Calculus

Problem 11401

Find critical points of $f(x)=\frac{2}{3} x^{3}+x^{2}-12 x$ and classify them using the first derivative test.

See Solution

Problem 11402

Zeigen Sie, dass der Grenzwert A der Unter- und Obersumme für $f(x)=x^{3}$ im Intervall $[0 ; b]$ $A=\frac{b^{4}}{4}$ ist.

See Solution

Problem 11403

Define the piecewise function $f(x)=\left|\left(x^{2}+1\right)\left(x^{2}-4\right)\right|$ , then find local and absolute extrema.

See Solution

Problem 11404

Find the first and second derivatives: (i) $y=(\tanh (x))^{x}$ , (ii) $x^{3}+\sin (x y)=x y^{2}$ , (iii) $x=3 \cosh (2 t)$ , $y=5 \sinh (2 t)$ .

See Solution

Problem 11405

Find the derivative of the function $y=x^{2} e^{3 x}$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 11406

Find the derivative of $y=3^{x^{2}-1}$ with respect to $x$ : $\frac{d y}{d x}=$

See Solution

Problem 11407

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{\ln x}{x^{2}}$ .

See Solution

Problem 11408

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin (3 x)-3 x \cos (3 x)}{3 x-\sin (3 x)}$ .

See Solution

Problem 11409

Find the critical numbers of the function $f(x)=x^{2} \sqrt{3-x}$ on the interval $[1,3]$ . List them or write DNE.

See Solution

Problem 11410

Find the derivative $\frac{d y}{d x}$ if $y=\tan^{-1}(\cos x)$ .

See Solution

Problem 11411

Find the critical numbers of $f(x)=x^{2} \sqrt{3-x}$ on $[1,3]$ . List them or write DNE if none exist. $x=$

See Solution

Problem 11412

Gegeben sind die Funktionen $h(x)=x(x-1)(x-3)$ und $i(x)=x^{3}-8x^{2}+15x$ .
a) Bestimme Nullstellen, Schnittpunkte mit Achsen und Verhalten für $x \to \pm \infty$ . Zeichne die Graphen. b) Untersuche die Symmetrie von $h$ . c) Beschreibe das Monotonieverhalten von $i$ mit Graph $I$ . d) Bestimme den Grad von $h$ und $i$ und erkläre, warum beide maximal 2 Extremstellen haben. e) Finde die Extrempunkte von $H$ und deren Art. f) Zeige, dass der Punkt $\left(\frac{8+\sqrt{19}}{3} \mid i\left(\frac{8+\sqrt{19}}{3}\right)\right)$ ein Tiefpunkt von $h$ ist. g) Untersuche die Abweichung der mittleren Steigung von $h$ im Intervall $[-0,5 \mid 0,5]$ zur lokalen Änderungsrate bei $x_{0}=0$ .

See Solution

Problem 11413

Find the time $t$ (0 ≤ $t$ ≤ 8) when a particle with initial velocity 27 ft/sec changes direction, given $a(t)=2t-12$ .

See Solution

Problem 11414

Find the derivative of $q(b)=\frac{b^{2}+4b+3}{b^{4}-1}$ .

See Solution

Problem 11415

Find the arclength function $s(t)$ for the curve $\mathbf{r}=\left(e^{-5 t} \cos (2 t), e^{-5 t} \sin (2 t), e^{-5 t}\right)$ starting at $t=0$ .

See Solution

Problem 11416

Gegeben ist $f_{a}(x)=-a x^{3}+4 a x$ . Zeigen Sie: a) punktsymmetrisch, b) durch $P(-2 \mid 0)$ und $Q(2 \mid 0)$ , c) einen Hoch- und Tiefpunkt, d) Wendetangente mit Steigung $m=8$ .

See Solution

Problem 11417

Find the indefinite integral using substitution: $\int 5 x^{4} e^{5 x^{5}} d x$

See Solution

Problem 11418

Find the second derivative $h^{\prime \prime}(x)$ for $h(x)=8 x^{-6}-4 x^{-8}$ . What is $\mathrm{h}^{\prime \prime}(\mathrm{x})=\square$ ?

See Solution

Problem 11419

Find the integral using substitution: $\int(1-t) e^{32 t-16 t^{2}} d t$

See Solution

Problem 11420

Find the indefinite integral using substitution: $\int(1-t) e^{22 t-11 t^{2}} d t$

See Solution

Problem 11421

Find the derivative $f^{\prime}(1)$ given the tangent line at $x=1$ is $y=6x+8$ .

See Solution

Problem 11422

Find the $y$ -intercept of the tangent line at $(3,8)$ for the function $f$ where $f^{\prime}(3)=6$ .

See Solution

Problem 11423

Bestimmen Sie den mittleren Anstieg von $f(x)=2-x^{2}$ im Intervall $[-0,5 \mid 0,5]$ und interpretieren Sie ihn geometrisch.

See Solution

Problem 11424

Given $f^{\prime}(-5)=0$ and $f^{\prime \prime}(-5)=2$ , does $f$ have a max or min at $x=-5$ ?

See Solution

Problem 11425

Given $g(x)=4 x^{3}+18 x^{2}-48 x$ , find $g^{\prime}(x)$ and $g^{\prime \prime}(x)$ . Determine concavity and local extrema at $x=1$ .

See Solution

Problem 11426

Find the critical value $c$ for $f(x)=(8+5x)^7$ and check if $f(x)$ is increasing or decreasing for $x<c$ and $x>c$ .

See Solution

Problem 11427

Find the critical numbers of the function $f(x)=2 x^{3}-33 x^{2}+180 x-3$ using derivatives.

See Solution

Problem 11428

Bestimmen Sie die Tangentensteigung der Funktion $f(x)=\frac{1}{3} x^{3}$ bei $P(-1, ?)$ und die Tangentengleichung.

See Solution

Problem 11429

Find the critical numbers of the function $f(x)=2 x^{3}-24 x^{2}+72 x-2$ . Identify the smaller and larger critical numbers.

See Solution

Problem 11430

Find the unique anti-derivative $F(x)$ of $f(x)=(4-x)^{2}-\frac{1}{(4-x)^{2}}$ with $F(0)=0$ and compute $F(3)$ .

See Solution

Problem 11431

Find the derivative of the function $f(x)=x+\frac{1}{x}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 11432

Find the local min and max of $f(x)=2x+4x^{-1}$ . Use calculus to find where the derivative equals zero.

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Problem 11433

Find the unique anti-derivative $F(x)$ of $f(x)=(4-x)^{2}-\frac{1}{(4-x)^{2}}$ with $F(0)=0$ and calculate $F(3)$ .

See Solution

Problem 11434

Find the critical number of the function $f(x)=(2 x-9) e^{5 x}$ . What is its exact value? $x=$

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Problem 11435

Evaluate the triple integral of $f(\rho, \theta, \phi)=\cos \phi$ in spherical coordinates over $0 \leq \theta \leq 2\pi$ , $\frac{\pi}{6} \leq \phi \leq \frac{\pi}{2}$ , $3 \leq \rho \leq 7$ .

See Solution

Problem 11436

Find the volume of the solid formed by rotating the area between $y=x^{2}+x$ and $y=2x$ around $y=-1$ . Also, calculate the volume of a triangular pyramid with vertices at $(0,0,0)$ , $(1,0,0)$ , $(0,2,0)$ , and $(0,0,3)$ using cross-sections.

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Problem 11437

Evaluate the integral of $f(x, y, z)=z\left(x^{2}+y^{2}+z^{2}\right)^{-3 / 2}$ over the region $x^{2}+y^{2}+z^{2} \leq 4$ , $z \geq 1$ .

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Problem 11438

Find the inflection point(s) of the function $f(x)=4 x^{3}+30 x^{2}+48 x+8$ . If none, type DNE.

See Solution

Problem 11439

Find the derivative of the function $f(x) = x^{2} - 4x + 6$ .

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Problem 11440

Find the linear approximation of $f(x)=\cos x$ at $x=\frac{\pi}{2}$ .

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Problem 11441

A stone creates a ripple with radius increasing at $3.1 \mathrm{ft/s}$ . Find area increase rate when radius is 5 ft and after 6 s.

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Problem 11442

Find the average rate of change of $f(x)=\log x$ on the interval $\left[\frac{1}{10}, 10\right]$ .

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Problem 11443

Find critical points of $f(x)=\frac{2 x^{2}}{3 x^{2}-2 x+12}$ . List them as comma-separated values or DNE if none exist.

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Problem 11444

Evaluate the integral $\int_{\pi}^{3 \pi / 2} x \sin x \, dx$ using areas of regions $R_2$ and $R_3$ : $1$ and $\pi + 1$ .

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Problem 11445

Find the derivative of $g(x)=6 x^{7}-3 x^{-1}+\ln x-2 e^{x}+\pi^{3}$ .

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Problem 11446

Find $A$ and $B$ for the curve $y=A x^{1/6}+B x^{-1/6}$ to have an inflection point at $(1,9)$ .

See Solution

Problem 11447

Find $\frac{d y}{d x}$ by using implicit differentiation on the equation $25 x = y^{2}$ .

See Solution

Problem 11448

Find where the curve $6 x + 3 y^{2} - y = 5$ has a vertical tangent line.

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Problem 11449

Find the points on the curve $6x + 2y^2 - y = 7$ where the tangent line is vertical.

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Problem 11450

Find how $\frac{d V}{d t}$ relates to $\frac{d r}{d t}$ and $\frac{d h}{d t}$ for a cylinder with $V=\pi r^{2} h$ .

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Problem 11451

Find the brain growth rate in nanograms/yr for fish at $L=18 \mathrm{~cm}$ , given $B=.007 W^{2/3}$ and $W=.12 L^{2.53}$ .

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Problem 11452

Find the average rate of change of the function $f(x)=1+3 \cos x$ on the interval $[0, \pi]$ .

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Problem 11453

Evaluate the integral: $\int \frac{\sec ^{2} t}{1+\tan t} d t$

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Problem 11454

Find the tangent line equation for $f$ at $x=-1$ given $f^{\prime}(x)=-3x+4$ and $f(-1)=6$ .

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Problem 11455

Find $\frac{d P}{d t}$ for $b=1.3, P=9 \mathrm{kPa}, V=100 \mathrm{cm}^{2}$ , $\frac{d V}{d t}=10 \mathrm{cm}^{3}/\mathrm{min}$ .

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Problem 11456

Find the tangent line equation to the graph of $f$ at $x=2$ , given $f^{\prime}$ and points $(2,6)$ , $(4,18)$ .

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Problem 11457

Consider the piecewise function $f(x)$ defined as:
1. $3x + 1$ for $x \leq 2$
2. $5x - 3$ for $x > 2$
Is $f$ continuous and/or differentiable at $x=2$ ? Choose the correct option: (A) Neither (B) Continuous, not differentiable (C) Differentiable, not continuous (D) Both

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Problem 11458

Find the derivative of $f(x)=\frac{1}{x^{7}}$ . Choices: (A) $\frac{1}{7 x^{6}}$ , (B) $-\frac{7}{x^{6}}$ , (C) $-\frac{1}{7 x^{8}}$ , (D) $-\frac{7}{x^{8}}$ .

See Solution

Problem 11459

Find the derivative of the function $f(x)=\sqrt[4]{x}$ . What is $f^{\prime}(x)$ ? (A) $\frac{1}{4} x^{\frac{1}{4}}$ (B) $x^{-\frac{3}{4}}$ (C) $\frac{1}{4} x^{-\frac{3}{4}}$ (D) $4 \cdot \sqrt[3]{x}$

See Solution

Problem 11460

Calculez les dérivées et évaluez le taux de variation instantané à $x=2$ pour : a) $f(x)=e^{-x}+e^{2 x} x^{3}$ , b) $f(x)=\ln(x^{4})-(\ln x)^{4}$ , c) $f(x)=\sin(3 x)-3 \sin(x)$ , d) $f(x)=\sin(2^{x}+\cos x)$ .

See Solution

Problem 11461

An object falls under gravity with distance $d(t)=16 t^{2}$ . Find $d^{\prime}(t)$ and compute height and speed after $7.3 \mathrm{~s}$ .

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Problem 11462

Find critical points of $f(x)=3 x^{3}-14 x^{2}+7 x$ on $[0,5]$ . List them or write DNE if none exist.

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Problem 11463

Find critical points of $f(x)=3 x^{3}-14 x^{2}+7 x$ on $[0,5]$ and extreme values $f_{\min}$ and $f_{\max}$ .

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Problem 11464

Evaluate the integral $\int_{-2}^{4} x f(x) d x$ for $f(x)=\frac{|x|}{10}$ when $-2 \leq x \leq 4$ .

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Problem 11465

Identify the limits where L'Hospital's Rule applies:
1. $\lim _{x \rightarrow \pi / 6} \frac{\sin (6 x)}{6 x-\pi}$
2. $\lim _{x \rightarrow-\infty} \frac{e^{-x}}{x}$
3. $\lim _{x \rightarrow 9} \frac{x-9}{\ln (10-x)}$
4. $\lim _{x \rightarrow \pi / 9} \frac{\sin x}{\pi / 9-x}$

See Solution

Problem 11466

Find the area for polar coordinates $r(\theta) = \sqrt{\theta (\pi^2 - \theta^2)}$ from $0 \leq \theta \leq \pi$ . Use $A = \frac{1}{2}\int_{0}^{\pi}(r(\theta))^2 d\theta$ .

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Problem 11467

Find the tangent line equation for $y=4 x \cos x$ at $(\pi,-4 \pi)$ . Write as $y=m x+b$ with $m=$ and $b=$ .

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Problem 11468

What is the velocity and acceleration of a ball at the peak of its height when thrown straight up?

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Problem 11469

Find dy/dx using implicit differentiation for the equation $e^{3x} = \sin(x + 5y)$ .

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Problem 11470

Evaluate the integral $\int_{0}^{5} \min (x, 4) f(x) d x$ where $f(x) = \frac{1}{5}$ for $0<x<5$ , else $0$ .

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Problem 11471

Find the tangent line equation for $y=4 x \cos x$ at $(\pi,-4 \pi)$ . Form: $y=m x+b$ , where $m=$ and $b=$ .

See Solution

Problem 11472

Find the inflection points of $f(x)=x^{2} e^{5 x}$ at $x=A$ and $x=B$ . Determine concavity in intervals $(-\infty, A)$ , $(A, B)$ , and $(B, \infty)$ .

See Solution

Problem 11473

Find the average velocity of a bowling ball described by $h(t)=-16 t^{2}+67 t+183$ over the interval $[1,3.6]$ .

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Problem 11474

Find $\frac{d y}{d t}$ when $y=\frac{\pi}{4}$ , given $\sin (x)+\cos (y)=\sqrt{2}$ and $\frac{d x}{d t}=5$ .

See Solution

Problem 11475

Find the intervals where $m(x)=-2 x^{4}+52 x^{2}+2$ is increasing and decreasing. Also, find $x$ for relative max and min.

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Problem 11476

Find the tangent line equation for $y=4 x \cos x$ at $(\pi,-4 \pi)$ in the form $y=m x+b$ . What are $m$ and $b$ ?

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Problem 11477

Find the growth rate of the mouse population given by $P=100(1+0.2t+0.02t^{2})$ at $t=13$ months.

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Problem 11478

Analyze the quadratic function $f(x)=3x^{2}-12x+13$ .
1. Does it have a minimum or maximum value?
2. Where does this value occur?
3. What is the value?

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Problem 11479

Differentiate the function $f(\theta)=\frac{\sin (\theta)}{1+\cos (\theta)}$ .

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Problem 11480

Find the tangent line equation to $y=\cos(x)$ at $x=\frac{1}{2} \pi$ .

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Problem 11481

A 16 cm wire is cut into two pieces to form squares.
(a) Find the area function $A(x)$ for side length $x$ .
(b) What side length $x$ minimizes the area?
(c) What is the minimum area?

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Problem 11482

Find $\mathrm{dy} / \mathrm{dx}$ using implicit differentiation for the equation $x^{4}=\cot y$ .

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Problem 11483

Find the angle the tangent line to $y=\frac{1}{\sqrt{3}} \sin 3 x$ at the origin makes with the $x$ -axis.

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Problem 11484

Given that $f^{\prime \prime}(x)$ is continuous and has an inflection point at $x=8$ , fill in values for $f^{\prime \prime}(3)$ , $f^{\prime \prime}(8)$ , and $f^{\prime \prime}(13)$ . Determine the behavior of the graph of $y=f(x)$ before and after $x=8$ .

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Problem 11485

Find the derivative of $y=(\csc x+\cot x)(\csc x-\cot x)$ .

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Problem 11486

Find the second derivative of $y=\frac{17 x^{3}}{6}-10$ .

See Solution

Problem 11487

Find where the function $f(x)=-2 x^{4}+28 x^{2}-5$ is increasing and decreasing, plus its relative max and min values.

See Solution

Problem 11488

Calculate the integral $\int 3 x(1+x)^{-4} dx$ .

See Solution

Problem 11489

Given the function $f(x)=-2 x^{4}+28 x^{2}-5$ , find where $f$ is increasing, decreasing, and the $x$ values for max/min.

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Problem 11490

Find the derivative of $y=(\ln 3 \theta)^{x}$ .

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Problem 11491

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

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Problem 11492

Find the slope of $f(x)=3+6 x e^{x}$ at the point $(0,3)$ .

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Problem 11493

Find the derivative of $y=(1+3x)e^{-3x}$ .

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Problem 11494

Find the tangent line equation for the curve $y=\frac{18}{x^{2}+2}$ at the point $(1,6)$ .

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Problem 11495

Find the second derivative $y^{\prime \prime}$ for the function $y=\sin(4x^{2}e^{x})$ .

See Solution

Problem 11496

Find the time $t$ when the particle changes direction, given $a(t)=2t-11$ and initial velocity 18 ft/sec, for $0 \leq t \leq 8$ .

See Solution

Problem 11497

A ball dropped from a building has height $s=144-16t^2$ meters after $t$ seconds. Find the time to reach the ground and impact velocity.

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Problem 11498

Is the integral $\int_{0}^{\infty} \frac{d x}{x^{2}+4}$ convergent or divergent? If convergent, find its value.

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Problem 11499

Find the derivative of $q=\sqrt{13r - r^{7}}$ .

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Problem 11500

If $f$ and $g$ are differentiable, show that $\frac{d}{d x} \ln (f(x) g(x))=\frac{f^{\prime}(x)}{f(x)}+\frac{g^{\prime}(x)}{g(x)}$ . Also, prove the Product Rule using $\frac{(f(x) g(x))^{\prime}}{f(x) g(x)}$ .

See Solution

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Calculus

Problem 11401

Find critical points of f(x)=23x3+x2−12xf(x)=\frac{2}{3} x^{3}+x^{2}-12 xf(x)=32​x3+x2−12x and classify them using the first derivative test.

Problem 11402

Zeigen Sie, dass der Grenzwert A der Unter- und Obersumme für f(x)=x3f(x)=x^{3}f(x)=x3 im Intervall [0;b][0 ; b][0;b] A=b44A=\frac{b^{4}}{4}A=4b4​ ist.

Problem 11403

Define the piecewise function f(x)=∣(x2+1)(x2−4)∣f(x)=\left|\left(x^{2}+1\right)\left(x^{2}-4\right)\right|f(x)=∣∣​(x2+1)(x2−4)∣∣​, then find local and absolute extrema.

Problem 11404

Find the first and second derivatives: (i) y=(tanh⁡(x))xy=(\tanh (x))^{x}y=(tanh(x))x, (ii) x3+sin⁡(xy)=xy2x^{3}+\sin (x y)=x y^{2}x3+sin(xy)=xy2, (iii) x=3cosh⁡(2t)x=3 \cosh (2 t)x=3cosh(2t), y=5sinh⁡(2t)y=5 \sinh (2 t)y=5sinh(2t).

Problem 11405

Find the derivative of the function y=x2e3xy=x^{2} e^{3 x}y=x2e3x. What is dydx\frac{d y}{d x}dxdy​?

Problem 11406

Find the derivative of y=3x2−1y=3^{x^{2}-1}y=3x2−1 with respect to xxx: dydx=\frac{d y}{d x}=dxdy​=

Problem 11407

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=ln⁡xx2y=\frac{\ln x}{x^{2}}y=x2lnx​.

Problem 11408

Find the limit: lim⁡x→0sin⁡(3x)−3xcos⁡(3x)3x−sin⁡(3x)\lim _{x \rightarrow 0} \frac{\sin (3 x)-3 x \cos (3 x)}{3 x-\sin (3 x)}limx→0​3x−sin(3x)sin(3x)−3xcos(3x)​.

Problem 11409

Find the critical numbers of the function f(x)=x23−xf(x)=x^{2} \sqrt{3-x}f(x)=x23−x​ on the interval [1,3][1,3][1,3]. List them or write DNE.

Problem 11410

Find the derivative dydx\frac{d y}{d x}dxdy​ if y=tan⁡−1(cos⁡x)y=\tan^{-1}(\cos x)y=tan−1(cosx).

Problem 11411

Find the critical numbers of f(x)=x23−xf(x)=x^{2} \sqrt{3-x}f(x)=x23−x​ on [1,3][1,3][1,3]. List them or write DNE if none exist. x=x=x=

Problem 11412

Problem 11413

Find the time ttt (0 ≤ ttt ≤ 8) when a particle with initial velocity 27 ft/sec changes direction, given a(t)=2t−12a(t)=2t-12a(t)=2t−12.

Problem 11414

Find the derivative of q(b)=b2+4b+3b4−1q(b)=\frac{b^{2}+4b+3}{b^{4}-1}q(b)=b4−1b2+4b+3​.

Problem 11415

Find the arclength function s(t)s(t)s(t) for the curve r=(e−5tcos⁡(2t),e−5tsin⁡(2t),e−5t)\mathbf{r}=\left(e^{-5 t} \cos (2 t), e^{-5 t} \sin (2 t), e^{-5 t}\right)r=(e−5tcos(2t),e−5tsin(2t),e−5t) starting at t=0t=0t=0.

Problem 11416

Gegeben ist fa(x)=−ax3+4axf_{a}(x)=-a x^{3}+4 a xfa​(x)=−ax3+4ax. Zeigen Sie: a) punktsymmetrisch, b) durch P(−2∣0)P(-2 \mid 0)P(−2∣0) und Q(2∣0)Q(2 \mid 0)Q(2∣0), c) einen Hoch- und Tiefpunkt, d) Wendetangente mit Steigung m=8m=8m=8.

Problem 11417

Find the indefinite integral using substitution: ∫5x4e5x5dx\int 5 x^{4} e^{5 x^{5}} d x∫5x4e5x5dx

Problem 11418

Find the second derivative h′′(x)h^{\prime \prime}(x)h′′(x) for h(x)=8x−6−4x−8h(x)=8 x^{-6}-4 x^{-8}h(x)=8x−6−4x−8. What is h′′(x)=□\mathrm{h}^{\prime \prime}(\mathrm{x})=\squareh′′(x)=□?

Problem 11419

Find the integral using substitution: ∫(1−t)e32t−16t2dt\int(1-t) e^{32 t-16 t^{2}} d t∫(1−t)e32t−16t2dt

Problem 11420

Find the indefinite integral using substitution: ∫(1−t)e22t−11t2dt\int(1-t) e^{22 t-11 t^{2}} d t∫(1−t)e22t−11t2dt

Problem 11421

Find the derivative f′(1)f^{\prime}(1)f′(1) given the tangent line at x=1x=1x=1 is y=6x+8y=6x+8y=6x+8.

Problem 11422

Find the yyy-intercept of the tangent line at (3,8)(3,8)(3,8) for the function fff where f′(3)=6f^{\prime}(3)=6f′(3)=6.

Problem 11423

Bestimmen Sie den mittleren Anstieg von f(x)=2−x2f(x)=2-x^{2}f(x)=2−x2 im Intervall [−0,5∣0,5][-0,5 \mid 0,5][−0,5∣0,5] und interpretieren Sie ihn geometrisch.

Problem 11424

Given f′(−5)=0f^{\prime}(-5)=0f′(−5)=0 and f′′(−5)=2f^{\prime \prime}(-5)=2f′′(−5)=2, does fff have a max or min at x=−5x=-5x=−5?

Problem 11425

Given g(x)=4x3+18x2−48xg(x)=4 x^{3}+18 x^{2}-48 xg(x)=4x3+18x2−48x, find g′(x)g^{\prime}(x)g′(x) and g′′(x)g^{\prime \prime}(x)g′′(x). Determine concavity and local extrema at x=1x=1x=1.

Problem 11426

Find the critical value ccc for f(x)=(8+5x)7f(x)=(8+5x)^7f(x)=(8+5x)7 and check if f(x)f(x)f(x) is increasing or decreasing for x<cx<cx<c and x>cx>cx>c.

Problem 11427

Find the critical numbers of the function f(x)=2x3−33x2+180x−3f(x)=2 x^{3}-33 x^{2}+180 x-3f(x)=2x3−33x2+180x−3 using derivatives.

Problem 11428

Bestimmen Sie die Tangentensteigung der Funktion f(x)=13x3f(x)=\frac{1}{3} x^{3}f(x)=31​x3 bei P(−1,?)P(-1, ?)P(−1,?) und die Tangentengleichung.

Problem 11429

Find the critical numbers of the function f(x)=2x3−24x2+72x−2f(x)=2 x^{3}-24 x^{2}+72 x-2f(x)=2x3−24x2+72x−2. Identify the smaller and larger critical numbers.

Problem 11430

Find the unique anti-derivative F(x)F(x)F(x) of f(x)=(4−x)2−1(4−x)2f(x)=(4-x)^{2}-\frac{1}{(4-x)^{2}}f(x)=(4−x)2−(4−x)21​ with F(0)=0F(0)=0F(0)=0 and compute F(3)F(3)F(3).

Problem 11431

Find the derivative of the function f(x)=x+1xf(x)=x+\frac{1}{x}f(x)=x+x1​. What is f′(x)f^{\prime}(x)f′(x)?

Problem 11432

Find the local min and max of f(x)=2x+4x−1f(x)=2x+4x^{-1}f(x)=2x+4x−1. Use calculus to find where the derivative equals zero.

Problem 11433

Find the unique anti-derivative F(x)F(x)F(x) of f(x)=(4−x)2−1(4−x)2f(x)=(4-x)^{2}-\frac{1}{(4-x)^{2}}f(x)=(4−x)2−(4−x)21​ with F(0)=0F(0)=0F(0)=0 and calculate F(3)F(3)F(3).

Problem 11434

Find the critical number of the function f(x)=(2x−9)e5xf(x)=(2 x-9) e^{5 x}f(x)=(2x−9)e5x. What is its exact value? x= x= x=

Problem 11435

Evaluate the triple integral of f(ρ,θ,ϕ)=cos⁡ϕf(\rho, \theta, \phi)=\cos \phif(ρ,θ,ϕ)=cosϕ in spherical coordinates over 0≤θ≤2π0 \leq \theta \leq 2\pi0≤θ≤2π, π6≤ϕ≤π2\frac{\pi}{6} \leq \phi \leq \frac{\pi}{2}6π​≤ϕ≤2π​, 3≤ρ≤73 \leq \rho \leq 73≤ρ≤7.

Problem 11436

Problem 11437

Evaluate the integral of f(x,y,z)=z(x2+y2+z2)−3/2f(x, y, z)=z\left(x^{2}+y^{2}+z^{2}\right)^{-3 / 2}f(x,y,z)=z(x2+y2+z2)−3/2 over the region x2+y2+z2≤4x^{2}+y^{2}+z^{2} \leq 4x2+y2+z2≤4, z≥1z \geq 1z≥1.

Problem 11438

Find the inflection point(s) of the function f(x)=4x3+30x2+48x+8f(x)=4 x^{3}+30 x^{2}+48 x+8f(x)=4x3+30x2+48x+8. If none, type DNE.

Problem 11439

Find the derivative of the function f(x)=x2−4x+6f(x) = x^{2} - 4x + 6f(x)=x2−4x+6.

Problem 11440

Find the linear approximation of f(x)=cos⁡xf(x)=\cos xf(x)=cosx at x=π2x=\frac{\pi}{2}x=2π​.

Problem 11441

Find critical points of $f(x)=\frac{2}{3} x^{3}+x^{2}-12 x$ and classify them using the first derivative test.

Zeigen Sie, dass der Grenzwert A der Unter- und Obersumme für $f(x)=x^{3}$ im Intervall $[0 ; b]$ $A=\frac{b^{4}}{4}$ ist.

Define the piecewise function $f(x)=\left|\left(x^{2}+1\right)\left(x^{2}-4\right)\right|$ , then find local and absolute extrema.

Find the first and second derivatives: (i) $y=(\tanh (x))^{x}$ , (ii) $x^{3}+\sin (x y)=x y^{2}$ , (iii) $x=3 \cosh (2 t)$ , $y=5 \sinh (2 t)$ .

Find the derivative of the function $y=x^{2} e^{3 x}$ . What is $\frac{d y}{d x}$ ?

Find the derivative of $y=3^{x^{2}-1}$ with respect to $x$ : $\frac{d y}{d x}=$

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{\ln x}{x^{2}}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin (3 x)-3 x \cos (3 x)}{3 x-\sin (3 x)}$ .

Find the critical numbers of the function $f(x)=x^{2} \sqrt{3-x}$ on the interval $[1,3]$ . List them or write DNE.

Find the derivative $\frac{d y}{d x}$ if $y=\tan^{-1}(\cos x)$ .

Find the critical numbers of $f(x)=x^{2} \sqrt{3-x}$ on $[1,3]$ . List them or write DNE if none exist. $x=$

Find the time $t$ (0 ≤ $t$ ≤ 8) when a particle with initial velocity 27 ft/sec changes direction, given $a(t)=2t-12$ .

Find the derivative of $q(b)=\frac{b^{2}+4b+3}{b^{4}-1}$ .

Find the arclength function $s(t)$ for the curve $\mathbf{r}=\left(e^{-5 t} \cos (2 t), e^{-5 t} \sin (2 t), e^{-5 t}\right)$ starting at $t=0$ .

Gegeben ist $f_{a}(x)=-a x^{3}+4 a x$ . Zeigen Sie: a) punktsymmetrisch, b) durch $P(-2 \mid 0)$ und $Q(2 \mid 0)$ , c) einen Hoch- und Tiefpunkt, d) Wendetangente mit Steigung $m=8$ .

Find the indefinite integral using substitution: $\int 5 x^{4} e^{5 x^{5}} d x$

Find the second derivative $h^{\prime \prime}(x)$ for $h(x)=8 x^{-6}-4 x^{-8}$ . What is $\mathrm{h}^{\prime \prime}(\mathrm{x})=\square$ ?

Find the integral using substitution: $\int(1-t) e^{32 t-16 t^{2}} d t$

Find the indefinite integral using substitution: $\int(1-t) e^{22 t-11 t^{2}} d t$

Find the derivative $f^{\prime}(1)$ given the tangent line at $x=1$ is $y=6x+8$ .

Find the $y$ -intercept of the tangent line at $(3,8)$ for the function $f$ where $f^{\prime}(3)=6$ .

Bestimmen Sie den mittleren Anstieg von $f(x)=2-x^{2}$ im Intervall $[-0,5 \mid 0,5]$ und interpretieren Sie ihn geometrisch.

Given $f^{\prime}(-5)=0$ and $f^{\prime \prime}(-5)=2$ , does $f$ have a max or min at $x=-5$ ?

Given $g(x)=4 x^{3}+18 x^{2}-48 x$ , find $g^{\prime}(x)$ and $g^{\prime \prime}(x)$ . Determine concavity and local extrema at $x=1$ .

Find the critical value $c$ for $f(x)=(8+5x)^7$ and check if $f(x)$ is increasing or decreasing for $x<c$ and $x>c$ .

Find the critical numbers of the function $f(x)=2 x^{3}-33 x^{2}+180 x-3$ using derivatives.

Bestimmen Sie die Tangentensteigung der Funktion $f(x)=\frac{1}{3} x^{3}$ bei $P(-1, ?)$ und die Tangentengleichung.

Find the critical numbers of the function $f(x)=2 x^{3}-24 x^{2}+72 x-2$ . Identify the smaller and larger critical numbers.

Find the unique anti-derivative $F(x)$ of $f(x)=(4-x)^{2}-\frac{1}{(4-x)^{2}}$ with $F(0)=0$ and compute $F(3)$ .

Find the derivative of the function $f(x)=x+\frac{1}{x}$ . What is $f^{\prime}(x)$ ?

Find the local min and max of $f(x)=2x+4x^{-1}$ . Use calculus to find where the derivative equals zero.

Find the unique anti-derivative $F(x)$ of $f(x)=(4-x)^{2}-\frac{1}{(4-x)^{2}}$ with $F(0)=0$ and calculate $F(3)$ .

Find the critical number of the function $f(x)=(2 x-9) e^{5 x}$ . What is its exact value? $x=$

Evaluate the triple integral of $f(\rho, \theta, \phi)=\cos \phi$ in spherical coordinates over $0 \leq \theta \leq 2\pi$ , $\frac{\pi}{6} \leq \phi \leq \frac{\pi}{2}$ , $3 \leq \rho \leq 7$ .

Evaluate the integral of $f(x, y, z)=z\left(x^{2}+y^{2}+z^{2}\right)^{-3 / 2}$ over the region $x^{2}+y^{2}+z^{2} \leq 4$ , $z \geq 1$ .

Find the inflection point(s) of the function $f(x)=4 x^{3}+30 x^{2}+48 x+8$ . If none, type DNE.

Find the derivative of the function $f(x) = x^{2} - 4x + 6$ .

Find the linear approximation of $f(x)=\cos x$ at $x=\frac{\pi}{2}$ .

A stone creates a ripple with radius increasing at $3.1 \mathrm{ft/s}$ . Find area increase rate when radius is 5 ft and after 6 s.

Find the average rate of change of $f(x)=\log x$ on the interval $\left[\frac{1}{10}, 10\right]$ .

Find critical points of $f(x)=\frac{2 x^{2}}{3 x^{2}-2 x+12}$ . List them as comma-separated values or DNE if none exist.

Evaluate the integral $\int_{\pi}^{3 \pi / 2} x \sin x \, dx$ using areas of regions $R_2$ and $R_3$ : $1$ and $\pi + 1$ .

Find the derivative of $g(x)=6 x^{7}-3 x^{-1}+\ln x-2 e^{x}+\pi^{3}$ .

Find $A$ and $B$ for the curve $y=A x^{1/6}+B x^{-1/6}$ to have an inflection point at $(1,9)$ .

Find $\frac{d y}{d x}$ by using implicit differentiation on the equation $25 x = y^{2}$ .

Find where the curve $6 x + 3 y^{2} - y = 5$ has a vertical tangent line.

Find the points on the curve $6x + 2y^2 - y = 7$ where the tangent line is vertical.

Find how $\frac{d V}{d t}$ relates to $\frac{d r}{d t}$ and $\frac{d h}{d t}$ for a cylinder with $V=\pi r^{2} h$ .

Find the brain growth rate in nanograms/yr for fish at $L=18 \mathrm{~cm}$ , given $B=.007 W^{2/3}$ and $W=.12 L^{2.53}$ .

Find the average rate of change of the function $f(x)=1+3 \cos x$ on the interval $[0, \pi]$ .

Evaluate the integral: $\int \frac{\sec ^{2} t}{1+\tan t} d t$

Find the tangent line equation for $f$ at $x=-1$ given $f^{\prime}(x)=-3x+4$ and $f(-1)=6$ .

Find $\frac{d P}{d t}$ for $b=1.3, P=9 \mathrm{kPa}, V=100 \mathrm{cm}^{2}$ , $\frac{d V}{d t}=10 \mathrm{cm}^{3}/\mathrm{min}$ .

Find the tangent line equation to the graph of $f$ at $x=2$ , given $f^{\prime}$ and points $(2,6)$ , $(4,18)$ .

Find the derivative of $f(x)=\frac{1}{x^{7}}$ . Choices: (A) $\frac{1}{7 x^{6}}$ , (B) $-\frac{7}{x^{6}}$ , (C) $-\frac{1}{7 x^{8}}$ , (D) $-\frac{7}{x^{8}}$ .

Find the derivative of the function $f(x)=\sqrt[4]{x}$ . What is $f^{\prime}(x)$ ? (A) $\frac{1}{4} x^{\frac{1}{4}}$ (B) $x^{-\frac{3}{4}}$ (C) $\frac{1}{4} x^{-\frac{3}{4}}$ (D) $4 \cdot \sqrt[3]{x}$

An object falls under gravity with distance $d(t)=16 t^{2}$ . Find $d^{\prime}(t)$ and compute height and speed after $7.3 \mathrm{~s}$ .

Find critical points of $f(x)=3 x^{3}-14 x^{2}+7 x$ on $[0,5]$ . List them or write DNE if none exist.

Find critical points of $f(x)=3 x^{3}-14 x^{2}+7 x$ on $[0,5]$ and extreme values $f_{\min}$ and $f_{\max}$ .

Evaluate the integral $\int_{-2}^{4} x f(x) d x$ for $f(x)=\frac{|x|}{10}$ when $-2 \leq x \leq 4$ .

Find the area for polar coordinates $r(\theta) = \sqrt{\theta (\pi^2 - \theta^2)}$ from $0 \leq \theta \leq \pi$ . Use $A = \frac{1}{2}\int_{0}^{\pi}(r(\theta))^2 d\theta$ .

Find the tangent line equation for $y=4 x \cos x$ at $(\pi,-4 \pi)$ . Write as $y=m x+b$ with $m=$ and $b=$ .

Find dy/dx using implicit differentiation for the equation $e^{3x} = \sin(x + 5y)$ .

Evaluate the integral $\int_{0}^{5} \min (x, 4) f(x) d x$ where $f(x) = \frac{1}{5}$ for $0<x<5$ , else $0$ .

Find the tangent line equation for $y=4 x \cos x$ at $(\pi,-4 \pi)$ . Form: $y=m x+b$ , where $m=$ and $b=$ .

Find the inflection points of $f(x)=x^{2} e^{5 x}$ at $x=A$ and $x=B$ . Determine concavity in intervals $(-\infty, A)$ , $(A, B)$ , and $(B, \infty)$ .

Find the average velocity of a bowling ball described by $h(t)=-16 t^{2}+67 t+183$ over the interval $[1,3.6]$ .

Find $\frac{d y}{d t}$ when $y=\frac{\pi}{4}$ , given $\sin (x)+\cos (y)=\sqrt{2}$ and $\frac{d x}{d t}=5$ .