Calculus

Problem 11601

Find the derivative of $y$ with respect to $v$ where $y=\ln (\cosh (6 v))-\frac{1}{2} \tanh ^{2}(6 v)$ .

See Solution

Problem 11602

Solve the equation $5 \sqrt{x y} \frac{d y}{d x}=4$ for $y$ where $x, y>0$ . Find $y=\square$ .

See Solution

Problem 11603

Find the derivative of $y$ where $y=3 \sinh \left(\frac{x}{2}\right)$ .

See Solution

Problem 11604

Berechnen Sie die Fläche zwischen den Funktionen und der $x$ -Achse für: a) $f_{1}(x)=-(x-4)(2+x)$ b) $f_{2}(x)=-x^{2}-6 x-5$ c) $f_{3}(x)=(x-4) \sqrt{x}$ d) $f_{4}(x)=x^{4}-10 x^{2}+9$ e) $f_{5}(x)=x^{3}+6 x^{2}+3 x-10$

See Solution

Problem 11605

Sketch a function $f(x)$ with: domain $\{x \in \mathbb{R} \mid x \neq-4,-2\}$ , $f^{\prime}(-3)=0$ , $f^{\prime}(0)$ non-existent, $f^{\prime}(x)>0$ on $(-\infty,-4) \cup(-3,-2) \cup(0, \infty)$ , $f^{\prime}(x)<0$ on $(-4,-3) \cup(-2,0)$ , $f(0)=0$ , $f(-3)=2$ , $\lim_{x \to -\infty} f(x)=1$ , $\lim_{x \to \infty} f(x)=1$ .

See Solution

Problem 11606

Differentiate the function: $f(x)=(3 x^{2}-5 x) e^{x}$ .

See Solution

Problem 11607

Find the inverse of $P(t)=\frac{1000}{1+9 e^{-t}}$ , then determine the time for $P$ to reach 800 and analyze $t \rightarrow \infty$ .

See Solution

Problem 11608

Find the derivative of $f(x)=x \cdot \sqrt{x+1}$ .

See Solution

Problem 11609

En kloss på $155 \mathrm{~g}$ henger i en fjær med stivhet $320 \mathrm{~N/m}$ . Hva er farten ved likevektsstillingen og $1.1 \mathrm{~cm}$ fra den? Svar i $\mathrm{m/s}$ .

See Solution

Problem 11610

Find the derivative $y^{\prime}(t)$ of $y(t) = 30 \cdot 2^{-t / 30}$ to analyze decay rate after $t$ years. Explain your steps.

See Solution

Problem 11611

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

See Solution

Problem 11612

Find the first derivative of $f(x)=x^{5/3} \ln(x)$ and determine where it has horizontal tangents. $x=$

See Solution

Problem 11613

Find the derivative of $G(x)=\frac{x^{2}-2}{2 x+1}$ .

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Problem 11614

Find the $x$ -coordinates of inflection points for the function $g(x)=\frac{1+x}{1+x^{2}}$ with $g^{\prime \prime}(x)=\frac{2(x-1)(x^{2}+4x+1)}{(1+x^{2})^{3}}$ .

See Solution

Problem 11615

Find the derivative of $H(u)=(u-\sqrt{u})(u+\sqrt{u})$ .

See Solution

Problem 11616

Find the limit: $\lim _{x \rightarrow \infty} \frac{5 x^{3 / 2}}{4 \sqrt{x}+1}$ .

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Problem 11617

Given the function $f(x)=x^{3}+9 x^{2}+4$ , find if $f^{\prime \prime}(x)$ is + or - in the intervals $(-\infty,-3)$ , -3, and $(-3, \infty)$ .

See Solution

Problem 11618

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

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Problem 11619

Find the derivative of $J(v)=(v^{3}-2v)(v^{-4}+v^{-2})$ .

See Solution

Problem 11620

Given $f(x) = x^3 + 9x^2 + 4$ , verify:
1. Is $x=-3$ a point of inflection?
2. Find critical value at $x=-6$ using the second derivative test.
3. Find critical value at $x=0$ using the second derivative test.

See Solution

Problem 11621

Find the differential $d y$ for each: 1) $y=-x^{3}-2$ 2) $y=-\frac{3}{x}$ .

See Solution

Problem 11622

15. Verkehrszählung: $f(x)=\frac{1}{10^{10}} x^{5}-\frac{1}{130000} x^{3}+\frac{1}{6} x+6$ .
a) Höchste Verkehrsdichte? b) Fahrzeuge zwischen 6 und 9 Uhr? c) Durchschnittlicher Fahrzeugstrom zwischen 6 und 8 Uhr?

See Solution

Problem 11623

Die Funktion $f(t)=\frac{1}{4} t^{3}-\frac{11}{4} t^{2}-4 t+44$ beschreibt die Wasseränderung.
a) Wann nimmt die Wassermenge zu? b) Bestimme den Zeitpunkt mit der geringsten Änderungsrate. c) Was bedeutet dieser Zeitpunkt für die Funktion $g(t)$ ?

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Problem 11624

Find the critical points of the function $f(x)=\frac{2 x^{2}}{x-9}$ . What are the $x$ -values?

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Problem 11625

Evaluate the triple integral $\iiint_{B}(2 x+3 y^{2}+4 z^{3}) d V$ over the box $B=\{(x, y, z) \mid 0 \leq x \leq 4,0 \leq y \leq 2,0 \leq z \leq 1\}$ .

See Solution

Problem 11626

Berechne die unbestimmten Integrale: a) $\int x^{6} d x$ , b) $\int 6 x^{2} d x$ , c) $\int n \cdot x^{2n-1} dx$ , d) $\int(4 x^{2}+2 x) dx$ , e) $\int(2 x^{3}-4 x+1) dx$ , f) $\int(a x^{2}+6 x) dx$ , g) $\int 3 x^{-2} dx$ , h) $\int(2 x+\frac{1}{x}) \cdot x dx$ , i) $\int(x+\frac{3}{x^{2}}) dx$ , j) $\int e^{x} \cdot e^{x+2} dx$ , k) $\int \frac{4}{e^{x}} dx$ , l) $\int(\sin x+2 \cos x) dx$ .

See Solution

Problem 11627

Find $c$ such that $\frac{f(9)-f(4)}{9-4}=f'(c)$ for $f(x)=\sqrt{x-4}$ on $[4,9]$ . $c=\square$

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Problem 11628

Find the derivative of $y=\ln(\tan(\frac{x}{2})) - \frac{1}{\sin x}$ .

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Problem 11629

Evaluate the triple integral $\iiint_{E} x^{2} dV$ for $E=\{(x, y, z) \mid 1-y^{2} \leq x \leq y^{2}-1, -1 \leq y \leq 1, 1 \leq z \leq 8\}$ .

See Solution

Problem 11630

Bestimme die Flächeninhaltsfunktion von $f$ mit unterer Grenze 0 für: a) $f(x)=4$ , b) $f(x)=x$ , c) $f(x)=3x+1$ , d) $f(x)=3x^2$ , e) $f(x)=\frac{1}{2}x^2$ , f) $f(x)=x^3+2x$ .

See Solution

Problem 11631

Solve $(\sec x) \frac{d y}{d x}=\mathrm{e}^{y} \cdot \mathrm{e}^{\sin x}$ using separation of variables.

See Solution

Problem 11632

A frog jumps at 30 degrees with a speed of $4.5 \mathrm{~m/s}$ . How far does it jump? $(1.8 \mathrm{~m})$

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Problem 11633

A conical cup has a diameter of 8 cm and depth of 10 cm. Liquid leaves at 7 cm³/s. Find the rate of level change when depth is 7 cm.

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Problem 11634

Evaluate the integral $\iiint_{E} z \, dV$ for the solid $E$ bounded by $y^{2}+z^{2}=4$ , $z=0$ , $x=0$ , and $x=5$ .

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Problem 11635

Find the max and min values of $g(x)=4 \csc x$ on $\left[\frac{\pi}{4}, \frac{3\pi}{4}\right]$ and graph it.

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Problem 11636

Evaluate the triple integral $\iiint_{E} f(x, y, z) d V$ where $f(x, y, z)=e^{\sqrt{x^{2}}+y^{2}}$ and $E$ is defined by the given conditions.

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Problem 11637

Leite die Funktion $f(x)=x^{2} \cdot x$ ab.

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Problem 11638

Leite die Funktion $f(x)=x^{2} \cdot x$ ab. Ergebnis: $f^{\prime}(x)=3 x^{2}$ .

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Problem 11639

Given the velocities (in mi/hr) of a car over 2 hours:
$t$ (hr): 0, 0.25, 0.5, 0.75, 1, 1.25, 1.5, 1.75, 2 $v$ (mi/hr): 30, 30, 50, 50, 45, 60, 30, 50, 80
a. Sketch a smooth curve through the points. b. Calculate the midpoint Riemann sum for displacement on $[0,2]$ with $n=2$ and $n=4$ .

See Solution

Problem 11640

Find $\frac{d x}{d t}$ when $\sqrt{x}+y=6$ and $\frac{d y}{d t}=2$ , given $x=4$ .

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Problem 11641

A kite is at 40 ft height and moves horizontally at 3 ft/sec. Find the rate of string release when its length is 50 ft.

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Problem 11642

Find the half-life of a substance that decays to $\frac{1}{11}$ of its original amount in 18 hours. Answer to 3 decimal places.

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Problem 11643

Find $f^{\prime}(0)$ if $f(x)=\sin ^{2}(3-x)$ .

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Problem 11644

Find formulas for $d y$ and $\Delta y$ for the functions: 3) $y=-x^{3}-2$ and 4) $y=\frac{2}{x}$ .

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Problem 11645

An archaeologist finds a wood fragment with $\frac{1}{6}$ of its original Carbon-14. How old is it? Use decay constant 0.00012.

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Problem 11646

Find the tangent line equation for $y=2 \arccos \sqrt{x}$ at $x=\frac{1}{2}$ .

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Problem 11647

Find $P^{\prime}(t)$ when $P(t)$ satisfies $y^{\prime}=12y$ , with $P(0)=500$ and $P(t)=900$ .

See Solution

Problem 11648

Find the tangent line equation for $y=2 \arccos \sqrt{x}$ at $x=\frac{1}{2}$ , where slope $y'=-\frac{1}{\sqrt{x(1-x)}}$ .

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Problem 11649

Calculate the work done when a gas with an initial volume of $5.29 \mathrm{~L}$ expands to double its volume at $0.987 \mathrm{~atm}$ .

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Problem 11650

Find $\lim_{x \rightarrow 1} \frac{f(x)}{g(x)+2}$ where $g(x)=\frac{1}{10}(4x^3 + 3x^2 - 10x - 17)$ and $f(1)=0$ .

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Problem 11651

Approximate $\sqrt{65}-\sqrt{63}$ using the tangent line of $f(x)=\sqrt{x}$ at $x=64$ . (A) 0 (B) 1

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Problem 11652

Find a linear approximation for: 7) $\sin 122^{\circ}$ 8) $6.99^{4}$

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Problem 11653

Find the total compression $\Delta L$ (in m) of an 8 m sandstone pillar with density 2250 kg/m³ and Young's Modulus 12.5 GPa.

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Problem 11654

Given the function $f(x)=\frac{x^{2}}{x^{2}+3}$ , find where $f$ is increasing, decreasing, local minimum, inflection points, and concavity.

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Problem 11655

Use differentials to estimate errors: 9) Sphere radius is $7 \mathrm{~cm}$ with error $\pm \frac{1}{10} \mathrm{~cm}$ . Find volume error. 10) Square sides are 4 in with error $\pm \frac{1}{5}$ in. Find area error.

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Problem 11656

Find the inflection point coordinates $(x, y)$ for the function $f(x)=\frac{e^{x}}{1+e^{x}}$ .

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Problem 11657

Find the coordinates of the inflection point for the function $f(x)=\frac{e^{x}}{1+e^{x}}$ . Determine $x$ and $y$ .

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Problem 11658

Find the formula for the $z$ coordinate of the centroid of the region defined by $x^{2}+y^{2}+z^{2} \leq 4$ and $z \geq 0$ .

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Problem 11659

Find the average speed $s(x)=\frac{3600}{60+x}$ for traveling one mile in 66 seconds. Approximate speed at $x=6$ .

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Problem 11660

A person travels $D$ miles at speed $(60+x) \mathrm{mi/hr}$ .
a. Use $L(x)=D\left(1-\frac{x}{60}\right)$ to approximate time for 77 miles at 58 mi/hr. b. Find the exact time.
a. Approximate time is 80 min. b. Exact time is $\square$ min.

See Solution

Problem 11661

Find the time $t$ when the oxygen level $f(t)=\frac{t^{2}+10 t+100}{t^{2}+36 t+324}$ is lowest.

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Problem 11662

A gardener has $850 \mathrm{~m}$ of fence to create a rectangular garden divided into 8 sub-gardens. What's the max area?

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Problem 11663

Find the monthly cost $C(90)$ , marginal cost at $x=90$ , estimate cost for 92 chairs, and actual cost increase for 92 chairs.

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Problem 11664

A coffee cup cools from $95^{\circ} \mathrm{C}$ to $60^{\circ} \mathrm{C}$ in 11 minutes. How long to cool to $30^{\circ} \mathrm{C}$ ? It takes approximately $\square$ minutes.

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Problem 11665

Evaluate the limit using the derivative definition: $\lim _{x \rightarrow e^{-1}} \frac{\ln x+1}{x-e^{-1}}=\square$

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Problem 11666

Find the second derivative of the cubic function $f(x)=13 x^{3}+19 x^{2}+10 x-9$ . What is $f^{\prime \prime}(x)$ ?

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Problem 11667

Find the integral that calculates the probability $\mathbb{P}(Y \geq X)$ for $0 \leq X, Y \leq 2$ .

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Problem 11668

Find the second derivative of $f(x)=11 x^{3}+17 x^{2}-13 x+10$ .

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Problem 11669

Find the inflection point of $f(x)=\frac{e^{x}}{1+e^{x}}$ . Determine $x=$ Number and $y=$ Number.

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Problem 11670

Evaluate the limit using the derivative definition: $\lim _{h \rightarrow 0} \frac{(3+h)^{4+h}-81}{h}$ .

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Problem 11671

Find the second derivative of $f(x)=x^{5} \ln x$ and identify its inflection point.

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Problem 11672

Find the derivative of $y$ with respect to $x$ , where $y=(x^{4}+1) \operatorname{sech}(2 \ln x)$ .

See Solution

Problem 11673

Find the derivative of $y$ with respect to $v$ for $y=\ln (\cosh (6 v))-\frac{1}{2} \tanh ^{2}(6 v)$ .

See Solution

Problem 11674

Find the derivative of $y$ with respect to $x$ : $y=(x^{4}+1) \operatorname{sech}(2 \ln x)$ . Simplify first.

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Problem 11675

Differentiate $g(\theta)=e^{\theta}(\tan (\theta)-\theta)$ and find $g^{\prime}(\theta)$ .

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Problem 11676

Find the time $t$ when the oxygen level $f(t)=\frac{t^{2}+10 t+100}{t^{2}+34 t+289}$ is lowest after waste is added.

See Solution

Problem 11677

Find the derivative of the function $y=(1-\theta) \tanh^{-1} \theta$ with respect to $\theta$ .

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Problem 11678

Find the minimum and maximum values of $f(4)$ given $f(0)=9$ and $4 \leq f'(x) \leq 7$ for $x \in [-5,5]$ .

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Problem 11679

Find the derivative of $y=(1-3 \theta) \operatorname{coth}^{-1}(3 \theta)$ with respect to $\theta$ : $\frac{d y}{d \theta}=\square$ .

See Solution

Problem 11680

Differentiate the function $s(t)=\frac{t^{2}+3 t}{(t^{2}-1)(t^{3}-9)}$ .

See Solution

Problem 11681

Differentiate the function $s(t)=\frac{t^{2}+4 t}{(t^{2}-1)(t^{3}-5)}$ .

See Solution

Problem 11682

Find the derivative of $f(t) = \sqrt[8]{2 + \tan(t)}$ .

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Problem 11683

Differentiate $f(x)=(2 x)^{\ln 2 x}$ using logarithmic differentiation to find $f^{\prime}(x)$ .

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Problem 11684

Find iterated integrals for the hyperboloid $x^{2}+y^{2}=z^{2}-1$ and cylinder $x^{2}+z^{2}=4$ over specified regions.

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Problem 11685

Set up an iterated integral for the solid $U$ defined by $x^2+y^2 \geq 1$ and $x^2+y^2 \leq z \leq 4$ to calculate $\iiint_U 2z \, dV$ . No need to evaluate.

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Problem 11686

How long would it take for two rocks to fall from a height of 177 feet? Use $d = \frac{1}{2}gt^2$ to find $t$ .

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Problem 11687

Find the surface area from revolving $x=\frac{y^{3}}{3}$ , $0 \leq y \leq 4$ , around the $y$ -axis. Answer in terms of $\pi$ .

See Solution

Problem 11688

Find the derivative of $y=(1-3 t) \operatorname{coth}^{-1} \sqrt{3 t}$ with respect to $t$ : $\frac{d y}{d t}=\square$ .

See Solution

Problem 11689

Calculate the deceleration of a bullet going from $400 \mathrm{~m} \mathrm{~s}^{-1}$ to rest after $10 \mathrm{~cm}$ .

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Problem 11690

How long does it take for the ruble to lose half its value if it falls by $12\%$ per year? Use the half-life formula.

See Solution

Problem 11691

Given $f\left(\frac{\pi}{6}\right)=-8$ , $f^{\prime}\left(\frac{\pi}{6}\right)=7$ , find $g^{\prime}(\pi / 6)$ and $h^{\prime}(\pi / 6)$ for $g(x)=f(x) \sin x$ , $h(x)=\frac{\cos x}{f(x)}$ .

See Solution

Problem 11692

Given $f\left(\frac{\pi}{6}\right)=-8$ , $f^{\prime}\left(\frac{\pi}{6}\right)=7$ , find $g^{\prime}(\frac{\pi}{6})$ and $h^{\prime}(\frac{\pi}{6})$ for $g(x)=f(x)\sin x$ , $h(x)=\frac{\cos x}{f(x)}$ .

See Solution

Problem 11693

Semi-infinite rod diffusion problem:
1. Find $c(x,t)$ for $-14.5<x<14.5$ , $D=1 \mathrm{~nm}^2 \mathrm{~s}^{-1}$ , and $t=0,5,\ldots,30$ s using $c(x, t)=\frac{c_{2}-c_{1}}{2}\left(1+\operatorname{erf}\left(\frac{x}{2 \sqrt{D t}}\right)\right)+c_{1}$ .
2. Simulate and plot the numerical solution for the same $D$ and time intervals.
3. Identify $X$ where $c(x,t)$ is closest to 0.92 from $t=1$ to $t=30$ and plot this characteristic length.

See Solution

Problem 11694

Determine the horizontal and vertical asymptotes for the function $f(x)=\frac{4}{x^{2}+3}$ .

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Problem 11695

Find the tangent line equation for $f(x)=(2 x-2)^{1/5}$ at $x=2$ . Also, find $x$ values for horizontal tangents.

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Problem 11696

Find the line equation for the linear approximation of $f(x)=\frac{x}{x+1}$ at $a=1$ , estimate $f(0.9)$ , and compute percent error.

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Problem 11697

A girl flies a kite at $300 \mathrm{ft}$ high, moving away at $25 \mathrm{ft/sec}$ . How fast to let out string when $500 \mathrm{ft}$ away?

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Problem 11698

Find the line equation for the linear approximation of $f(x)=e^{-x}$ at $a=0$ , estimate $e^{-0.04}$ , and compute percent error.

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Problem 11699

Use a linear approximation to estimate $\ln(0.95)$ . Choose a suitable $a$ for minimal error without a calculator. What is the estimate?

See Solution

Problem 11700

Estimate $\ln(0.95)$ using linear approximation with a suitable value of $a$ for minimal error (no calculator).

See Solution

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Calculus

Problem 11601

Find the derivative of yyy with respect to vvv where y=ln⁡(cosh⁡(6v))−12tanh⁡2(6v)y=\ln (\cosh (6 v))-\frac{1}{2} \tanh ^{2}(6 v)y=ln(cosh(6v))−21​tanh2(6v).

Problem 11602

Solve the equation 5xydydx=45 \sqrt{x y} \frac{d y}{d x}=45xy​dxdy​=4 for yyy where x,y>0x, y>0x,y>0. Find y=□y=\squarey=□.

Problem 11603

Find the derivative of yyy where y=3sinh⁡(x2)y=3 \sinh \left(\frac{x}{2}\right)y=3sinh(2x​).

Problem 11604

Problem 11605

Problem 11606

Differentiate the function: f(x)=(3x2−5x)exf(x)=(3 x^{2}-5 x) e^{x}f(x)=(3x2−5x)ex.

Problem 11607

Find the inverse of P(t)=10001+9e−tP(t)=\frac{1000}{1+9 e^{-t}}P(t)=1+9e−t1000​, then determine the time for PPP to reach 800 and analyze t→∞t \rightarrow \inftyt→∞.

Problem 11608

Find the derivative of f(x)=x⋅x+1f(x)=x \cdot \sqrt{x+1}f(x)=x⋅x+1​.

Problem 11609

En kloss på 155 g155 \mathrm{~g}155 g henger i en fjær med stivhet 320 N/m320 \mathrm{~N/m}320 N/m. Hva er farten ved likevektsstillingen og 1.1 cm1.1 \mathrm{~cm}1.1 cm fra den? Svar i m/s\mathrm{m/s}m/s.

Problem 11610

Find the derivative y′(t)y^{\prime}(t)y′(t) of y(t)=30⋅2−t/30y(t) = 30 \cdot 2^{-t / 30}y(t)=30⋅2−t/30 to analyze decay rate after ttt years. Explain your steps.

Problem 11611

Find the derivative of y=ex1−exy=\frac{e^{x}}{1-e^{x}}y=1−exex​.

Problem 11612

Find the first derivative of f(x)=x5/3ln⁡(x)f(x)=x^{5/3} \ln(x)f(x)=x5/3ln(x) and determine where it has horizontal tangents. x=x=x=

Problem 11613

Find the derivative of G(x)=x2−22x+1G(x)=\frac{x^{2}-2}{2 x+1}G(x)=2x+1x2−2​.

Problem 11614

Find the xxx-coordinates of inflection points for the function g(x)=1+x1+x2g(x)=\frac{1+x}{1+x^{2}}g(x)=1+x21+x​ with g′′(x)=2(x−1)(x2+4x+1)(1+x2)3g^{\prime \prime}(x)=\frac{2(x-1)(x^{2}+4x+1)}{(1+x^{2})^{3}}g′′(x)=(1+x2)32(x−1)(x2+4x+1)​.

Problem 11615

Find the derivative of H(u)=(u−u)(u+u)H(u)=(u-\sqrt{u})(u+\sqrt{u})H(u)=(u−u​)(u+u​).

Problem 11616

Find the limit: lim⁡x→∞5x3/24x+1\lim _{x \rightarrow \infty} \frac{5 x^{3 / 2}}{4 \sqrt{x}+1}limx→∞​4x​+15x3/2​.

Problem 11617

Given the function f(x)=x3+9x2+4f(x)=x^{3}+9 x^{2}+4f(x)=x3+9x2+4, find if f′′(x)f^{\prime \prime}(x)f′′(x) is + or - in the intervals (−∞,−3)(-\infty,-3)(−∞,−3), -3, and (−3,∞)(-3, \infty)(−3,∞).

Problem 11618

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 11619

Find the derivative of J(v)=(v3−2v)(v−4+v−2)J(v)=(v^{3}-2v)(v^{-4}+v^{-2})J(v)=(v3−2v)(v−4+v−2).

Problem 11620

Given f(x)=x3+9x2+4f(x) = x^3 + 9x^2 + 4f(x)=x3+9x2+4, verify:1. Is x=−3x=-3x=−3 a point of inflection?2. Find critical value at x=−6x=-6x=−6 using the second derivative test.3. Find critical value at x=0x=0x=0 using the second derivative test.

Problem 11621

Find the differential dyd ydy for each: 1) y=−x3−2y=-x^{3}-2y=−x3−2 2) y=−3xy=-\frac{3}{x}y=−x3​.

Problem 11622

15. Verkehrszählung: f(x)=11010x5−1130000x3+16x+6f(x)=\frac{1}{10^{10}} x^{5}-\frac{1}{130000} x^{3}+\frac{1}{6} x+6f(x)=10101​x5−1300001​x3+61​x+6. a) Höchste Verkehrsdichte? b) Fahrzeuge zwischen 6 und 9 Uhr? c) Durchschnittlicher Fahrzeugstrom zwischen 6 und 8 Uhr?

Problem 11623

Problem 11624

Find the critical points of the function f(x)=2x2x−9f(x)=\frac{2 x^{2}}{x-9}f(x)=x−92x2​. What are the xxx-values?

Problem 11625

Evaluate the triple integral ∭B(2x+3y2+4z3)dV\iiint_{B}(2 x+3 y^{2}+4 z^{3}) d V∭B​(2x+3y2+4z3)dV over the box B={(x,y,z)∣0≤x≤4,0≤y≤2,0≤z≤1}B=\{(x, y, z) \mid 0 \leq x \leq 4,0 \leq y \leq 2,0 \leq z \leq 1\}B={(x,y,z)∣0≤x≤4,0≤y≤2,0≤z≤1}.

Problem 11626

Problem 11627

Find ccc such that f(9)−f(4)9−4=f′(c)\frac{f(9)-f(4)}{9-4}=f'(c)9−4f(9)−f(4)​=f′(c) for f(x)=x−4f(x)=\sqrt{x-4}f(x)=x−4​ on [4,9][4,9][4,9]. c=□ c=\square c=□

Problem 11628

Find the derivative of y=ln⁡(tan⁡(x2))−1sin⁡xy=\ln(\tan(\frac{x}{2})) - \frac{1}{\sin x}y=ln(tan(2x​))−sinx1​.

Problem 11629

Evaluate the triple integral ∭Ex2dV\iiint_{E} x^{2} dV∭E​x2dV for E={(x,y,z)∣1−y2≤x≤y2−1,−1≤y≤1,1≤z≤8}E=\{(x, y, z) \mid 1-y^{2} \leq x \leq y^{2}-1, -1 \leq y \leq 1, 1 \leq z \leq 8\}E={(x,y,z)∣1−y2≤x≤y2−1,−1≤y≤1,1≤z≤8}.

Problem 11630

Bestimme die Flächeninhaltsfunktion von fff mit unterer Grenze 0 für: a) f(x)=4f(x)=4f(x)=4, b) f(x)=xf(x)=xf(x)=x, c) f(x)=3x+1f(x)=3x+1f(x)=3x+1, d) f(x)=3x2f(x)=3x^2f(x)=3x2, e) f(x)=12x2f(x)=\frac{1}{2}x^2f(x)=21​x2, f) f(x)=x3+2xf(x)=x^3+2xf(x)=x3+2x.

Problem 11631

Solve (sec⁡x)dydx=ey⋅esin⁡x(\sec x) \frac{d y}{d x}=\mathrm{e}^{y} \cdot \mathrm{e}^{\sin x}(secx)dxdy​=ey⋅esinx using separation of variables.

Problem 11632

A frog jumps at 30 degrees with a speed of 4.5 m/s4.5 \mathrm{~m/s}4.5 m/s. How far does it jump? (1.8 m)(1.8 \mathrm{~m})(1.8 m)

Problem 11633

A conical cup has a diameter of 8 cm and depth of 10 cm. Liquid leaves at 7 cm³/s. Find the rate of level change when depth is 7 cm.

Problem 11634

Evaluate the integral ∭Ez dV\iiint_{E} z \, dV∭E​zdV for the solid EEE bounded by y2+z2=4y^{2}+z^{2}=4y2+z2=4, z=0z=0z=0, x=0x=0x=0, and x=5x=5x=5.

Problem 11635

Find the max and min values of g(x)=4csc⁡xg(x)=4 \csc xg(x)=4cscx on [π4,3π4]\left[\frac{\pi}{4}, \frac{3\pi}{4}\right][4π​,43π​] and graph it.

Problem 11636

Evaluate the triple integral ∭Ef(x,y,z)dV\iiint_{E} f(x, y, z) d V∭E​f(x,y,z)dV where f(x,y,z)=ex2+y2f(x, y, z)=e^{\sqrt{x^{2}}+y^{2}}f(x,y,z)=ex2​+y2 and EEE is defined by the given conditions.

Problem 11637

Leite die Funktion f(x)=x2⋅xf(x)=x^{2} \cdot xf(x)=x2⋅x ab.

Problem 11638

Leite die Funktion f(x)=x2⋅xf(x)=x^{2} \cdot xf(x)=x2⋅x ab. Ergebnis: f′(x)=3x2f^{\prime}(x)=3 x^{2}f′(x)=3x2.

Problem 11639

Problem 11640

Find dxdt\frac{d x}{d t}dtdx​ when x+y=6\sqrt{x}+y=6x​+y=6 and dydt=2\frac{d y}{d t}=2dtdy​=2, given x=4x=4x=4.

Problem 11641

A kite is at 40 ft height and moves horizontally at 3 ft/sec. Find the rate of string release when its length is 50 ft.

Problem 11642

Find the half-life of a substance that decays to 111\frac{1}{11}111​ of its original amount in 18 hours. Answer to 3 decimal places.

Find the derivative of $y$ with respect to $v$ where $y=\ln (\cosh (6 v))-\frac{1}{2} \tanh ^{2}(6 v)$ .

Solve the equation $5 \sqrt{x y} \frac{d y}{d x}=4$ for $y$ where $x, y>0$ . Find $y=\square$ .

Find the derivative of $y$ where $y=3 \sinh \left(\frac{x}{2}\right)$ .

Differentiate the function: $f(x)=(3 x^{2}-5 x) e^{x}$ .

Find the inverse of $P(t)=\frac{1000}{1+9 e^{-t}}$ , then determine the time for $P$ to reach 800 and analyze $t \rightarrow \infty$ .

Find the derivative of $f(x)=x \cdot \sqrt{x+1}$ .

En kloss på $155 \mathrm{~g}$ henger i en fjær med stivhet $320 \mathrm{~N/m}$ . Hva er farten ved likevektsstillingen og $1.1 \mathrm{~cm}$ fra den? Svar i $\mathrm{m/s}$ .

Find the derivative $y^{\prime}(t)$ of $y(t) = 30 \cdot 2^{-t / 30}$ to analyze decay rate after $t$ years. Explain your steps.

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

Find the first derivative of $f(x)=x^{5/3} \ln(x)$ and determine where it has horizontal tangents. $x=$

Find the derivative of $G(x)=\frac{x^{2}-2}{2 x+1}$ .

Find the $x$ -coordinates of inflection points for the function $g(x)=\frac{1+x}{1+x^{2}}$ with $g^{\prime \prime}(x)=\frac{2(x-1)(x^{2}+4x+1)}{(1+x^{2})^{3}}$ .

Find the derivative of $H(u)=(u-\sqrt{u})(u+\sqrt{u})$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{5 x^{3 / 2}}{4 \sqrt{x}+1}$ .

Given the function $f(x)=x^{3}+9 x^{2}+4$ , find if $f^{\prime \prime}(x)$ is + or - in the intervals $(-\infty,-3)$ , -3, and $(-3, \infty)$ .

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

Find the derivative of $J(v)=(v^{3}-2v)(v^{-4}+v^{-2})$ .

Given $f(x) = x^3 + 9x^2 + 4$ , verify:
1. Is $x=-3$ a point of inflection?
2. Find critical value at $x=-6$ using the second derivative test.
3. Find critical value at $x=0$ using the second derivative test.

Find the differential $d y$ for each: 1) $y=-x^{3}-2$ 2) $y=-\frac{3}{x}$ .

15. Verkehrszählung: $f(x)=\frac{1}{10^{10}} x^{5}-\frac{1}{130000} x^{3}+\frac{1}{6} x+6$ .
a) Höchste Verkehrsdichte? b) Fahrzeuge zwischen 6 und 9 Uhr? c) Durchschnittlicher Fahrzeugstrom zwischen 6 und 8 Uhr?

Find the critical points of the function $f(x)=\frac{2 x^{2}}{x-9}$ . What are the $x$ -values?

Evaluate the triple integral $\iiint_{B}(2 x+3 y^{2}+4 z^{3}) d V$ over the box $B=\{(x, y, z) \mid 0 \leq x \leq 4,0 \leq y \leq 2,0 \leq z \leq 1\}$ .

Find $c$ such that $\frac{f(9)-f(4)}{9-4}=f'(c)$ for $f(x)=\sqrt{x-4}$ on $[4,9]$ . $c=\square$

Find the derivative of $y=\ln(\tan(\frac{x}{2})) - \frac{1}{\sin x}$ .

Evaluate the triple integral $\iiint_{E} x^{2} dV$ for $E=\{(x, y, z) \mid 1-y^{2} \leq x \leq y^{2}-1, -1 \leq y \leq 1, 1 \leq z \leq 8\}$ .

Bestimme die Flächeninhaltsfunktion von $f$ mit unterer Grenze 0 für: a) $f(x)=4$ , b) $f(x)=x$ , c) $f(x)=3x+1$ , d) $f(x)=3x^2$ , e) $f(x)=\frac{1}{2}x^2$ , f) $f(x)=x^3+2x$ .

Solve $(\sec x) \frac{d y}{d x}=\mathrm{e}^{y} \cdot \mathrm{e}^{\sin x}$ using separation of variables.

A frog jumps at 30 degrees with a speed of $4.5 \mathrm{~m/s}$ . How far does it jump? $(1.8 \mathrm{~m})$

Evaluate the integral $\iiint_{E} z \, dV$ for the solid $E$ bounded by $y^{2}+z^{2}=4$ , $z=0$ , $x=0$ , and $x=5$ .

Find the max and min values of $g(x)=4 \csc x$ on $\left[\frac{\pi}{4}, \frac{3\pi}{4}\right]$ and graph it.

Evaluate the triple integral $\iiint_{E} f(x, y, z) d V$ where $f(x, y, z)=e^{\sqrt{x^{2}}+y^{2}}$ and $E$ is defined by the given conditions.

Leite die Funktion $f(x)=x^{2} \cdot x$ ab.

Leite die Funktion $f(x)=x^{2} \cdot x$ ab. Ergebnis: $f^{\prime}(x)=3 x^{2}$ .

Find $\frac{d x}{d t}$ when $\sqrt{x}+y=6$ and $\frac{d y}{d t}=2$ , given $x=4$ .

Find the half-life of a substance that decays to $\frac{1}{11}$ of its original amount in 18 hours. Answer to 3 decimal places.

Find $f^{\prime}(0)$ if $f(x)=\sin ^{2}(3-x)$ .

Find formulas for $d y$ and $\Delta y$ for the functions: 3) $y=-x^{3}-2$ and 4) $y=\frac{2}{x}$ .

An archaeologist finds a wood fragment with $\frac{1}{6}$ of its original Carbon-14. How old is it? Use decay constant 0.00012.

Find the tangent line equation for $y=2 \arccos \sqrt{x}$ at $x=\frac{1}{2}$ .

Find $P^{\prime}(t)$ when $P(t)$ satisfies $y^{\prime}=12y$ , with $P(0)=500$ and $P(t)=900$ .

Find the tangent line equation for $y=2 \arccos \sqrt{x}$ at $x=\frac{1}{2}$ , where slope $y'=-\frac{1}{\sqrt{x(1-x)}}$ .

Calculate the work done when a gas with an initial volume of $5.29 \mathrm{~L}$ expands to double its volume at $0.987 \mathrm{~atm}$ .

Find $\lim_{x \rightarrow 1} \frac{f(x)}{g(x)+2}$ where $g(x)=\frac{1}{10}(4x^3 + 3x^2 - 10x - 17)$ and $f(1)=0$ .

Approximate $\sqrt{65}-\sqrt{63}$ using the tangent line of $f(x)=\sqrt{x}$ at $x=64$ . (A) 0 (B) 1

Find a linear approximation for: 7) $\sin 122^{\circ}$ 8) $6.99^{4}$

Find the total compression $\Delta L$ (in m) of an 8 m sandstone pillar with density 2250 kg/m³ and Young's Modulus 12.5 GPa.

Given the function $f(x)=\frac{x^{2}}{x^{2}+3}$ , find where $f$ is increasing, decreasing, local minimum, inflection points, and concavity.

Use differentials to estimate errors: 9) Sphere radius is $7 \mathrm{~cm}$ with error $\pm \frac{1}{10} \mathrm{~cm}$ . Find volume error. 10) Square sides are 4 in with error $\pm \frac{1}{5}$ in. Find area error.

Find the inflection point coordinates $(x, y)$ for the function $f(x)=\frac{e^{x}}{1+e^{x}}$ .

Find the coordinates of the inflection point for the function $f(x)=\frac{e^{x}}{1+e^{x}}$ . Determine $x$ and $y$ .

Find the formula for the $z$ coordinate of the centroid of the region defined by $x^{2}+y^{2}+z^{2} \leq 4$ and $z \geq 0$ .

Find the average speed $s(x)=\frac{3600}{60+x}$ for traveling one mile in 66 seconds. Approximate speed at $x=6$ .

A person travels $D$ miles at speed $(60+x) \mathrm{mi/hr}$ .
a. Use $L(x)=D\left(1-\frac{x}{60}\right)$ to approximate time for 77 miles at 58 mi/hr. b. Find the exact time.
a. Approximate time is 80 min. b. Exact time is $\square$ min.

Find the time $t$ when the oxygen level $f(t)=\frac{t^{2}+10 t+100}{t^{2}+36 t+324}$ is lowest.

A gardener has $850 \mathrm{~m}$ of fence to create a rectangular garden divided into 8 sub-gardens. What's the max area?

Find the monthly cost $C(90)$ , marginal cost at $x=90$ , estimate cost for 92 chairs, and actual cost increase for 92 chairs.

A coffee cup cools from $95^{\circ} \mathrm{C}$ to $60^{\circ} \mathrm{C}$ in 11 minutes. How long to cool to $30^{\circ} \mathrm{C}$ ? It takes approximately $\square$ minutes.

Evaluate the limit using the derivative definition: $\lim _{x \rightarrow e^{-1}} \frac{\ln x+1}{x-e^{-1}}=\square$

Find the second derivative of the cubic function $f(x)=13 x^{3}+19 x^{2}+10 x-9$ . What is $f^{\prime \prime}(x)$ ?

Find the integral that calculates the probability $\mathbb{P}(Y \geq X)$ for $0 \leq X, Y \leq 2$ .

Find the second derivative of $f(x)=11 x^{3}+17 x^{2}-13 x+10$ .

Find the inflection point of $f(x)=\frac{e^{x}}{1+e^{x}}$ . Determine $x=$ Number and $y=$ Number.

Evaluate the limit using the derivative definition: $\lim _{h \rightarrow 0} \frac{(3+h)^{4+h}-81}{h}$ .

Find the second derivative of $f(x)=x^{5} \ln x$ and identify its inflection point.

Find the derivative of $y$ with respect to $x$ , where $y=(x^{4}+1) \operatorname{sech}(2 \ln x)$ .

Find the derivative of $y$ with respect to $v$ for $y=\ln (\cosh (6 v))-\frac{1}{2} \tanh ^{2}(6 v)$ .

Find the derivative of $y$ with respect to $x$ : $y=(x^{4}+1) \operatorname{sech}(2 \ln x)$ . Simplify first.