Browse Derivatives problems Studdy Solved!

Problem 301

Suppose that $f(x)=4 x^{2}(2-2 x)^{4}$ (A) Find an equation for the tangent line to the graph of $f$ at $x=1$ .
Tangent line: $y=$ $\square$ (B) Find the average of all values of $x$ where the tangent line is horizontal. If there are no such values, enter -1000 .
Average of $x$ values $=$ $\square$

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Problem 302

Problem 9. (1 point)
A table of values for $f, g, f^{\prime}$ , and $g^{\prime}$ is given below. \begin{tabular}{|c|c|c|c|c|} \hline $x$ & $f(x)$ & $g(x)$ & $f^{\prime}(x)$ & $g^{\prime}(x)$ \\ \hline 1 & 3 & 1 & 2 & 2 \\ \hline 2 & 2 & 2 & 2 & 2 \\ \hline 3 & 1 & 1 & 2 & 3 \\ \hline \end{tabular} (A) If $h(x)=f(g(x))$ , then $h^{\prime}(2)=$ $\square$ (B) If $H(x)=g(f(x))$ , then $H^{\prime}(1)=$ $\square$

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Problem 303

Find the derivative of the function $y=\sqrt{5+3 x}$ . $\frac{d y}{d x}=$

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Problem 304

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Differentiate the function. $f(x)=\frac{1}{\sqrt[4]{x}}$

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Problem 305

Problem (13) A cylinderical vessel without a lid is to be made of a metallic lamina of surface area $462 \mathrm{~cm}^{2}$ . Find the length of its base radius when its capacity is maximum $\left(\pi=\frac{22}{7}\right)$ . Answer

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Problem 306

$\triangle X R R C I C E 4$ A Dans chacun des cas suivants, étudier la dérivabilited a gauche et la derivabilite id droite en $x_{0}$ . 1) $f(x)=|2-2 x| ; x_{0}=1$ , 2) $f(x)=\left\{\begin{array}{l}x^{2}-5 \text { si } x>-2 \\ -4 x-9 \text { si } x \text { - } y^{2}\end{array} ; x_{0}=-2\right.$ $\mathrm{B} /$ Soit $f$ la fonction définie sur $\mathbb{R}$ par $f(x)=2 x^{2}-|x|+1$ . Etudier la dérivabilité de $f$ en 0 , puis donner une interprutution graphique du résultat obtenu.

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Problem 307

Use the Fundamental Theorem of Calculus to find the derivative of $f(x)=\int_{\sqrt{x}}^{x^{2}} \arctan \left(t^{2}+4\right) d t$

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Problem 308

A rectangle has one side of 6 cm . How fast is the area of the rectangle changing at the instant when the other side is 18 cm and increasing at 2 cm per minute? (Give units.)
Answer: $216 \frac{\mathrm{~cm}^{2}}{\mathrm{~min}}$

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Problem 309

If $f(x)=x \cos ^{-1}(3 x+2)-\sqrt{6-3 x^{2}}$ , find $f^{\prime}(x)$

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Problem 310

The function $g(t)$ models the number of books sold in a store on day $t$ , where $t$ is the number of days after January 1,2020 . Which of the following is the best interpretation of the statement $g^{\prime}(7)=-11$ ? (A) On January 8, 2020, approximately 11 books were sold. (B) On January 8,2020 , the number of books sold was decreasing at a rate of 11 books per day.
C On January 8,2020 , the rate at which books were sold was decreasing at a rate of 11 books per day per day. (D) From January 1, 2020, to January 8,2020 , the number of books sold was decreasing at an average rate of 11 books per day.

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Problem 311

```latex The domain of the function $h$ graphed below is all real numbers, and all of its extreme values occur when $-3<x<3$ . Use the graph to answer the following questions.
Identify all of the values of $c$ for which $h^{\prime}(c)=0$ . If there is more than one value, enter the values as a comma-separated list. If there are none, enter DNE. values: $\square$
Identify all of the values of $c$ for which $h^{\prime}(c)$ does not exist. If there is more than one value, enter the values as a comma-separated list. If there are none, enter DNE. values: $\square$ ```

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Problem 312

$f(x)=\frac{4+6 x}{4-7 x}$
For each of the following, enter DNE if no such answer exist. Enter multiple values in a comma-separated list. Enter intervals using interval notation, including the union symbol when entering multiple intervals if necesary.
2. Determine all critical $x$ -value $(s)$ of $f(x)$ .
Critical value(s): $\square$ b. Determine the interval(s) where $f(x)$ is increasing.
Increasing: $\square$ c. Deternine the interval(s) where $f(x)$ is decreasing.
Decreasing: $\square$ d. Deremine the $x$ -coordinate(s) of all local maxima of $f(x)$ . z-value(s) of local maximas: $\square$ e. Detertmine the $x$ -coordinate of all local minima of $f(x)$ .
I valuer(s) of local minimas $\square$ f. Determine che interval(s) where $f(x)$ is concave up.
Concave up $\square$ g. Deermine the interval(s) where $f(x)$ is concave down.
Concave down: $\square$ h. Deernine the $x$ -value(s) of all inflection point(s) of $f(x)$ . z-value(s) of inflection point(s): $\square$ i. Deternine all horizonal asymptote(s) of $f(x)$ . $y=$ $\square$ j. Decermine all vertical asymprote( $(x)$ of $f(x)$ . $\square$
1. Use all of the preceding information to shecth a graph of $f(x)$ on your own paper. Change the following to Yes when you're done.
Graph complete: $\square$ 1

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Problem 313

Consider the function $f(x)=2 x^{3}+6 x^{2}-90 x+7, \quad-5 \leq x \leq 4$ .
Find the absolute minimum value of this function. Answer: $\square$ Find the absolute maximum value of this function. Answer: $\square$
Note: You can earn partial credit on this problem.

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Problem 314

Find the critical points and determine if the function is increasing or decreasing on the given intervals. $y=3 x^{4}+6 x^{3}$
Left critical point: $c_{1}=$ $\square$ Right critical point: $c_{2}=$ $\square$ The function is: ? on $\left(-\infty, c_{1}\right)$ . ? on $\left(c_{1}, c_{2}\right)$ . ? on $\left(c_{2}, \infty\right)$ .

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Problem 315

Find the critical point and determine if the function is increasing or decreasing on the given intervals. $y=-x^{2}+2 x+7$
Critical point: $c=$ $\square$ The function is: ? on $(-\infty, c)$ . ? on $(c, \infty)$ .

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Problem 316

Find the critical numbers of the function $f(x)=12 x^{5}-15 x^{4}-20 x^{3}+9$ and classify them using a graph. $x=$ $\square$ $x=$ $\square$ is a Select an answer $x=$ $\square$ is a Select an answer is a Select an answer

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Problem 317

3. Find the absolute maximum and absolute minimum of the functions on the given intervals. (a) $\qquad$ $3-6 x^{2}+5$ (d) $f(x)=x+\frac{1}{x}$ , $[1 / 4,4]$ (C)

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Problem 318

If $x^{3}+6-3 x y^{2}=-1-\sin \left(2 x^{5}\right)$ , find $\frac{d y}{d x}$ when $x=1, y=2$ .

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Problem 319

Find the inverse of the function and differentiate the inverse in two ways. (i) Differentiate the inverse function directly. (ii) Use $\frac{d}{d x} f^{-1}(x)=\frac{1}{f^{\prime}\left[f^{-1}(x)\right]}$ to find the derivative of the inverse. $f(x)=\sqrt{4 x+9}, x \geq-\frac{9}{4}$
The inverse of $f(x)$ is $f^{-1}(x)=\frac{y^{2}-9}{4}$ , $\square$ $x \geq 0 \text {. }$ for all $x$ . $x \geq 90$

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Problem 320

ore: $2 / 3$ Penalty: 1 off
2uestion Watch Video Show Examples The radius of a cylinder is increasing at a constant rate of 3 meters per second, and the volume is increasing at a rate of 108 cubic meters per second. It the instant when the height of the cylinder is 6 meters and the volume is 33 cubic meters, what is the rate of change of the height? The volume of a ylinder can be found with the equation $V=\pi r^{2} h$ . Round your answer to three decimal places. Answer Attempt 3 out of 3

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Problem 321

6. [-/1 Points]
DETAILS MY NOTES MARSVECTORCALC6 2.6.013.
Find a unit vector normal to the surface $\cos (x y)=e^{z}-2$ at $(1, \pi, 0)$ . $\square$
Additional Materials eBook

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Problem 322

Consider the following function. $h(z)=\frac{1}{z}+5 z^{2} \text { for } z>0$
Select the exact global maximum and minimum values of the function. The global maximum of $h(z)$ on $z>0$ is $\frac{1}{10}+125$ , the global minimum is $\sqrt[3]{10}+\sqrt[3]{\frac{5}{4}}$ The global maximum of $h(z)$ on $z>0$ does not exist, the global minimum is $\sqrt[3]{10}+\sqrt[3]{\frac{5}{2}}$ The global maximum of $h(z)$ on $z>0$ is $\frac{1}{10}+125$ , the global minimum is $\sqrt[3]{5}+\sqrt[3]{\frac{5}{4}}$ The global maximum of $h(z)$ on $z>0$ does not exist, the global minimum is $\sqrt[3]{10}+\sqrt[3]{\frac{5}{4}}$

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Problem 323

23. If $f(1)=10$ and $f^{\prime}(x) \geqslant 2$ for $1 \leqslant x \leqslant 4$ , how small can $f(4)$ possibly be?
24. Suppose that $3 \leqslant f^{\prime}(x) \leqslant 5$ for all values of $x$ . Show that $18 \leqslant f(8)-f(2) \leqslant 30$ .
25. Does there exist a function $f$ such that $f(0)=-1, f(2)=4$ , and $f^{\prime}(x) \leqslant 2$ for all $x$ ?

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Problem 324

السؤال 9 $y=\tan \left(4 x^{2}+1\right) \quad$ اذا كان لدينا الداله $\begin{aligned} y^{\prime}=\sec ^{2}(8 x) & \text {.a } \bigcirc \\ y^{\prime}=(8 x+1) \sec ^{2}\left(4 x^{2}+1\right) & \text {.b } \bigcirc \\ y^{\prime}=(8 x) \sec ^{2}\left(4 x^{2}+1\right) & . c \bigcirc \end{aligned}$

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Problem 325

السؤال 1
ازا كان لدينا الدالة y=3x5 فإن : $\begin{array}{rr} y^{\prime \prime \prime}=15 x^{4}-2 x-4 & . \mathrm{a} \bigcirc \\ y^{\prime \prime \prime}=180 x^{2} & . \mathrm{b} \bigcirc \\ y^{\prime \prime \prime}=60 x^{3}-2 & . c \bigcirc \end{array}$

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Problem 326

السؤال 8 y=3x5 -x²-4x+2 اذا كان لدينا الدالة فإن : $\begin{array}{r} y^{\prime \prime \prime}=15 x^{4}-2 x-4 \quad \text {.a } \bigcirc \\ y^{\prime \prime \prime}=180 x^{2} \quad . \mathrm{b} \bigcirc \\ y^{\prime \prime \prime}=60 x^{3}-2 \quad . c \bigcirc \end{array}$

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Problem 327

السؤال 6
اذا كان لاينا الداله $y=\tan \left(4 x^{2}+1\right)$ فإن : $y^{\prime}=(8 x) \sec ^{2}\left(4 x^{2}+1\right)$ .a $y^{\prime}=(8 x+1) \sec ^{2}\left(4 x^{2}+1\right)$ $y^{\prime}=\sec ^{2}(8 x)$ .c
1 درجات حفظ الإجا

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Problem 328

السؤال 5 $y=\sin \left(x^{3}+x-4\right) \quad$ اذا كان لدينا الدالة فإن : $\begin{aligned} y^{\prime}=\left(3 x^{2}+1\right) \cos \left(x^{3}+x-4\right) & . \mathrm{a} \bigcirc \\ y^{\prime}=\cos \left(3 x^{2}+1\right) & \text {.b } \bigcirc \\ y^{\prime}=\left(3 x^{2}\right) \cos \left(x^{3}+x-4\right) & \text {.c } \bigcirc \end{aligned}$

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Problem 329

Exercice 7. Étudier la dérivabilité des fonctions suivantes aux points indiqués : $f(x)=\left\{\begin{array}{lll} \left|x^{2}-4\right| & \text { si } & x \leq 2 \\ \frac{x^{2}-x-2}{x^{2}+1} & \text { si } & x>2 . \end{array} \quad x_{0}=2, \quad g(x)=\left\{\begin{array}{ll} \frac{2^{x+2}-16}{4^{x}-16} & \text { si } x \neq 2 \\ \frac{1}{2} & \text { si } x=2 . \end{array}\right.\right.$

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Problem 330

3 Mark for Review $\frac{d}{d x}\left(x \cos \left(x^{2}\right)\right)=$ (A) $\sum_{n=0}^{\infty}(-1)^{n} \frac{2 n+1}{(2 n)!} x^{2 n}$
B $\sum_{n=0}^{\infty}(-1)^{n} \frac{4 n+1}{(2 n)!} x^{4 n}$ (C) $\sum_{n=0}^{\infty}(-1)^{n} \frac{4 n+3}{(2 n+1)!} x^{4 n+2}$ (D) $\sum_{n=1}^{\infty}(-1)^{n+1} \frac{4 n}{(2 n)!} x^{4 n-1}$

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Problem 331

Find the derivative of $y$ with respect to $x$ . $\begin{array}{l} y=\log _{5}\left(\left(\frac{x+3}{x-3}\right)^{\ln 5}\right) \\ \frac{d y}{d x}=\square \end{array}$

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Problem 332

Find the indicated relative minimum or maximum.
Minimum of $f(x, y)=x^{2}+4 y^{2}+6$ , subject to $2 x-8 y=20$

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Problem 333

The marginal revenue (in thousands of dollars) from the sale of $x$ handheld gaming devices is given by the following function. $R^{\prime}(x)=4 x\left(x^{2}+25,000\right)^{-\frac{2}{3}}$ (a) Find the total revenue function if the revenue from 125 devices is $\$ 36,260$ . (b) How many devices must be sold for a revenue of at least $\$ 41,000$ ? (a) The total revenue function is $R(x)=6\left(x^{2}+25,000\right)^{\frac{1}{3}}-170$ , given that the revenue from 125 devices is $\$ 36,260$ . (Round to the nearest integer as needed.) (b) $\square$ devices must be sold to generate a revenue of at least $\$ 41,000$ . (Type a whole number.)

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Problem 334

2. (a) Find the $x$ -coordinate of the stationary point on the curve with the equation $y=18 x-4 \sqrt{x^{3}}$ (b) Hence, determine the greatest and least values of $y$ in the interval

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Problem 335

Evaluate dz. $\begin{array}{l} z=\frac{2 x}{y}+\sqrt{5 x y} \\ x=6, y=16, d x=0.03, d y=0.02 \end{array}$

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Problem 336

Consider the following function. $h(z)=\frac{1}{z}+5 z^{2} \text { for } z>0$
Select the exact global maximum and minimum values of the function. The global maximum of $h(z)$ on $z>0$ is $\frac{1}{10}+125$ , the global minimum is $\sqrt[3]{10}+\sqrt[3]{\frac{5}{4}}$ The global maximum of $h(z)$ on $z>0$ does not exist, the global minimum is $\sqrt[3]{10}+\sqrt[3]{\frac{5}{2}}$ The global maximum of $h(z)$ on $z>0$ is $\frac{1}{10}+125$ , the global minimum is $\sqrt[3]{5}+\sqrt[3]{\frac{5}{4}}$ The global maximum of $h(z)$ on $z>0$ does not exist, the global minimum is $\sqrt[3]{10}+\sqrt[3]{\frac{5}{4}}$ The global maximum of $h(z)$ on $z>0$ is $\frac{1}{5}+125$ , the global minimum is $\sqrt[3]{5}+\sqrt[3]{\frac{5}{4}}$ eTextbook and Media

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Problem 337

A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at a rate of 4 feet per second, how fast is the circumference changing when the radius is 15 feet?
Change in circumference $=$

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Problem 338

Use derivatives to find the critical points and inflection points. $f(x)=5 x-6 \ln x$
Enter the exact answers in increasing order. If there is only one critical point, enter NA in the second area. If there are no inflection points, enter NA.
Critical points: $\begin{array}{l} x=\square \\ x=\square \end{array}$
Inflection point: $x=$ $\square$

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Problem 339

Find the derivative of $f^{\prime}(x)=\square \quad f(x)=\sinh (x) \tanh (x)$

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Problem 340

16. Gegeben ist die Funktion $E[0 ; 10] \mapsto \mathbb{R} . E(t)$ beschreibt die Anzahl der an einem Grippevirus erkrankten Einwohner in dieser Stadt zum Zeitpunkt $t$ (in Tagen). Wofür steht der Ausdruck $\frac{E(6)-E(3)}{E(3)} \cdot 100 ?$ Kreuze die zutreffende Antwort an! [1 aus 6] \begin{tabular}{|l|c|} \hline Der Ausdruck beschreibt die mittlere Änderungsrate von $E$ in $[3 ; 6]$ . & $\square$ \\ \hline \begin{tabular}{l} Der Ausdruck gibt den prozentuellen Zuwachs an Erkrankungen vom 3. bis zum \\
6. Tag an. \end{tabular} & $\square$ \\ \hline \begin{tabular}{l} Der Ausdruck beschreibt den durchschnittlichen Zuwachs an Erkrankungen pro \\ Tag in $[3 ; 6]$ . \end{tabular} & $\square$ \\ \hline Der Ausdruck beschreibt den absoluten Zuwachs der Erkrankten in $[3 ; 6]$ . & $\square$ \\ \hline Der Ausdruck gibt die Anzahl der neu Erkrankten am 6. Tag an. & $\square$ \\ \hline Der Ausdruck gibt die prozentuelle Änderung von E pro Tag an. & $\square$ \\ \hline \end{tabular}

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Problem 341

Let $h(x)=\sqrt{3 x-3}$ and let $c$ be the number that satisfies the Mean Value Theorem for $h$ on the interval $4 \leq x \leq 13$ .
What is $c$ ?
Choose 1 answer: (A) 5.25 (B) 7.5 (c) 7.75 (D) 11.5

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Problem 342

$h(z)=\frac{1}{z}+9 z^{2} \text { for } z>0$
Select the exact global maximum and minimum values of the function. The global maximum of $h(z)$ on $z>0$ does not exist, the global minimum is $\sqrt[3]{18}+\sqrt[3]{\frac{9}{4}}$ The global maximum of $h(z)$ on $z>0$ is $\frac{1}{9}+729$ , the global minimum is $\sqrt[3]{9}+\sqrt[3]{\frac{9}{4}}$ The global maximum of $h(z)$ on $z>0$ is $\frac{1}{18}+729$ , the global minimum is $\sqrt[3]{18}+\sqrt[3]{\frac{9}{4}}$ The global maximum of $h(z)$ on $z>0$ does not exist, the global minimum is $\sqrt[3]{18}+\sqrt[3]{\frac{9}{2}}$ The global maximum of $h(z)$ on $z>0$ is $\frac{1}{18}+729$ , the global minimum is $\sqrt[3]{9}+\sqrt[3]{\frac{9}{4}}$

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Problem 343

The function $f(x)=9 x+7 x^{-1}$ has one local minimum and one local maximum. This function has a local maximum at $x=$ $\square$ with value $\square$ and a local minimum at $x=$ $\square$ with value $\square$

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Problem 344

Determine the signs (positive, negative, or zero) of $y=f(x)$ (shown in the graph), $f^{\prime}(x)$ , and $f^{\prime \prime}(x)$ when $x=1$ .
The sign of $f(1)$ is Select an answer $v$ The sign of $f^{\prime}(1)$ is Select an answer $\vee$ The sign of $f^{\prime \prime}(1)$ is Select an answer $v$

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Problem 345

Find $f^{\prime}(x)$ . $\begin{array}{l} f(x)=\left(5 x^{6}+6\right)^{3} \\ f^{\prime}(x)=\square \end{array}$

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Problem 346

Differentiate. $\begin{array}{c} y=\ln \left[(x+6)^{3}(x+7)^{6}(x+2)^{4}\right] \\ \frac{d}{d x}\left[\ln \left[(x+6)^{3}(x+7)^{6}(x+2)^{4}\right]\right]= \end{array}$

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Problem 347

Differentiate the following function. $\begin{array}{l} y=(\ln x)^{8}+\ln \left(x^{8}\right) \\ \frac{d y}{d x}=\square \end{array}$ $\square$

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Problem 348

Given the function $g(x)=6 x^{3}+18 x^{2}-144 x$ , find the first derivative, $g^{\prime}(x)$ . $g^{\prime}(x)=$ $\square$ Notice that $g^{\prime}(x)=0$ when $x=-4$ , that is, $g^{\prime}(-4)=0$ . Now, we want to know whether there is a local minimum or local maximum at $x=-4$ , so we will use the second derivative test. Find the second derivative, $g^{\prime \prime}(x)$ . $g^{\prime \prime}(x)=$ $\square$ Evaluate $g^{\prime \prime}(-4)$ . $g^{\prime \prime}(-4)=$ $\square$ Based on the sign of this number, does this mean the graph of $g(x)$ is concave up or concave down at $x=-4$ ? At $x=-4$ the graph of $g(x)$ is Select an answer $v$ Based on the concavity of $g(x)$ at $x=-4$ , does this mean that there is a local minimum or local maximum at $x=-4$ ? At $x=-4$ there is a local Select an answer $\checkmark$

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Problem 349

$\mathrm{t}=18$ days? Answer in appropriate units.

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Problem 350

2. Let $S(t)=\frac{1}{1+e^{-t}}$ . a. Find $S^{\prime}(t)$ . Show your work completely to justify your response. b. Which of the following equations hold true? Explain your thinking fully. (Note: Only one equation is true.) 1) $S^{\prime}(t)=S(t)$ 2) $S^{\prime}(t)=(S(f))^{2}$ 3) $S^{\prime}(t)=S(t)(1-S(t))$ 4) $S^{\prime}(t)=-S(-t)$
Note: The function $S(f)$ is called the "Sigmoid activation function" and is extremely important in machine learning and artificial intelligence.

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Problem 351

$1 \leftarrow \quad$ Find $\frac{d y}{d x}$ $\begin{array}{l} y=\int_{x}^{1} \sqrt{t^{4}+5} d t \\ \frac{d y}{d x}=\square \end{array}$ $\square$

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Problem 352

(1 point)
Differentiate $y=\sqrt{1-x^{2}} \sin ^{-1} x$ $y^{\prime}=$

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Problem 353

Find $f^{\prime}$ , given $f^{\prime}(x)=1, f(x)=\frac{x \sin ^{3}\left(x^{2}\right)}{\sqrt{x^{5}+1}}$

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Problem 354

Consider the curve given by the equation $(2 y+1)^{3}-24 x=-3 . \quad 3(2 y$ (a) Show that $\frac{d y}{d x}=\frac{4}{(2 y+1)^{2}}$ . (b) Write an equation for the line tangent to the curve at the point $(-1,-2)$ (c) Evaluate $\frac{d^{2} y}{d z^{2}}$ at the point $(-1,-2)$ . (d) The point $\left(\frac{1}{6}, 0\right)$ is on the curve. Find the value of $\left(y^{-1}\right)^{\prime}(0)$ .

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Problem 355

$2 W$ Mark for Review $4 \%$
Consider the curve in the $x y$ -plane defined by $x^{2}-\frac{y^{2}}{5}=1$ . It is known that $\frac{d y}{d x}=\frac{5 x}{y}$ and $\frac{d^{2} y}{d x^{2}}=-\frac{25}{y^{3}}$ . Which of the following statements is true about the curve in Quadrant IV? A) The curve is concave up because $\frac{d y}{d x}>0$ .
B The curve is concave down because $\frac{d y}{d x}<0$ .
C The curve is concave up because $\frac{d^{2} y}{d x^{2}}>0$ .
D The curve is concave down because $\frac{d^{2} y}{d x^{2}}<0$ .

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Problem 356

This resource is marked with invisible traceable identification associated with your SID\# Read the Disclaimer [키 for details. Getting a job on ... another planet (Y-Planet to be exact!) is a big deal but... YOU GOT IT! (your license \# is $313601330 . .$ . Hooray!)
Getting used to such new environment is also a huge challenge. For example, the 3D space around you is curved and shaped in a weird way, it might be hard to just fit into a big door opening until you learn the tricks of "Y-Planet" geometry. The Y-Planet inhabitants are... funny and friendly creatures... but we don't have time right now to chat about that.
Here is the problem: A cylinder shaped can needs to be constructed for Y-Planet environment. This can is supposed to hold 600 cubic centimeters of... soup (how the aliens eat soup... that's something... but don't laugh at them... you'll be fired).
The material for the sides of the can costs 0.04 cents per square centimeter. The material for the top and bottom of the can need to be thicker, and costs 0.07 cents per square centimeter.
Find the dimensions for the can that will minimize production cost. Helpful YP information: $h$ : height of can, $r$ : radius of can $V=\sqrt{\pi r^{5} h} \quad\left(V=\pi r^{2} h\right. \text { on Earth, don't forget "Mother Land") }$
Area of the sides: $A=2 \pi r h$ (same as on Earth) Area of the top/bottom: $A=\pi r^{2}$ (same as on Earth) To minimize the cost of the can: Radius of the can, $r=$ $\square$ Height of the can, $h=$ $\square$ and Minimum Cost is: $\square$ cents
Use calculator, round to 3 decimal places.

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Problem 357

Determine the signs (positive, negative, or zero) of $y=f(x)$ (shown in the graph), $f^{\prime}(x)$ , and $f^{\prime \prime}(x)$ when $x=1$ .
The sign of $f(1)$ is Select an answer $\vee$ The sign of $f^{\prime}(1)$ is Select an answer $\checkmark$ The sign of $f^{\prime \prime}(1)$ is Select an answer $\vee$

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Problem 358

3. Each side of a square is increasing at a rate of $6 \mathrm{~cm} / \mathrm{s}$ . At what rate is the area of the square increasing when the area of the square is $16 \mathrm{~cm}^{2}$ ?

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Problem 359

9 Mark for Review 40
The function $f$ is defined by $f(x)=x^{2} e^{-x^{2}}$ . At what values of $x$ does $f$ have a relative maximum? (A) -2 (B) 0 (C) 1 only (D) -1 and 1

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Problem 360

Consider the function $f(x)=5 x+5 x^{-1}$ . For this function there are four important intervals: $(-\infty, A],[A, B),(B, C]$ , and $[C, \infty)$ where $A$ , and $C$ are the critical numbers and the function is not defined at $B$ . Find $A$ $\square$ and $B$ $\square$ and $C$ $\square$ For each of the following open intervals, tell whether $f(x)$ is increasing or decreasing. $(-\infty, A)$ : Select an answer $\checkmark$ $(A, B)$ : Select an answer $(B, C)$ : Select an answer $(C, \infty)$ Select an answer $\vee$ Note that this function has no inflection points, but we can still consider its concavity. For each of the following intervals, tell whether $f(x)$ is concave up or concave down. $(-\infty, B):$ Select an answer $\vee$ $(B, \infty)$ : Select an answer $\vee$

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Problem 361

1 Mark for Review If $x y^{2}+\frac{x^{2}}{y}=5$ , then $\frac{d y}{d x}=$ (A) $\frac{-2 x y}{x\left(2 y^{3}-x\right)}$
B $\frac{-y\left(y^{3}+2 x\right)}{x\left(2 y^{3}-x\right)}$ (C) $\frac{-y^{4}}{x\left(2 y^{3}-x\right)}$ (D) $\frac{5 y^{2}-y^{4}-2 x y}{x\left(2 y^{3}+x\right)}$

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Problem 362

Find the derivative of the function $f(x) = 4x e^{2x}$ , and then find the critical points of the function.

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Problem 363

6 Mark for Review AP
The function $f$ is defined by $f(x)=e^{-x}\left(x^{2}+2 x\right)$ . At what values of $x$ does $f$ have a relative maximum?
A $x=-2+\sqrt{2}$ and $x=-2-\sqrt{2}$
B $x=-\sqrt{2}$ only (C) $x=-2$ and $x=0$ (D) $x=\sqrt{2}$ only

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Problem 364

7 ■ Mark for Review
Let $f$ be a differentiable function with a domain of $(0,10)$ . It is known that $f^{\prime}(x)$ , the derivative of $f(x)$ , is negative on the intervals $(0,2)$ and $(4,6)$ and positive on the intervals $(2,4)$ and $(6,10)$ . Which of the following statements is true? A) $f$ has no relative minima and three relative maxima. (B) $f$ has one relative minimum and two relative maxima. (C) $f$ has two relative minima and one relative maximum.
D $f$ has three relative minima and no relative maxima.

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Problem 365

8 Mark for Review
Let $f$ be the function with derivative $f^{\prime}(x)=x^{3}-3 x-2$ . Which of the following statements is true?
A $f$ has no relative minima and one relative maximum. (B) $f$ has one relative minimum and no relative maxima. (C) $f$ has one relative minimum and one relative maximum. (D) $f$ has two relative minima and one relative maximum.

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Problem 366

At the point shown on the function above, which of the following is true? $f^{\prime}<0, f^{\prime \prime}>0$ $f^{\prime}>0, f^{\prime \prime}>0$ $f^{\prime}<0, f^{\prime \prime}<0$ $f^{\prime}>0, f^{\prime \prime}<0$

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Problem 367

Assume $x$ and $y$ are functions of $t$ . Evaluate $\frac{d y}{d t}$ for $2 x y-2 x+6 y^{3}=-84$ , with the conditions $\frac{d x}{d t}=-12, x=6, y=-2$ .

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Problem 368

15 Mark for Review $4 \%$ \begin{tabular}{|c||c|c|c|c|} \hline $x$ & 0 & 1 & 2 & 3 \\ \hline $f(x)$ & 15 & 14 & 12 & 9 \\ \hline \end{tabular}
Let $f$ be a function with selected values given in the table above. Which of the following statements must be true? I. By the Intermediate Value Theorem, there is a value $c$ in the interval $(0,3)$ such that $f(c)=10$ . II. By the Mean Value Theorem, there is a value $c$ in the interval $(0,3)$ such that $f^{\prime}(c)=-2$ . III. By the Extreme Value Theorem, there is a value $c$ in the interval $[0,3]$ such that $f(c) \leq f(x)$ for all $x$ in the interval $[0,3]$ . (A) None (B) Ionly (C) 11 only (D) $1, I I$ , and III

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Problem 369

A ladder 30 ft long-rests against a vertical wall. If the top of the ladder is being pulled up the wall at a rate of $4 \mathrm{ft} / \mathrm{s}$ , at what rate is the bottom of the ladder moving towards the wall when the top of the ladder is 6 ft from the ground?
Do not include units in your answer. Your answer can be exact or approximate. If it is approximate, round to three decimal places.

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Problem 370

Find the equation of the tangent line at the given point on the curve. $x-\sin (\pi y)=5 ;(5,4)$

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Problem 371

Find the derivative of the function. $\begin{array}{l} h(t)=11 \cot ^{-1}(t)+11 \cot ^{-1}\left(\frac{1}{t}\right) \\ h^{\prime}(t)=\square \end{array}$

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Problem 372

17 Mark for Review Aps.
Let $f$ be a twice-differentiable function. Which of the following statements are individually sufficient to conclude that $x=2$ is the location of the absolute maximum of $f$ on the interval $[-5,5]$ ? I. $f^{\prime}(2)=0$ II. $x=2$ is the only critical point of $f$ on the interval $[-5,5]$ , and $f^{\prime \prime}(2)<0$ . III. $x=2$ is the only critical point of $f$ on the interval $[-5,5]$ , and $f(-5)<f(5)<f(2)$ . (A) II only (B) Ill only (C) I and II only
D DI and III only

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Problem 373

[-/0.5 Points] DETAILS MY NOTES TANAPMATH7 10.2.058.
Find the inflection point(s), if any, of the function. (If an answer does not exist, enter DNE.) $g(x)=4 x^{4}-8 x^{3}+6$ smaller $x$ -value $(x, y)=$ $\square$ ) larger $x$ -value $(x, y)=$ $\square$ ) Need Help? Read it

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Problem 374

Use difference quotients with $\Delta x=0.1$ and $\Delta y=0.1$ to estimate $f_{x}(4,2)$ and $f_{y}(4,2)$ where $f(x, y)=e^{-x} \sin (y) .$ $\begin{array}{l} f_{x}(4,2) \approx \\ f_{y}(4,2) \approx \end{array}$ $\square$ $\square$ Then give better estimates by using $\Delta x=0.01$ and $\Delta y=0.01$ . $\begin{array}{l} f_{x}(4,2) \approx \square \\ f_{y}(4,2) \approx \square \end{array}$ Submit answer Next item

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Problem 375

18 Mark for Review \begin{tabular}{|c|c|c|c|c|c|c|} \hline $x$ & 0 & 2 & 4 & 6 & 8 & 10 \\ \hline $f^{\prime}(x)$ & -1 & 0 & -2 & 3 & 0 & -1 \\ \hline $f^{\prime \prime}(x)$ & 8.333 & -1.900 & 0.971 & -0.304 & 0.400 & -4.167 \\ \hline \end{tabular}
Let $f$ be a twice-differentiable function. Selected values of $f^{\prime}$ and $f^{\prime \prime}$ are shown in the table above. Which of the following statements are true? I. $f$ has neither a relative minimum nor a relative maximum at $x=2$ . II. $f$ has a relative maximum $x=2$ . III. $f$ has a relative maximum $x=8$ . (A) Ionly (B) II only (C) III only (D) I and III only

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Problem 376

Find $d y / d x$ by implicit differentiation. $e^{y} \cos (x)=9+\sin (x y)$ $\frac{d y}{d x}$ $\square$

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Problem 377

Find $d y / d x$ using the method of logarithmic differehtiation when $y=x^{4 \cos (x)}$ . $d y / d x=4 x^{\wedge}(4 \cos x)(\operatorname{co}$

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Problem 378

Consider the following equation. $6 x^{2}-y^{2}=4$ (a) Find $y^{\prime}$ by implicit differentiation. $y^{\prime}=$ $\square$ (b) Solve the equation explicitly for $y$ and differentiate to get $y^{\prime}$ in terms of $x$ . $y^{\prime}= \pm$ $\square$ Need Help? Read It Watch It Submit Answer 14. [-/4.76 Points] DETAILS MY NOTES SCALCET9 3.5.009.
Find $\frac{d y}{d x}$ by implicit differentiation. $\begin{array}{l} \frac{x^{2}}{x+y}=y^{2}+2 \\ \frac{d y}{d x}=\square \end{array}$

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Problem 379

Let $f$ be the function defined by $f(x)=3 x^{2}-x^{3}$ . What is the absolute minimum value of $f$ on the closed interval $\left[1, \frac{5}{2}\right]$ ? (A) 0 (B) 2 (C) $\frac{25}{8}$ (D) 4

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Problem 380

20 Mark for Review
Let $g$ be the function given by $g(x)=3 x^{4}-8 x^{3}$ . At what value of $x$ on the closed interval $[-2,2]$ does $g$ have an absolute maximum? (A) -2 (B) 0 (C) 2 (D) $\frac{8}{3}$

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Problem 381

21 Mark for Review 420
Let $f$ be the function defined by $f(x)=12 x-x^{3}$ . What is the absolute minimum value of $f$ on the closed interval $[0,3]$ ? (A) -16 (B) 0 (C) 9 (D) 16

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Problem 382

22 W Mark for Review ape
The function $f$ is differentiable and increasing on the interval $0 \leq x \leq 6$ , and the graph of $f$ has exactly two points of inflection on this interval. Which of the following could be the graph of $f^{\prime}$ , the derivative of $f$ ? (A) (B) (C) (D)

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Problem 383

Let $\begin{array}{l} f(x)=\frac{x}{\cos x^{3}} . \\ f^{\prime}(x)=\square \end{array}$

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Problem 384

28 Mark for Review
The first derivative of the function $h$ is given by $h^{\prime}(x)=x^{4}-x^{3}+x$ . On which of the following intervals is the graph of $h$ concave down? (A) $(-0.755,0)$ (B) $(0,0.5)$ only (C) $(-0.455, \infty)$ (D) $(-\infty,-0.455)$

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Problem 385

29 Mark for Review Auto saved at: 21 :40:44
The first derivative of the function $h$ is given by $h^{\prime}(x)=x^{5}-3 x^{2}+x$ . What are all intervals on which the graph of $h$ is concave down? (A) $(-\infty, 0)$ and $(0.338,1.307)$ (B) $(-\infty, 0.669)$ (C) $(-\infty, 0.167)$ and $(1, \infty)$ (D) $(0.167,1)$

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Problem 386

$31 \square$ Mark for Review Let $f$ be the function with derivative given by $f^{\prime}(x)=\sin x+\cos (2 x)-\frac{\pi}{4}$ for $0 \leq x \leq \pi$ . On which of the following intervals is $f$ increasing? (A) $[0,0,724]$ only (B) $[0,0.724]$ and $[2.418,3.142]$ (C) $[0,0.253]$ and $[1.571,2.889]$ (D) $[0.724,2.418]$

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Problem 387

32 Mark for Review
Let $f$ be the function with derivative given by $f^{\prime}(x)=\sin x+x \cos x$ for $0<x \leq \pi$ . On which of the following intervals is $f$ increasing? (A) $[0,1.077]$ only (B) $[0,2.029]$ (C) (D) $[2.029, \pi]$ only

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Problem 388

Find $y$ " by implicit differentiation. Simplify where possible. $y^{\prime \prime}=\square$

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Problem 389

Submit Answer
14. [0.38/0.5 Points]
DETAILS MY NOTES
TANAPMATH7 10.1.029.EP. PREVIOUS ANSWERS
Consider the following function. $g(t)=\frac{9 t}{t^{2}+4}$
Find the derivative of the function. $g^{\prime}(t)=\frac{-9 t^{2}+36}{\left(t^{2}+4\right)^{2}} \quad$ Nice job. Find all the values of $t$ for which $g^{\prime}(t)=0$ or $g^{\prime}(t)$ is discontinuous. (Enter your answers as a comma-separated list.) $t=$ $\square$ Please remove the grouping symbols from around your list. Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. (Enter your answers using interval notation. If the answer cannot be expressed as an interval increasing $\quad(-2,2)$ Terrific! decreasing $(-\infty,-2) \cup(2, \infty)$ - Good job. Need Help? Read II

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Problem 390

Help | Baran Uyan (6791123176) |Logout Gradebook External
12024 Remaining Time: 12:41:31
A water tank has the shape of a right circular cone, with the tip pointing downwards. The tank is 25 cm tall, and the radius (at the top) is 11 cm . If water is being drained from the tank at a rate of $6 \mathrm{~cm}^{3} / \mathrm{s}$ , find the rate at which the water level is changing when the water is 6 cm deep.
Do not include units in your answer. Your answer can be exact or approximate. If it is approximate, round to three decimal places.

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Problem 391

Differentiate the function with respect to the independent variable. $\begin{array}{l} h(t)=\frac{\ln t}{5+4 t^{2}} \\ h^{\prime}(t)=\square \end{array}$ $\square$

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Problem 392

Consider the partial derivatives $\begin{array}{c} f_{x}(x, y)=5 x^{4} y^{6}-8 x^{3} y \\ f_{y}(x, y)=6 x^{5} y^{5}-2 x^{4} \end{array}$
Is there a function $f$ which has these partial derivatives? Yes
If so, what is it? $f=$ (Enter none if there is no such function.) Are there any others? Yes Submit answer Next item

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Problem 393

4. [-/1 Points] DETAILS MY NOTES MARSVECTORCALC6 3.1.009.
Can there exist a $C^{2}$ function $f(x, y)$ with $f_{x}=2 x-3 y$ and $f_{y}=4 x+y$ ? Yes No
Additional Materials eBook Submit Answer
5. [-/2 Points]
DETAILS MY NOTES MARSVECTORCALC6 3.1.025.
A function $u=f(x, y)$ with continuous second partial derivatives satisfying Laplace's equation $\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0$ is called a harmonic function. Show that the function $u(x, y)=9 x^{3}-27 x y^{2}$ is harmonic. Since $\frac{\partial^{2} u}{\partial x^{2}}=$ $\square$ and $\frac{\partial^{2} u}{\partial y^{2}}=$ $\square$ , $\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0$ .
Additional Materials $\square$ eBook Submit Answer Home My Assignments

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Problem 394

Davecsizme: The $g$ yph of $f(x)$ is given below. Use the graph to answer the following questions. If not enoug' information is given to answer a question, write "Cannot be determined".
1. At which $x$ value $(s)$ is $f(x)=0$ ? Explain your reasoning. at $x=-6, x=-1, x=6$ be those are all points where $f(x)$ is
2. At which $x$ value $(s)$ is $f^{\prime}(x)=0$ ? Explain your reasoning. the $x$ axis. at $x=-4, x=2, x=6$ bc the slpe at those point,
3. On what interval $(s)$ is $f(x)$ increasing? Explain your reasoning. $-4 \leq x \leq 2, \quad 6 \leq x \leq 10$
4. At which $x$ value(s) does $f(x)$ have a relative minimum? Explain your reasoning. $-4$ 6
5. At which $x$ value(s) does $f^{\prime \prime}(x)=0$ ? Explain your reasoning.
6. On what interval(s) is $f(x)$ concave down? Explain your reasoning.
7. At which $x$ value(s) does $f(x)$ have a point of inflection? Explain your reasoning.

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Problem 395

Remaining Time: 09:05:11
Air is being removed from a spherical balloon so that its volume decreases at a rate of $190 \mathrm{~cm}^{3} / \mathrm{s}$ . How fast is the radius of the balloon decreasing when the diameter is 10 cm ?
Do not include units in your answer. Your answer can be exact or approxinate, Ifitis approximate, round to three decimal places. 돈

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Problem 396

Remaining Time: 09:03:27
A ladder 17 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of $8 \mathrm{ft} / \mathrm{s}$ , how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 2 ft from the wall?
Do not include units in your answer. Your answer can be exact or approximate. If it is approximate, round to three decimal places.

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Problem 397

Find $\frac{d y}{d x}$ by implicit differentiation. $x^{2}+4 x y-y^{3}=6$

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Problem 398

(2 points)
Let $f(x)=2 x^{2}+3 x+4$ and let $g(h)=\frac{f(2+h)-f(2)}{h}$ . Determine each of the following: (a) $g(1)=$ $\square$ (b) $g(0.1)=$ $\square$ (c) $g(0.01)=$ $\square$ You will notice that the values that you entered are getting closer and closer to a number $L$ . This number is called the limit of $g(h)$ as $h$ approaches 0 and is also called the derivative of $f(x)$ at the point when $x=2$ . We will see more of this when we get to the calculus textbook.
Enter the value of $L$ : $\square$

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Problem 399

P2. In each case, find $y$ . Do NOT simplify your answer. (3 Mar (a) $y=\left(4 x^{3}+3 x^{4}\right)^{100}$ (b) $y=\tan ^{2}\left(e^{x}\right)+\arctan (2 x)$ (c) $y=\frac{\ln (\ln x)}{x}$ (d) $y=(\cos 2 x)^{x}$
Hint: Use logarithmic differe

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Problem 400

We wish to find the length of the shortest ladder that will reach from the ground over the fence to the wall of the building. [A] First, find a formula for the length of the ladder in terms of $\theta$ . (Hint: split the ladder into 2 parts.) Type theta for $\theta$ . $L(\theta)=\frac{15}{\sin (\theta)}+\frac{5}{\cos (\theta)}$ [B] Now, find the derivative, $L^{\prime}(\theta)$ . Type theta for $\theta$ . $L^{\prime}(\theta)=-\frac{15 \cos (\theta)}{\sin ^{2}(\theta)}+\frac{5 \sin (\theta)}{\cos ^{2}(\theta)}$ [C] Once you find the value of $\theta$ that makes $L^{\prime}(\theta)=0$ , substitute that into your original function to find the length of the shortest ladder. (Give your answer accurate to 5 decimal places.) $L\left(\theta_{\min }\right) \approx 1 \text { feet }$ $\square$ Enter an integer or decimal number [more.-1] Submit Question

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Derivatives

Problem 301

Problem 302

Problem 303

Find the derivative of the function y=5+3xy=\sqrt{5+3 x}y=5+3x​. dydx=\frac{d y}{d x}=dxdy​=

Problem 304

SaveDifferentiate the function. f(x)=1x4f(x)=\frac{1}{\sqrt[4]{x}}f(x)=4x​1​

Problem 305

Problem (13) A cylinderical vessel without a lid is to be made of a metallic lamina of surface area 462 cm2462 \mathrm{~cm}^{2}462 cm2. Find the length of its base radius when its capacity is maximum (π=227)\left(\pi=\frac{22}{7}\right)(π=722​). Answer

Problem 306

Problem 307

Use the Fundamental Theorem of Calculus to find the derivative of f(x)=∫xx2arctan⁡(t2+4)dtf(x)=\int_{\sqrt{x}}^{x^{2}} \arctan \left(t^{2}+4\right) d tf(x)=∫x​x2​arctan(t2+4)dt

Problem 308

A rectangle has one side of 6 cm . How fast is the area of the rectangle changing at the instant when the other side is 18 cm and increasing at 2 cm per minute? (Give units.)Answer: 216 cm2 min216 \frac{\mathrm{~cm}^{2}}{\mathrm{~min}}216 min cm2​

Problem 309

If f(x)=xcos⁡−1(3x+2)−6−3x2f(x)=x \cos ^{-1}(3 x+2)-\sqrt{6-3 x^{2}}f(x)=xcos−1(3x+2)−6−3x2​, find f′(x)f^{\prime}(x)f′(x)

Problem 310

Problem 311

Problem 312

Problem 313

Problem 314

Problem 315

Find the critical point and determine if the function is increasing or decreasing on the given intervals. y=−x2+2x+7y=-x^{2}+2 x+7y=−x2+2x+7Critical point: c=c=c= □\square□ The function is: ? on (−∞,c)(-\infty, c)(−∞,c). ? on (c,∞)(c, \infty)(c,∞).

Problem 316

Find the critical numbers of the function f(x)=12x5−15x4−20x3+9f(x)=12 x^{5}-15 x^{4}-20 x^{3}+9f(x)=12x5−15x4−20x3+9 and classify them using a graph. x=x=x= □\square□ x=x=x= □\square□ is a Select an answer x=x=x= □\square□ is a Select an answer is a Select an answer

Problem 317

3. Find the absolute maximum and absolute minimum of the functions on the given intervals. (a) \qquad 3−6x2+53-6 x^{2}+53−6x2+5 (d) f(x)=x+1xf(x)=x+\frac{1}{x}f(x)=x+x1​, [1/4,4][1 / 4,4][1/4,4] (C)

Problem 318

If x3+6−3xy2=−1−sin⁡(2x5)x^{3}+6-3 x y^{2}=-1-\sin \left(2 x^{5}\right)x3+6−3xy2=−1−sin(2x5), find dydx\frac{d y}{d x}dxdy​ when x=1,y=2x=1, y=2x=1,y=2.

Problem 319

Problem 320

Problem 321

6. [-/1 Points]DETAILS MY NOTES MARSVECTORCALC6 2.6.013.Find a unit vector normal to the surface cos⁡(xy)=ez−2\cos (x y)=e^{z}-2cos(xy)=ez−2 at (1,π,0)(1, \pi, 0)(1,π,0). □\square□Additional Materials eBook

Problem 322

Problem 323

Problem 324

Problem 325

Problem 326

Problem 327

Problem 328

Problem 329

Problem 330

Problem 331

Find the derivative of yyy with respect to xxx. y=log⁡5((x+3x−3)ln⁡5)dydx=□\begin{array}{l} y=\log _{5}\left(\left(\frac{x+3}{x-3}\right)^{\ln 5}\right) \\ \frac{d y}{d x}=\square \end{array}y=log5​((x−3x+3​)ln5)dxdy​=□​

Problem 332

Find the indicated relative minimum or maximum.Minimum of f(x,y)=x2+4y2+6f(x, y)=x^{2}+4 y^{2}+6f(x,y)=x2+4y2+6, subject to 2x−8y=202 x-8 y=202x−8y=20

Problem 333

Problem 334

2. (a) Find the xxx-coordinate of the stationary point on the curve with the equation y=18x−4x3y=18 x-4 \sqrt{x^{3}}y=18x−4x3​ (b) Hence, determine the greatest and least values of yyy in the interval

Problem 335

Evaluate dz. z=2xy+5xyx=6,y=16,dx=0.03,dy=0.02\begin{array}{l} z=\frac{2 x}{y}+\sqrt{5 x y} \\ x=6, y=16, d x=0.03, d y=0.02 \end{array}z=y2x​+5xy​x=6,y=16,dx=0.03,dy=0.02​

Problem 336

Problem 337

A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at a rate of 4 feet per second, how fast is the circumference changing when the radius is 15 feet?Change in circumference ===

Problem 338

Problem 339

Find the derivative of f′(x)=□f(x)=sinh⁡(x)tanh⁡(x)f^{\prime}(x)=\square \quad f(x)=\sinh (x) \tanh (x)f′(x)=□f(x)=sinh(x)tanh(x)

Problem 340

Problem 341

Let h(x)=3x−3h(x)=\sqrt{3 x-3}h(x)=3x−3​ and let ccc be the number that satisfies the Mean Value Theorem for hhh on the interval 4≤x≤134 \leq x \leq 134≤x≤13.What is ccc ?Choose 1 answer: (A) 5.25 (B) 7.5 (c) 7.75 (D) 11.5

Problem 342

Problem 343

The function f(x)=9x+7x−1f(x)=9 x+7 x^{-1}f(x)=9x+7x−1 has one local minimum and one local maximum. This function has a local maximum at x=x=x= □\square□ with value □\square□ and a local minimum at x=x=x= □\square□ with value □\square□

Problem 344

Problem 345

Find f′(x)f^{\prime}(x)f′(x). f(x)=(5x6+6)3f′(x)=□\begin{array}{l} f(x)=\left(5 x^{6}+6\right)^{3} \\ f^{\prime}(x)=\square \end{array}f(x)=(5x6+6)3f′(x)=□​

Problem 346

Differentiate. y=ln⁡[(x+6)3(x+7)6(x+2)4]ddx[ln⁡[(x+6)3(x+7)6(x+2)4]]=\begin{array}{c} y=\ln \left[(x+6)^{3}(x+7)^{6}(x+2)^{4}\right] \\ \frac{d}{d x}\left[\ln \left[(x+6)^{3}(x+7)^{6}(x+2)^{4}\right]\right]= \end{array}y=ln[(x+6)3(x+7)6(x+2)4]dxd​[ln[(x+6)3(x+7)6(x+2)4]]=​

Problem 347

Differentiate the following function. y=(ln⁡x)8+ln⁡(x8)dydx=□\begin{array}{l} y=(\ln x)^{8}+\ln \left(x^{8}\right) \\ \frac{d y}{d x}=\square \end{array}y=(lnx)8+ln(x8)dxdy​=□​ □\square□

Problem 348

Problem 349

t=18\mathrm{t}=18t=18 days? Answer in appropriate units.

Problem 350

Problem 351

1←1 \leftarrow \quad1← Find dydx\frac{d y}{d x}dxdy​ y=∫x1t4+5dtdydx=□\begin{array}{l} y=\int_{x}^{1} \sqrt{t^{4}+5} d t \\ \frac{d y}{d x}=\square \end{array}y=∫x1​t4+5​dtdxdy​=□​ □\square□

Problem 352

(1 point)Differentiate y=1−x2sin⁡−1xy=\sqrt{1-x^{2}} \sin ^{-1} xy=1−x2​sin−1x y′=y^{\prime}=y′=

Problem 353

Find f′f^{\prime}f′, given f′(x)=1,f(x)=xsin⁡3(x2)x5+1f^{\prime}(x)=1, f(x)=\frac{x \sin ^{3}\left(x^{2}\right)}{\sqrt{x^{5}+1}}f′(x)=1,f(x)=x5+1​xsin3(x2)​

Find the derivative of the function $y=\sqrt{5+3 x}$ . $\frac{d y}{d x}=$

Save
Differentiate the function. $f(x)=\frac{1}{\sqrt[4]{x}}$

Problem (13) A cylinderical vessel without a lid is to be made of a metallic lamina of surface area $462 \mathrm{~cm}^{2}$ . Find the length of its base radius when its capacity is maximum $\left(\pi=\frac{22}{7}\right)$ . Answer

Use the Fundamental Theorem of Calculus to find the derivative of $f(x)=\int_{\sqrt{x}}^{x^{2}} \arctan \left(t^{2}+4\right) d t$

A rectangle has one side of 6 cm . How fast is the area of the rectangle changing at the instant when the other side is 18 cm and increasing at 2 cm per minute? (Give units.)
Answer: $216 \frac{\mathrm{~cm}^{2}}{\mathrm{~min}}$

If $f(x)=x \cos ^{-1}(3 x+2)-\sqrt{6-3 x^{2}}$ , find $f^{\prime}(x)$

Find the critical point and determine if the function is increasing or decreasing on the given intervals. $y=-x^{2}+2 x+7$
Critical point: $c=$ $\square$ The function is: ? on $(-\infty, c)$ . ? on $(c, \infty)$ .

Find the critical numbers of the function $f(x)=12 x^{5}-15 x^{4}-20 x^{3}+9$ and classify them using a graph. $x=$ $\square$ $x=$ $\square$ is a Select an answer $x=$ $\square$ is a Select an answer is a Select an answer

3. Find the absolute maximum and absolute minimum of the functions on the given intervals. (a) $\qquad$ $3-6 x^{2}+5$ (d) $f(x)=x+\frac{1}{x}$ , $[1 / 4,4]$ (C)

If $x^{3}+6-3 x y^{2}=-1-\sin \left(2 x^{5}\right)$ , find $\frac{d y}{d x}$ when $x=1, y=2$ .

6. [-/1 Points]
DETAILS MY NOTES MARSVECTORCALC6 2.6.013.
Find a unit vector normal to the surface $\cos (x y)=e^{z}-2$ at $(1, \pi, 0)$ . $\square$
Additional Materials eBook

Find the derivative of $y$ with respect to $x$ . $\begin{array}{l} y=\log _{5}\left(\left(\frac{x+3}{x-3}\right)^{\ln 5}\right) \\ \frac{d y}{d x}=\square \end{array}$

Find the indicated relative minimum or maximum.
Minimum of $f(x, y)=x^{2}+4 y^{2}+6$ , subject to $2 x-8 y=20$

2. (a) Find the $x$ -coordinate of the stationary point on the curve with the equation $y=18 x-4 \sqrt{x^{3}}$ (b) Hence, determine the greatest and least values of $y$ in the interval

Evaluate dz. $\begin{array}{l} z=\frac{2 x}{y}+\sqrt{5 x y} \\ x=6, y=16, d x=0.03, d y=0.02 \end{array}$

A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at a rate of 4 feet per second, how fast is the circumference changing when the radius is 15 feet?
Change in circumference $=$

Find the derivative of $f^{\prime}(x)=\square \quad f(x)=\sinh (x) \tanh (x)$

Let $h(x)=\sqrt{3 x-3}$ and let $c$ be the number that satisfies the Mean Value Theorem for $h$ on the interval $4 \leq x \leq 13$ .
What is $c$ ?
Choose 1 answer: (A) 5.25 (B) 7.5 (c) 7.75 (D) 11.5

The function $f(x)=9 x+7 x^{-1}$ has one local minimum and one local maximum. This function has a local maximum at $x=$ $\square$ with value $\square$ and a local minimum at $x=$ $\square$ with value $\square$

Find $f^{\prime}(x)$ . $\begin{array}{l} f(x)=\left(5 x^{6}+6\right)^{3} \\ f^{\prime}(x)=\square \end{array}$

Differentiate. $\begin{array}{c} y=\ln \left[(x+6)^{3}(x+7)^{6}(x+2)^{4}\right] \\ \frac{d}{d x}\left[\ln \left[(x+6)^{3}(x+7)^{6}(x+2)^{4}\right]\right]= \end{array}$

Differentiate the following function. $\begin{array}{l} y=(\ln x)^{8}+\ln \left(x^{8}\right) \\ \frac{d y}{d x}=\square \end{array}$ $\square$

$\mathrm{t}=18$ days? Answer in appropriate units.

$1 \leftarrow \quad$ Find $\frac{d y}{d x}$ $\begin{array}{l} y=\int_{x}^{1} \sqrt{t^{4}+5} d t \\ \frac{d y}{d x}=\square \end{array}$ $\square$

(1 point)
Differentiate $y=\sqrt{1-x^{2}} \sin ^{-1} x$ $y^{\prime}=$

Find $f^{\prime}$ , given $f^{\prime}(x)=1, f(x)=\frac{x \sin ^{3}\left(x^{2}\right)}{\sqrt{x^{5}+1}}$

3. Each side of a square is increasing at a rate of $6 \mathrm{~cm} / \mathrm{s}$ . At what rate is the area of the square increasing when the area of the square is $16 \mathrm{~cm}^{2}$ ?

9 Mark for Review 40
The function $f$ is defined by $f(x)=x^{2} e^{-x^{2}}$ . At what values of $x$ does $f$ have a relative maximum? (A) -2 (B) 0 (C) 1 only (D) -1 and 1

Find the derivative of the function $f(x) = 4x e^{2x}$ , and then find the critical points of the function.

Assume $x$ and $y$ are functions of $t$ . Evaluate $\frac{d y}{d t}$ for $2 x y-2 x+6 y^{3}=-84$ , with the conditions $\frac{d x}{d t}=-12, x=6, y=-2$ .

Find the equation of the tangent line at the given point on the curve. $x-\sin (\pi y)=5 ;(5,4)$

Find the derivative of the function. $\begin{array}{l} h(t)=11 \cot ^{-1}(t)+11 \cot ^{-1}\left(\frac{1}{t}\right) \\ h^{\prime}(t)=\square \end{array}$