find the average cost function:$ \bar{C}(x)=\square $Find the production level that will minimize the average cost:$ x=\square$ items
Find the minimal average cost: $\square$ Question Help: $\square$ Video Submit Question

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Problem 204

a. Find $z_{t t}$ for $z_{t t}=\quad z(x, y, t)=\cos (\sqrt{4+49 t}) \cdot \sin (2 x) \cdot \sin (7 y)$ $a^{b}$ $\sin (a) \quad \frac{\partial}{\partial x} f$ : $\infty$ $\alpha$ $\Omega$ ? b. Does $u=\sin (\sqrt{53} \cdot t) \cdot \sin (2 x) \cdot \sin (7 y)$ satisfy the membrane equation $u_{t t}=u_{x x}+u_{y y} ?$ Yes No

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Problem 205

6. Find the exact absolute maximum and minimum of $h(x)=x e^{-x^{2}}$ on the interval $[-1,1]$ .
7. Let $f(x)=\ln \left(2 x^{3}-3 x^{2}\right)$ . Find all values of $x$ for which $f^{\prime}(x)$ is 0 or undefined. Determine

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Problem 206

Suppose that $w=x^{2} \cdot \exp (2 y) \cdot \cos (6 z)$ with $\begin{array}{c} x=\sin \left(t+\frac{\pi}{2}\right) \\ y=\ln (t+7) \\ z=t \end{array}$ a. Find $\frac{\mathrm{d} w}{\mathrm{~d} t}$ in terms of $t$ . $\frac{\mathrm{d} w}{\mathrm{~d} t}=$ $a^{b}$ $\sin (a) \quad \frac{\partial}{\partial x} f$ : $\infty$ $\alpha$ $\Omega$

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Problem 207

Use Taylor's formula for $f(x, y)$ at $(0,0)$ to find the quadratic approximations of $f$ near the origin when $f(x, y) \approx \quad f(x, y)=\frac{5}{7-x-2 y+7 x y} .$

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Problem 208

Suppose $y$ is a function of $x$ , i.e. $y=y(x)$ , and $-12 \cdot x \cdot y+4 \cdot e^{x+y}=-16$ a. Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ in terms of $x$ and $y$ . $\frac{\mathrm{d} y}{\mathrm{~d} x}=$

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Problem 209

Example 37
For what values of $x$ in $[0,2 \pi]$ does the graph of $f(x)=x+2 \sin x$ have a horizontal tangent line $\Rightarrow f^{\prime}(x)=0$

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Problem 210

Suppose that $w=x^{3} \cdot \exp (2 y) \cdot \cos (7 z)$ with $\begin{array}{c} x=\sin \left(t+\frac{\pi}{2}\right) \\ y=\ln (t+3) \\ z=t \end{array}$ a. Find $\frac{\mathrm{d} w}{\mathrm{~d} t}$ in terms of $t$ . $\frac{\mathrm{d} w}{\mathrm{~d} t}=$

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Problem 211

If $f(x)=\frac{\tan x-2}{\sec x}$ find $f^{\prime}(x)$ . $(1+2 \tan (x)) \cos (x)$
Find $f^{\prime}\left(\frac{\pi}{2}\right)$ . $\square$

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Problem 212

13. Find the derivative of $f(x)=\sqrt{1-3 x}$ at $a \neq 1 / 3$ using the definition. Use this to find an equation for the tangent line at $(a, f(a))$ .

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Problem 213

$-2 \cdot x \cdot y+e^{x+y}=20$ a. Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ in terms of $x$ and $y$ . $\frac{\mathrm{d} y}{\mathrm{~d} x}=$ $a^{b}$ $\sin (a)$ $\frac{\partial}{\partial x} f$ : $\infty$ $\alpha$ $\Omega$ $\frac{2 \cdot y-e^{x+y}}{e^{x+y}-2 \cdot x}$ b. Find the value of $\frac{d y}{d x}$ at the point $P(\sqrt{2},-\sqrt{2})$ . $\left.\frac{\mathrm{d} y}{\mathrm{~d} x}\right|_{(\sqrt{2},-\sqrt{2})}=$ $a^{b}$ $\sin (a)$ $\frac{\partial}{\partial x} f$ $\infty$ $\alpha$ $\Omega$

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Problem 214

Zeit zu überprüfen aut lange Sicht einstellen?
4: Eine Kleinstadt hat im Jahre 2006 mehrere Neubaugebiete eingerichtet. Man rechnet damit, dass die Einwohnerzahl der Kleinstadt in den folgenden Jahren zunimmt. Zählungen haben ergeben, dass sich die Zunahme der Einwohner mit der Funktion $f$ mit $f(x)=1000 \cdot x^{2} \cdot e^{-x}$ modellieren lässt, wobei x = 0 dem Jahr 2006 entspricht. a) Berechnen Sie, wann die Anzahl der Einwohner in der Kleinstadt am stärksten zunimmt, wenn man die Funktion $f$ als Modellfunktion verwendet. b) Berechnen Sie, wie sich die Einwohnerzahl der Kleinstadt von 2006 bis 2014 verändert hat. c) Berechnen Sie den Durchschnitt der jährlichen Zunahme der Einwohnerzahl von 2006 bis 2014.
5 Der Temperaturverlauf während eines Tages kann durch die Funktion t modelliert werden. Hinweis:

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Problem 215

8. [-/1 Points] DETAILS MY NOTES TANAPCALC10 4.4.045.MI. ASK YOUR TEACHER PRACTICE ANOTHER
Flight of a Rocket The altitude in feet attained by a model rocket $t$ seconds into flight is given by the function $h(t)$ . Find the maximum altitude (in ft) attained by the rocket. (Round your answer to the nearest foot.) $h(t)=-\frac{1}{3} t^{3}+4 t^{2}+20 t+20 \quad(t \geq 0)$ $\qquad$ ft Need Help? Read It Master It Submit Answer View Previous Question Question 8 of 10 VieynNext Question Home My Assignments Request Extension

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Problem 216

11 Mark for Review $\frac{d}{d x}\left(\cos ^{-1} x\right)=$ (A) $-\frac{1}{\sqrt{1-x^{2}}}$ (B) $\frac{1}{\sqrt{1-x^{2}}}$ (C) $-\sin ^{-1} x$ (D) $-\cos ^{-2} x$

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Problem 217

3. [0/1 Points] DETAILS MYNOTES
A plane flying horizontally at an altitude of 3 miles and a speed of $440 \mathrm{mi} / \mathrm{h}$ passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it has a total distance of 4 miles away from the station. (Round your answer to the nearest whole number.)
292 $x$ mi/h
Enhanced Feedback Please try again. Keep in mind that distance $=\sqrt{(\text { altitude })^{2}+(\text { horizontal distance })^{2}}$ (or $y^{2}=x^{2}+h^{2}$ .) Differentiate with respect to $t$ on both sides of the equation, using the Chain Rule, to solve for $\frac{d y}{d t}$ . The given speed of the plane is $\frac{d x}{d t}$ . Need Help? Read It

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Problem 218

Given a $W=10$ inch by $L=13$ inch piece of paper, we will cut out squares (size $x$ by $x$ ) from each corner and fold to create an (open top) box. Our goal is to find the size of the cut out square $(x)$ , that maximizes the volume of the box.
The length of the box, as a function of $\tau$ , is $l=13-2 x \vee \sigma^{6}$
The width of the box, as a function of $x$ , is $w=10-2 x \vee 0^{6}$
The volume of the box, as a function of $x$ , is $V=13 \cdot 10 \cdot x \times 0^{6} x(13-2 x)(10-2 x)$ which, after distributing, simplifies to $V=130 x-46 x^{2}+4 x^{3} \quad o^{6}$ To determine the value of $x$ that corresponds to a maximum volume, we need to find $V^{\prime}$ . $V^{\prime}=130-92 x+12 x^{2}$
The $\boldsymbol{x}$ that corresponds to a maximum volume is $\boldsymbol{x}=1.8712 \times 0$ ob 1.8683623129081 inches and the maximum volume is $V=$ $\square$ cubic inches

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Problem 219

Differentiate the following function. $\begin{array}{l} y=\frac{e^{x}-2 e^{-x}}{e^{x}+2 e^{-x}} \\ y^{\prime}=\square \end{array}$

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Problem 220

Next question Get a similar question
Find the differential and evaluate for the given $x$ and $d x$ . Answer exactly. $\begin{array}{l} y=\tan (x), x=\frac{7 \pi}{4}, d x=\frac{\pi}{11} \\ d y=\frac{2 \pi}{11} \end{array}$ 0 Submit Question

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Problem 221

Find $y^{\prime}$ by (a) applying the Product Rule and (b) multiplying the factors to produce a sum of simpler terms to differentiate. $y=\left(3-x^{2}\right)\left(x^{3}-4 x+2\right)$ a. Apply the Product Rule. Let $u=\left(3-x^{2}\right)$ and $v=\left(x^{3}-4 x+2\right)$ . $\frac{d}{d x}(u v)=\left(3-x^{2}\right)\left(3 x^{2}-4\right)+\left(x^{3}-4 x+2\right)(-2 x)$ b. Multiply the factors of the original expression, $u$ and $v$ , to produce a sum of simpler terms. $\mathrm{y}=\square$ (Simplify your answer.)

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Problem 222

We want to find the relative extrema of $f(x)$ . Here is the graph of its derivative $f^{\prime}(x)$ .
Use the graph of $f^{\prime}(x)$ to identify locations of any relative maximum(s), relative minimum(s) and Horizontal Points of Inflection of $f(x)$ .

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Problem 223

2 Mark for Review
The derivative of the function $A$ is given by $A^{\prime}(t)=2+9 e^{0.4 \text { sint }}$ , and $A(1.2)=7.5$ . If the linear approximation to $A(t)$ at $t=1.2$ is used to estimate $A(t)$ , at what value of $t$ does the linear approximation estimate that $A(t)=15$ ? (A) 0.498 (B) 1.166 (C) 1.698 (D) 2.400

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Problem 224

Find the value or values of $c$ that satisfy the equation $\frac{f(b)-f(a)}{b-a}=f^{\prime}(c)$ in the conclus of the Mean Value Theorem for the function and interval. $f(x)=x+\frac{12}{x},[3,4]$ A) $-2 \sqrt{3}, 2 \sqrt{3}$ B) 3,4 C) $2 \sqrt{3}$ D) $0,2 \sqrt{3}$

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Problem 225

4 Mark for Review 420
The figure above shows the graph of the differentiable function $f$ for $1 \leq x \leq 11$ and the secant line through the points $(1, f(1))$ and (11, $f(11))$ . For how many values of $x$ in the closed interval $[1,11]$ does the instantaneous rate of change of $f$ at $x$ equal the average rate of change of $f$ over that interval? (A) Zero (B) Two (C) Three (D) Four

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Problem 226

Question Watch Video Show Examples
The derivative of the function $f$ is defined by $f^{\prime}(x)=x^{2}-2 x-5 \sin (2 x-2)$ for $-2<x<4$ . Find all intervals in the given domain where the function $f$ is increasing. You may use a calculator and round all values to 3 decimal places. Answer Attempt 2 out of 3

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Problem 227

Let $f$ be the function defined by $f(x)=\sin (h(x))$ , where $h$ is a differentiable function. Which of the following is equivalent to the derivative of $f$ with respect to $x$ ? (A) $\cos (h(x))$ (B) $\cos (h \prime(x))$ (C) $\cos (h(x)) h \prime(x)$ (D) $\sin (h(x)) h \prime(x)$

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Problem 228

Question Watch Video Show Exam
Write a degree 3 Taylor Polynomial for $f(x)=-4 e^{x}$ centered at $x=-3$ . Let $f(x)=-4 e^{x}$ . Find the function value and first 3 derivatives of $f(x)$ at $x=-3$ $f(x)=$ $\square$ $f(-3)=$ $\square$ $f^{\prime}(x)=$ $\square$ $f^{\prime}(-3)=$ $\square$ $f^{\prime \prime}(x)=$ $\square$ $f^{\prime \prime}(-3)=$ $\square$ $f^{\prime \prime \prime}(x)=$ $\square$ $f^{\prime \prime \prime}(-3)=$ $\square$ thy

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Problem 229

If $y=x^{2} \sin 2 x$ , then $\frac{d y}{d x}=$ (A) $2 x \cos 2 x$ (B) $4 x \cos 2 x$ (C) $2 x(\sin 2 x+\cos 2 x)$ (D) $2 x(\sin 2 x-x \cos 2 x)$ (E) $2 x(\sin 2 x+x \cos 2 x)$

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Problem 230

Question Watch Video Show Examples
The function $f$ is defined by $f(x)=x^{3}-\cos (3 x)$ . Use a calculator to write the equation of the line tangent to the graph of $f$ when $x=-0.5$ . You should round all decimals to 3 places.
Answer Attempt 1 out of 3 $\square$ Sulmit Answer

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Problem 231

3. Local maximum: $(0,1)$ ; local minima: $(-1,-1)$ and $(1,-1) ;-1$ In Exercises 1-6, use the First Derivative Test to determine th extreme values of the function, and identify any absolute extr Support your answers graphically.
1. $y=x^{2}-x-1$
2. $y=-2 x^{3}+6 x^{2}-3$
3. $y=2 x^{4}-4 x^{2}+1$
4. $y=x e^{1 / x}$ Local minimu
Local matime $(-18,0)$ and $(2,4)$ :
5. $y=x \sqrt{8-x^{2}}$
Locat minimum (0.1) local minims $(-2,4)$ and $(\sqrt{8} 8,0)$
6. $y=\left\{\begin{array}{ll}3-x^{2}, & x<0 \\ x^{2}+1, & x \geq 0\end{array}\right.$
4 is an absolutc maximum and, 4 tis an absolute ninimum

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Problem 232

Where on $(-\infty, 0)$ does $f(x)=\frac{25 x^{2}+3 x+9}{x}$ take its maximum value, and what is this maximum value?

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Problem 233

Question Watch Video Show Examples
The function $f$ is defined by $f(x)=x^{2}-5 x+4 \cos (3 x)$ . The graph of $f$ crosses the $x$ -axis at one point in the interval $[-0.5,3.5]$ . Use a calculator to determine the slope of the tangent line to $f$ at this point, rounded to the nearest thousandth.
Answer Attempt 1 out of 3

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Problem 234

At least one of the answers above is NOT correct. (1 point) Find the coordinates of all extrema of $f(t)=4 t^{3}-60 t^{2}$ with domain $[-5, \infty)$ .

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Problem 235

Determine the open intervals on which the graph is concave upward or concave downward. (Enter your answers using interval notation. If an answer does not exist, enter DNE.) $y=-x^{3}+6 x^{2}-8$ concave upward $\square$ concave downward $\square$

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Problem 236

Question 3
If $f(x)=\frac{2-x^{2}}{7+x^{2}}$ , find: $f^{\prime}(x)=$ $\square$ Question Help: Video Submit Question

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Problem 237

Question 5
Suppose a product's revenue function is given by $R(q)=-5 q^{2}+500 q$ . Find an expression for the marginal revenue function, simplify it, and record your result in the box below. Be sure to use the proper variable in your answer. (Use the preview button to check your syntax before submitting your answer.) $M R(q)=$ $\square$ Question Help: Video
Submit Question

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Problem 238

If $f(x)=\ln \left(x+4+e^{-3 x}\right)$ , then $f^{\prime}(0)$ is (A) $-\frac{2}{5}$ (B) $\frac{1}{5}$ (C) $\frac{1}{4}$ (D) $\frac{2}{5}$ (E) nonexistent

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Problem 239

20. Thermodynamics texts ${ }^{4}$ use the relationship $\left(\frac{\partial y}{\partial x}\right)\left(\frac{\partial z}{\partial y}\right)\left(\frac{\partial x}{\partial z}\right)=-1 .$
Explain the meaning of this equation and prove that it is true. [HinT: Start with a relationship $F(x, y, z)=0$ that defines $x=f(y, z), y=g(x, z)$ , and $z=h(x, y)$ and differentiate implicitly.]

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Problem 240

Let $f(x)$ be a function whose derivative exists everywhere, and let $T_{1}(x)$ be the first-order Taylor polynomial to $f(x)$ about $x=a$ . Which of the following statements are guaranteed to be true? Select all that apply. $T_{1}(a)=f(a)$ $T_{1}(0)=f(0)$ $T_{1}^{\prime}(a)=f^{\prime}(a)$ $T_{1}^{\prime}(0)=f^{\prime}(0)$ $T_{1}^{\prime \prime}(a)=f^{\prime \prime}(a)$ $T_{1}^{\prime \prime}(0)=f^{\prime \prime}(0)$ $T_{1}^{\prime \prime \prime}(a)=f^{\prime \prime \prime}(a)$ $T_{1}^{\prime \prime \prime}(0)=f^{\prime \prime \prime}(0)$

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Problem 241

Problem 3. Given a function $f(x)=e^{-x^{2}+4 x}$ (a) (6 points) find the minimum and maximum values attained by $f$ over an interval $[0,3]$ , (b) (2 points) find the equation of the tangent line to the graph of $f$ at a point $(0, f(0))$ .

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Problem 242

$f(x) = x^{\sqrt{x}}$
Apply the logarithmic differentiation method to find the derivative of the function $f(x) = x^{\sqrt{x}}$ .

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Problem 243

(1 point) Find the coordinates of all extrema of $f(x)=7 x^{4}-9 x^{3}$ with domain $[-1, \infty)$ .
Absolute maximum: none
Absolute minimum: $(0,0)$
Relative maxima: $(-1,16)$
Relative minima: none

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Problem 244

(1 point) Find the coordinates of all extrema of $k(x)=\frac{4 x}{9}-(x-3)^{4 / 9}$ with domain $[0, \infty)$ .
Note: WeBWorK, like many computer programs, does not know how to calculate $a^{b}$ if $a$ is negative and $b$ is not an integer. Rewrite any answers of this form in a different way.
Absolute maximum:
Absolute minimum:
Relative maxima:
Relative minima:

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Problem 245

Find the critical points of the function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ defined by $f(x, y) = 2xy - x^2 - 2y^2 + 3x + 2024$ .

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Problem 246

If $f(x)=\sqrt{x}+\frac{3}{\sqrt{x}}$ , then $f^{\prime}(4)=$ (A) $\frac{1}{16}$ (B) $\frac{5}{16}$ (C) 1 (D) $\frac{7}{2}$ (E) $\frac{49}{4}$

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Problem 247

If $y=5 x \sqrt{x^{2}+1}$ , then $\frac{d y}{d x}$ at $x=3$ is

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Problem 248

At least one of the answers above is NOT correct.
A street light is at the top of a 22 ft pole. A 6 ft tall girl walks along a straight path away from the pole with a speed of $5 \mathrm{ft} / \mathrm{sec}$ . At what rate is the tip of her shadow moving away from the light (ie. away from the top of the pole) when the girl is 40 ft away from the pole? Answer: $\left(\frac{11}{8}\right)^{2}\left[\frac{40}{\sqrt{22^{2}+\left(\frac{11}{8}\right)^{2}+40^{2}}}(5)\right.$
How fast is her shadow lengthening? Answer: $\frac{15}{8}$
Note: You can earn partial credit on this problem. Preview My Answers Submin' Answers

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Problem 249

on 4 of 6
Find the tangent line to the curve $y=4 e^{x}$ at the point $(0,4)$ . $y=$ $\qquad$
Find the normal line to the curve $y=4 e^{x}$ at the point $(0,4)$ . $y=$ $\square$

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Problem 250

Use the power rule for functions to find the derivative of $h$ . $h(x)=\left(7 x^{3}-17 e^{x}\right)^{5}$
First, find the value of $n$ used when applying the power rule for functions to $h$ . $\begin{array}{r} \frac{d}{d x}[f(x)]^{n}=n[f(x)]^{n-1} \cdot f^{\prime}(x) \\ n=\quad \end{array}$ $\square$
Now, use the power rule for functions to find the derivative of $h$ . $h^{\prime}(x)=$ $\square$ Search

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Problem 251

1 of 6
Determine the derivative of $f$ using the product rule. $\begin{array}{l} f(x)=-3.1 x^{4} e^{x} \\ f^{\prime}(x)=\left(4\left(-3.1 x^{3}(-3.1)\right)\right)\left(e^{x}\right)+\left(\frac{1}{x}\right)\left(-3.1 x^{4}\right) \end{array}$
Incorrect Answer

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Problem 252

Suppose $f^{\prime}(8)=7$ and $g^{\prime}(8)=4$ . Find $h^{\prime}(8)$ where $h(x)=4 f(x)+3 g(x)+9$ $h^{\prime}(8)=$

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Problem 253

An inverted cylindrical cone, 40 ft deep and 20 ft across at the top, is being filled with water at a rate of $19 \mathrm{ft}^{3} / \mathrm{min}$ . At what rate is the water rising in the tank when the depth of the water is:
1 foot? Answer= $\square$ 10 feet? Answer= $\square$ 39 feet? Answer= $\square$

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Problem 254

Find $\frac{d}{d x}\left(\frac{7 x^{2}}{8}-\frac{7}{5 x^{2}}\right)$ $\frac{d}{d x}\left(\frac{7 x^{2}}{8}-\frac{7}{5 x^{2}}\right)=$

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Problem 255

Find the directional derivative of the function $f(x, y, z) = xy^2z^3$ at the point $P(2,1,1)$ in the direction of the point $Q(0,-3,5)$ .

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Problem 256

Find an equation of a tangent line to the graph of a $y=f(x)$ at the given point. $y=x \ln (x-5) \text { at }(6,0)$ equation: $\square$

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Problem 257

Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) $\begin{array}{l} f(x)=4 x^{3}-6 x^{2}-24 x \\ x=\square \end{array}$ Need Help? Read It

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Problem 258

Consider the following. (If an answer does not exist, enter DNE.) $f(x)=x^{3}-3 x^{2}-7 x+5$
Find the interval(s) on which $f$ is concave up. (Enter your answer using interval notation.) $\square$ Find the interval(s) on which $f$ is concave down. (Enter your answer using interval notation.) $\square$ Find the inflection point of $f$ . $\square$ $(x, y)=(\square)$ Need Help? Read It

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Problem 259

stion 8 of 11
Find $f^{\prime}$ and $f^{\prime \prime}$ for the function $f(x)=3 x^{8} e^{x}$ (Express numbers in exact form. Use symbolic notation and fractions where needed.) $\begin{array}{l} f^{\prime}(x)=\left(24 x^{7}\right)\left(e^{x}\right)+\left(e^{x}\right)\left(3 x^{8}\right) \\ f^{\prime \prime}(x)= \end{array}$ $\square$

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Problem 260

A ladder 23 ft long rests against a vertical wall. If the top of the ladder is being pulled up the wall at a rate of $19 \mathrm{ft} / \mathrm{s}$ , at what rate is the bottom of the ladder moving towards the wall when the top of the ladder is 7 ft from the ground?

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Problem 261

Question 9 of 11
Find the derivative of the function. $g(x)=\left(\log _{2}(x)\right)^{9 / 8}$ (Express numbers in exact form. Use symbolic notation and fractions where needed.) $g^{\prime}(x)=$

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Problem 262

Find the derivative of the function $f(x)=\log _{2}\left(x^{2}-3\right)$ . (Express numbers in exact form. Use symbolic notation and fractions where needed.) $f^{\prime}(x)=\sqrt{\frac{1}{x^{2}-3} \cdot 2 x}$
Incorrect Answer

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Problem 263

\text{As I left home in the morning, I put on a light jacket because, although the temperature was dropping, it seemed that the temperature would not go much lower. But I was wrong. Around noon a northerly wind blew up and the temperature began to drop faster and faster. The worst was around 10 pm when, fortunately, the temperature started going back up. Let } t=0 \text{ hours at 7 am.} \\
\text{(a) When was there a critical point in the graph of temperature as a function of time?} \\
\text{There was a critical point at } t= \, \square \text{ hours after 7 am.} \\
\text{(b) When was there an inflection point in the graph of temperature as a function of time?} \\
\text{There was an inflection point } \square \\

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Problem 264

Find $\frac{d y}{d x}$ by implicit differentiation. $e^{y}=4 x-8 y$
Answer: $\frac{d y}{d x}=$

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Problem 265

Current Attempt in Progress The figure below is the graph of a second derivative $f^{\prime \prime}$ . What are the $x$ -values that are inflection points of $f$ ?
Enter your answers in increasing order. $\begin{array}{l} x=\mathbf{i} \\ x=i \end{array}$

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Problem 266

Previous Problem Problem List Next Problem
NYA Module 5: Problem 4 (1 point)
Two carts, A and B , are connected by a rope 39 ft long that passes over a pulley $P$ . The point $Q$ is on the floor 12 ft directly beneath $P$ and between the carts. Cart A is being pulled away from $Q$ at a speed of $2 \mathrm{ft} / \mathrm{s}$ . How fast is cart B moving toward $Q$ at the instant when cart A is 5 ft from $Q$ ? $\square$ $\mathrm{ft} / \mathrm{s}$ Preview My Answers Submit Answers

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Problem 267

349. 9709 m22_qp_12 Q: 11
It is given that a curve has equation $y=k(3 x-k)^{-1}+3 x$ , where $k$ is a constant. (a) Find, in terms of $k$ , the values of $x$ at which there is a stationary point.

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Problem 268

For $f(x)=x-\ln x$ , and $0.1 \leq x \leq 2$ , find the following. (a) Find the values of $x$ for which $f(x)$ has a local maximum.
Enter your answers in the increasing order. $\begin{array}{l} x=i \\ x=i \end{array}$ eTextbook and Media (b) Find the value of $x$ for which $f(x)$ has a local minimum. $x=\mathbf{i}$ eTextbook and Media (c) Find the value of $x$ for which $f(x)$ has a global maximum. $x=$ $\square$

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Problem 269

Current Attempt in Progress
For $f(x)=x-\ln x$ , and $0.1 \leq x \leq 2$ , find the following. (a) Find the values of $x$ for which $f(x)$ has a local maximum.
Enter your answers in the increasing order. $\begin{array}{l} x=\mathbf{i}|l| \\ \hline x=i \end{array}$

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Problem 270

For $f(x) = x - \ln x$ , and $0.1 \leq x \leq 2$ , find the value of $x$ for which $f(x)$ has a global minimum.
$x =$ $\square$

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Problem 271

Let $f(x)=\csc (g(x))$ . If $g(3)=\frac{1}{6} \pi \quad$ and $g^{\prime}(3)=1$ then determine $f^{\prime}(3)$ .
Your answer must be exact. Use sqrt(x) for $\sqrt{x}$ or switch to equation editor mode. $\begin{array}{l} a^{b} \quad \sin (a) \\ f^{\prime}(3)= \end{array}$

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Problem 272

Find the absolute maximum and minimum values of: $f(x)=x^{3}+3 x^{2}-9 x-7 \text { on }[-6,4]$

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Problem 273

b. $y=\log _{6}(x) \Rightarrow \frac{d y}{d x}=$

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Problem 274

A voltage $V$ across a resistance $R$ generates a current $I=V / R$ . If a constant voltage of 8 volts is put across a resistance that is increasing at a rate of 0.1 ohms per second when the resistance is 7 ohms, at what rate is the current changing? (Give units..) rate $=$ $\square$

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Problem 275

Find the second-order partial derivative. Find $f_{x y}$ when $f(x, y)=10 x^{2} y^{4}-7 x^{3} y^{5}$ . $160 x y^{3}-21 x^{2} y^{4}$ $80 x y^{3}-21 x^{2} y^{4}$ $80 x y^{3}-105 x^{2} y^{4}$ $160 x y^{3}-105 x^{2} y^{4}$

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Problem 276

$\underline{27}, \underline{28}, \underline{29}, \underline{30}, \underline{31}, \underline{32}, \underline{33}, \underline{34}, \underline{35}$ , and $\underline{36}$ Use implicit differentiation to find an equation of the tangent line the curve at the given point.
27. $y e^{\sin x}=x \cos y,(0,0)$
Answer

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Problem 277

Find the equation of the tangent line to $y=2^{x^{2}-3 x+1}$ at $x=4$ . $y=$

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Problem 278

$27,28,29,30,31,32,33,34,35$ , and 36 Use implicit differentiation to find an equation of the tangent line to the curve at the given point.
33. $x^{2}+y^{2}=\left(2 x^{2}+2 y^{2}-x\right)^{2},\left(0, \frac{1}{2}\right)$ (cardioid)

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Problem 279

$\underline{27}, \underline{28}, \underline{29}, \underline{30}, \underline{31}, \underline{32}, \underline{33}, \underline{34}, \underline{35}$ , and $\underline{36}$ Use implicit differentiation to find an equation of the tangent line to the curve at the given point. 3.
31. $x^{2}-x y-y^{2}=1,(2,1)$ (hyperbola) Answer 4

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Problem 280

Find the Taylor polynomial $T_{3}(x)$ for the function $f$ centered at the number $a$ . $\begin{array}{r} f(x)=x e^{-7 x}, \quad a=0 \\ T_{3}(x)=\square \end{array}$
Graph $f$ and $T_{3}$ in the same viewing rectangle.

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Problem 281

(1 pt) Find the inflection points of thefunction $f(x)=e^{-x}\left(34-x^{2}\right)$ . INTERMEDIATE WORK $\begin{array}{l} f^{\prime}(x)=e^{\wedge}-x\left(x^{\wedge} 2-2 x-34\right) \\ f^{\prime \prime}(x)=e^{\wedge}-x\left(-x^{\wedge} 2+32\right) \end{array}$
On what intervals is $f$ concave upward? On what intervals is $f$ concave downward? $\square$ (Give your answer in interval notation, for example: [a,b) U (c,d]. If needed enter $-\infty$ as - infinity and $\infty$ as infinity. ) FINAL ANSWER Inflection points of $f(x)$ are $\square$ (Give your answer as a comma separated list, for example: $\mathbf{a}, \mathbf{b}, \ldots$ )

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Problem 282

5. [-/5 Points] DETAILS MY NOTES SCALCCC5 8.8.010.
Find the Taylor polynomial $T_{n}(x)$ for the function $f$ at the number $a$ . Graph $f$ and $T_{3}$ on the same paper. $\begin{aligned} & f(x)=\frac{\ln (3 x)}{2 x}, a=\frac{1}{3}, n=3 \\ & T_{3}(x)=\square \end{aligned}$ Need Help? Read it Submit Answer

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Problem 283

A graphing calculator is recommended. Find the Taylor polynomial $T_{3}(x)$ for the function $f$ centered at the number $a$ . $\begin{array}{l} f(x)=\ln (x), a=1 \\ T_{3}(x)=\square \end{array}$
Graph $f$ and $T_{3}$ in the same viewing rectangle.

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Problem 284

Differentiate the following function. $\begin{array}{l} f(x)=\frac{x-e^{-x}}{1+x e^{-x}} \\ f^{\prime}(x)=\square \end{array}$

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Problem 285

Differentiate the following function. $y=\frac{3 \ln (4 x)}{\sqrt{x}}$
$\frac{d}{d x}\left(\frac{3 \ln (4 x)}{\sqrt{x}}\right)=$ $\square$

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Problem 286

Differentiate the function $y=\ln \left(2 x^{2}-7 x+1\right)$ $y^{\prime}=\frac{1}{\frac{d}{d x}\left[2 x^{2}-7 x+1\right]}$ D. $y^{\prime}=\frac{1}{2 x^{2}-7 x+1} \cdot \frac{d}{d x}\left[2 x^{2}-7 x+1\right]$ $y^{\prime}=$ $\square$

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Problem 287

Differentiate the function $y=\left(x^{2}+9\right) \ln \left(x^{2}+9\right)$ $y^{\prime}=$

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Problem 288

Differentiate the following function. $y=\frac{x^{4} \ln x}{4}$ D. $\frac{d}{d x}\left[\frac{x^{4} \ln x}{4}\right]=\frac{d}{d x}\left[\frac{x^{4}}{4}\right] \cdot \ln x+\frac{d}{d x}\left[\frac{x^{4}}{4}\right] \cdot \ln x$ $\frac{d}{d x}\left(\frac{x^{4} \ln x}{4}\right)=$ $\square$

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Problem 289

Differentiate the following function. $y=\frac{4 \ln (3 x)}{\sqrt{x}}$ D. $\frac{d}{d x}\left[\frac{4 \ln (3 x)}{\sqrt{x}}\right]=\frac{v^{\wedge} d x^{[\cdots l \text { I } /]}}{(\sqrt{x})^{2}}$ $\frac{d}{d x}\left(\frac{4 \ln (3 x)}{\sqrt{x}}\right)=$

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Problem 290

Differentiate the following function. $y=\frac{4 \ln (3 x)}{\sqrt{x}}$ $\frac{d}{d x}\left(\frac{4 \ln (3 x)}{\sqrt{x}}\right)=$

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Problem 291

devivatic $\frac{\left(x^{2}\right)(-\pi-4)}{8}=$

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Problem 292

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for the function $f(x)=x^{3}+6 x^{2}+9 x-6$ on the domain $[-6,0]$ .

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Problem 293

Use the graph to answer the following questions. (a) Over which intervals is the function decreasing? Choose all that apply. $(-\infty,-4)$ $\square$ $(-4,-2)$ $\square$ $(-2,2)$ $\square$ $(-4,2)$ $\square$ $(2,6)$ $\square$ $(8, \infty)$ (b) At which $x$ -values does the function have local maxima? If there is more than one value, separate them with commas. $\square$

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Problem 294

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for the function $f(x)=2 x-224 x^{\frac{2}{7}}$ (a) on the interval $[-128,122]$ and $(b)$ on the interval $[122,384]$ . (a) Identify the absolute maximum on the interval $[-128,122]$ if it exists. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.

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Problem 295

What is the approximate slope of the curve with equation $f(x)=\frac{1}{x-1}$ at $x=1.2 ?$ a) -25 b) 25 c) $-\frac{1}{25}$ d) $\frac{1}{25}$

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Problem 296

Find the derivative with respect to the independent variable $\begin{array}{l} f(x)=7 \tan \left(3-x^{2}\right) \\ f^{\prime}(x)=\square \end{array}$ $\square$

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Problem 297

Find the derivative with respect to the independent variable. $\begin{array}{l} g(x)=\frac{1}{\csc ^{2}(9 x)} \\ g^{\prime}(x)=\square \end{array}$

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Problem 298

Find the derivative with respect to the independent variable. $\begin{array}{l} f(x)=\frac{\cos x^{9}}{\cos ^{2} x} \\ f^{\prime}(x)=\square \end{array}$

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Problem 299

Find the derivative of the following function. $\begin{array}{l} y=-8^{8 x^{2}-5} \\ \frac{d y}{d x}=\square \end{array}$

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Problem 300

A particle moves according to the law of motion $s(t)=t^{3}-7 t^{2}+4 t, t \geq 0$ , where $t$ is measured in seconds and $s$ in feet. c.) When is the particle at rest? Enter your answer as a comma separated list. Enter None if the particle is never at rest. At $t_{1}=$ $\square$ and $t_{2}=$ $\square$ with $t_{1}<t_{2}$ . d.) When is the particle moving in the positive direction?
When $0 \leq t<$ $\square$ and $t>$ $\square$

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Derivatives

Problem 201

AD1 \& AD2: Product and Quotient Rules Differentiate the following functions. Show the [⋅]′[\cdot]^{\prime}[⋅]′ step in your work.1. f(x)=(x4+5x−1)⋅(3x+2)f(x)=\left(x^{4}+5 x-1\right) \cdot(3 x+2)f(x)=(x4+5x−1)⋅(3x+2)2. g(x)=2x4+7x5x+1g(x)=\frac{2 x^{4}+7 x}{5 x+1}g(x)=5x+12x4+7x​

Problem 202

Problem 203

Problem 204

Problem 205

Problem 206

Problem 207

Use Taylor's formula for f(x,y)f(x, y)f(x,y) at (0,0)(0,0)(0,0) to find the quadratic approximations of fff near the origin when f(x,y)≈f(x,y)=57−x−2y+7xy.f(x, y) \approx \quad f(x, y)=\frac{5}{7-x-2 y+7 x y} .f(x,y)≈f(x,y)=7−x−2y+7xy5​.

Problem 208

Suppose yyy is a function of xxx, i.e. y=y(x)y=y(x)y=y(x), and −12⋅x⋅y+4⋅ex+y=−16-12 \cdot x \cdot y+4 \cdot e^{x+y}=-16−12⋅x⋅y+4⋅ex+y=−16 a. Find dy dx\frac{\mathrm{d} y}{\mathrm{~d} x} dxdy​ in terms of xxx and yyy. dy dx=\frac{\mathrm{d} y}{\mathrm{~d} x}= dxdy​=

Problem 209

Example 37For what values of xxx in [0,2π][0,2 \pi][0,2π] does the graph of f(x)=x+2sin⁡xf(x)=x+2 \sin xf(x)=x+2sinx have a horizontal tangent line ⇒f′(x)=0\Rightarrow f^{\prime}(x)=0⇒f′(x)=0

Problem 210

Problem 211

If f(x)=tan⁡x−2sec⁡xf(x)=\frac{\tan x-2}{\sec x}f(x)=secxtanx−2​ find f′(x)f^{\prime}(x)f′(x). (1+2tan⁡(x))cos⁡(x)(1+2 \tan (x)) \cos (x)(1+2tan(x))cos(x)Find f′(π2)f^{\prime}\left(\frac{\pi}{2}\right)f′(2π​). □\square□

Problem 212

13. Find the derivative of f(x)=1−3xf(x)=\sqrt{1-3 x}f(x)=1−3x​ at a≠1/3a \neq 1 / 3a=1/3 using the definition. Use this to find an equation for the tangent line at (a,f(a))(a, f(a))(a,f(a)).

Problem 213

Problem 214

Problem 215

Problem 216

11 Mark for Review ddx(cos⁡−1x)=\frac{d}{d x}\left(\cos ^{-1} x\right)=dxd​(cos−1x)= (A) −11−x2-\frac{1}{\sqrt{1-x^{2}}}−1−x2​1​ (B) 11−x2\frac{1}{\sqrt{1-x^{2}}}1−x2​1​ (C) −sin⁡−1x-\sin ^{-1} x−sin−1x (D) −cos⁡−2x-\cos ^{-2} x−cos−2x

Problem 217

Problem 218

Problem 219

Differentiate the following function. y=ex−2e−xex+2e−xy′=□\begin{array}{l} y=\frac{e^{x}-2 e^{-x}}{e^{x}+2 e^{-x}} \\ y^{\prime}=\square \end{array}y=ex+2e−xex−2e−x​y′=□​

Problem 220

Problem 221

Problem 222

We want to find the relative extrema of f(x)f(x)f(x). Here is the graph of its derivative f′(x)f^{\prime}(x)f′(x).Use the graph of f′(x)f^{\prime}(x)f′(x) to identify locations of any relative maximum(s), relative minimum(s) and Horizontal Points of Inflection of f(x)f(x)f(x).

Problem 223

Problem 224

Problem 225

Problem 226

Problem 227

Problem 228

Problem 229

Problem 230

Problem 231

Problem 232

Where on (−∞,0)(-\infty, 0)(−∞,0) does f(x)=25x2+3x+9xf(x)=\frac{25 x^{2}+3 x+9}{x}f(x)=x25x2+3x+9​ take its maximum value, and what is this maximum value?

Problem 233

Problem 234

At least one of the answers above is NOT correct. (1 point) Find the coordinates of all extrema of f(t)=4t3−60t2f(t)=4 t^{3}-60 t^{2}f(t)=4t3−60t2 with domain [−5,∞)[-5, \infty)[−5,∞).

Problem 235

Determine the open intervals on which the graph is concave upward or concave downward. (Enter your answers using interval notation. If an answer does not exist, enter DNE.) y=−x3+6x2−8y=-x^{3}+6 x^{2}-8y=−x3+6x2−8 concave upward □\square□ concave downward □\square□

Problem 236

Question 3If f(x)=2−x27+x2f(x)=\frac{2-x^{2}}{7+x^{2}}f(x)=7+x22−x2​, find: f′(x)=f^{\prime}(x)=f′(x)= □\square□ Question Help: Video Submit Question

Problem 237

Problem 238

If f(x)=ln⁡(x+4+e−3x)f(x)=\ln \left(x+4+e^{-3 x}\right)f(x)=ln(x+4+e−3x), then f′(0)f^{\prime}(0)f′(0) is (A) −25-\frac{2}{5}−52​ (B) 15\frac{1}{5}51​ (C) 14\frac{1}{4}41​ (D) 25\frac{2}{5}52​ (E) nonexistent

Problem 239

Problem 240

Problem 241

Problem 3. Given a function f(x)=e−x2+4xf(x)=e^{-x^{2}+4 x}f(x)=e−x2+4x (a) (6 points) find the minimum and maximum values attained by fff over an interval [0,3][0,3][0,3], (b) (2 points) find the equation of the tangent line to the graph of fff at a point (0,f(0))(0, f(0))(0,f(0)).

Problem 242

f(x)=xxf(x) = x^{\sqrt{x}}f(x)=xx​Apply the logarithmic differentiation method to find the derivative of the function f(x)=xx f(x) = x^{\sqrt{x}} f(x)=xx​.

Problem 243

(1 point) Find the coordinates of all extrema of f(x)=7x4−9x3f(x)=7 x^{4}-9 x^{3}f(x)=7x4−9x3 with domain [−1,∞)[-1, \infty)[−1,∞).Absolute maximum: noneAbsolute minimum: (0,0)(0,0)(0,0)Relative maxima: (−1,16)(-1,16)(−1,16)Relative minima: none

Problem 244

Problem 245

Find the critical points of the function f:R2→R f: \mathbb{R}^{2} \rightarrow \mathbb{R} f:R2→R defined by f(x,y)=2xy−x2−2y2+3x+2024 f(x, y) = 2xy - x^2 - 2y^2 + 3x + 2024 f(x,y)=2xy−x2−2y2+3x+2024.

Problem 246

If f(x)=x+3xf(x)=\sqrt{x}+\frac{3}{\sqrt{x}}f(x)=x​+x​3​, then f′(4)=f^{\prime}(4)=f′(4)= (A) 116\frac{1}{16}161​ (B) 516\frac{5}{16}165​ (C) 1 (D) 72\frac{7}{2}27​ (E) 494\frac{49}{4}449​

Problem 247

If y=5xx2+1y=5 x \sqrt{x^{2}+1}y=5xx2+1​, then dydx\frac{d y}{d x}dxdy​ at x=3x=3x=3 is

Problem 248

Problem 249

on 4 of 6Find the tangent line to the curve y=4exy=4 e^{x}y=4ex at the point (0,4)(0,4)(0,4). y=y=y= \qquadFind the normal line to the curve y=4exy=4 e^{x}y=4ex at the point (0,4)(0,4)(0,4). y=y=y= □\square□

Problem 250

Problem 251

Problem 252

Suppose f′(8)=7f^{\prime}(8)=7f′(8)=7 and g′(8)=4g^{\prime}(8)=4g′(8)=4. Find h′(8)h^{\prime}(8)h′(8) where h(x)=4f(x)+3g(x)+9h(x)=4 f(x)+3 g(x)+9h(x)=4f(x)+3g(x)+9 h′(8)=h^{\prime}(8)=h′(8)=

Problem 253

Problem 254

Find ddx(7x28−75x2)\frac{d}{d x}\left(\frac{7 x^{2}}{8}-\frac{7}{5 x^{2}}\right)dxd​(87x2​−5x27​) ddx(7x28−75x2)=\frac{d}{d x}\left(\frac{7 x^{2}}{8}-\frac{7}{5 x^{2}}\right)=dxd​(87x2​−5x27​)=

Problem 255

Find the directional derivative of the function f(x,y,z)=xy2z3 f(x, y, z) = xy^2z^3 f(x,y,z)=xy2z3 at the point P(2,1,1) P(2,1,1) P(2,1,1) in the direction of the point Q(0,−3,5) Q(0,-3,5) Q(0,−3,5).

AD1 \& AD2: Product and Quotient Rules Differentiate the following functions. Show the $[\cdot]^{\prime}$ step in your work.
1. $f(x)=\left(x^{4}+5 x-1\right) \cdot(3 x+2)$
2. $g(x)=\frac{2 x^{4}+7 x}{5 x+1}$

Use Taylor's formula for $f(x, y)$ at $(0,0)$ to find the quadratic approximations of $f$ near the origin when $f(x, y) \approx \quad f(x, y)=\frac{5}{7-x-2 y+7 x y} .$

Suppose $y$ is a function of $x$ , i.e. $y=y(x)$ , and $-12 \cdot x \cdot y+4 \cdot e^{x+y}=-16$ a. Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ in terms of $x$ and $y$ . $\frac{\mathrm{d} y}{\mathrm{~d} x}=$

Example 37
For what values of $x$ in $[0,2 \pi]$ does the graph of $f(x)=x+2 \sin x$ have a horizontal tangent line $\Rightarrow f^{\prime}(x)=0$

If $f(x)=\frac{\tan x-2}{\sec x}$ find $f^{\prime}(x)$ . $(1+2 \tan (x)) \cos (x)$
Find $f^{\prime}\left(\frac{\pi}{2}\right)$ . $\square$

13. Find the derivative of $f(x)=\sqrt{1-3 x}$ at $a \neq 1 / 3$ using the definition. Use this to find an equation for the tangent line at $(a, f(a))$ .

11 Mark for Review $\frac{d}{d x}\left(\cos ^{-1} x\right)=$ (A) $-\frac{1}{\sqrt{1-x^{2}}}$ (B) $\frac{1}{\sqrt{1-x^{2}}}$ (C) $-\sin ^{-1} x$ (D) $-\cos ^{-2} x$

Differentiate the following function. $\begin{array}{l} y=\frac{e^{x}-2 e^{-x}}{e^{x}+2 e^{-x}} \\ y^{\prime}=\square \end{array}$

We want to find the relative extrema of $f(x)$ . Here is the graph of its derivative $f^{\prime}(x)$ .
Use the graph of $f^{\prime}(x)$ to identify locations of any relative maximum(s), relative minimum(s) and Horizontal Points of Inflection of $f(x)$ .

Where on $(-\infty, 0)$ does $f(x)=\frac{25 x^{2}+3 x+9}{x}$ take its maximum value, and what is this maximum value?

At least one of the answers above is NOT correct. (1 point) Find the coordinates of all extrema of $f(t)=4 t^{3}-60 t^{2}$ with domain $[-5, \infty)$ .

Determine the open intervals on which the graph is concave upward or concave downward. (Enter your answers using interval notation. If an answer does not exist, enter DNE.) $y=-x^{3}+6 x^{2}-8$ concave upward $\square$ concave downward $\square$

Question 3
If $f(x)=\frac{2-x^{2}}{7+x^{2}}$ , find: $f^{\prime}(x)=$ $\square$ Question Help: Video Submit Question

If $f(x)=\ln \left(x+4+e^{-3 x}\right)$ , then $f^{\prime}(0)$ is (A) $-\frac{2}{5}$ (B) $\frac{1}{5}$ (C) $\frac{1}{4}$ (D) $\frac{2}{5}$ (E) nonexistent

Problem 3. Given a function $f(x)=e^{-x^{2}+4 x}$ (a) (6 points) find the minimum and maximum values attained by $f$ over an interval $[0,3]$ , (b) (2 points) find the equation of the tangent line to the graph of $f$ at a point $(0, f(0))$ .

$f(x) = x^{\sqrt{x}}$
Apply the logarithmic differentiation method to find the derivative of the function $f(x) = x^{\sqrt{x}}$ .

(1 point) Find the coordinates of all extrema of $f(x)=7 x^{4}-9 x^{3}$ with domain $[-1, \infty)$ .
Absolute maximum: none
Absolute minimum: $(0,0)$
Relative maxima: $(-1,16)$
Relative minima: none

Find the critical points of the function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ defined by $f(x, y) = 2xy - x^2 - 2y^2 + 3x + 2024$ .

If $f(x)=\sqrt{x}+\frac{3}{\sqrt{x}}$ , then $f^{\prime}(4)=$ (A) $\frac{1}{16}$ (B) $\frac{5}{16}$ (C) 1 (D) $\frac{7}{2}$ (E) $\frac{49}{4}$

If $y=5 x \sqrt{x^{2}+1}$ , then $\frac{d y}{d x}$ at $x=3$ is

on 4 of 6
Find the tangent line to the curve $y=4 e^{x}$ at the point $(0,4)$ . $y=$ $\qquad$
Find the normal line to the curve $y=4 e^{x}$ at the point $(0,4)$ . $y=$ $\square$

Suppose $f^{\prime}(8)=7$ and $g^{\prime}(8)=4$ . Find $h^{\prime}(8)$ where $h(x)=4 f(x)+3 g(x)+9$ $h^{\prime}(8)=$

Find $\frac{d}{d x}\left(\frac{7 x^{2}}{8}-\frac{7}{5 x^{2}}\right)$ $\frac{d}{d x}\left(\frac{7 x^{2}}{8}-\frac{7}{5 x^{2}}\right)=$

Find the directional derivative of the function $f(x, y, z) = xy^2z^3$ at the point $P(2,1,1)$ in the direction of the point $Q(0,-3,5)$ .

Find an equation of a tangent line to the graph of a $y=f(x)$ at the given point. $y=x \ln (x-5) \text { at }(6,0)$ equation: $\square$

Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) $\begin{array}{l} f(x)=4 x^{3}-6 x^{2}-24 x \\ x=\square \end{array}$ Need Help? Read It

A ladder 23 ft long rests against a vertical wall. If the top of the ladder is being pulled up the wall at a rate of $19 \mathrm{ft} / \mathrm{s}$ , at what rate is the bottom of the ladder moving towards the wall when the top of the ladder is 7 ft from the ground?

Question 9 of 11
Find the derivative of the function. $g(x)=\left(\log _{2}(x)\right)^{9 / 8}$ (Express numbers in exact form. Use symbolic notation and fractions where needed.) $g^{\prime}(x)=$

Find $\frac{d y}{d x}$ by implicit differentiation. $e^{y}=4 x-8 y$
Answer: $\frac{d y}{d x}=$

Current Attempt in Progress The figure below is the graph of a second derivative $f^{\prime \prime}$ . What are the $x$ -values that are inflection points of $f$ ?
Enter your answers in increasing order. $\begin{array}{l} x=\mathbf{i} \\ x=i \end{array}$

349. 9709 m22_qp_12 Q: 11
It is given that a curve has equation $y=k(3 x-k)^{-1}+3 x$ , where $k$ is a constant. (a) Find, in terms of $k$ , the values of $x$ at which there is a stationary point.

For $f(x) = x - \ln x$ , and $0.1 \leq x \leq 2$ , find the value of $x$ for which $f(x)$ has a global minimum.
$x =$ $\square$

Find the absolute maximum and minimum values of: $f(x)=x^{3}+3 x^{2}-9 x-7 \text { on }[-6,4]$

b. $y=\log _{6}(x) \Rightarrow \frac{d y}{d x}=$

A voltage $V$ across a resistance $R$ generates a current $I=V / R$ . If a constant voltage of 8 volts is put across a resistance that is increasing at a rate of 0.1 ohms per second when the resistance is 7 ohms, at what rate is the current changing? (Give units..) rate $=$ $\square$