Calculus

Problem 19601

Gegeben ist die Funktion $f(x)=0,1 x^{2}$ . Berechne die Ableitung an $x_{0}=3$ , die allgemeine Ableitung und die Tangentengleichung an $x_{0}=5$ .

See Solution

Problem 19602

A ball is thrown up at $30 \mathrm{ft/s}$ from a $145 \mathrm{ft}$ cliff. Find its speed after 3 seconds. Gravity is $-32 \mathrm{ft/s}^2$ .

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Problem 19603

Which one indicates an upward direction: $y \rightarrow \infty$ or $y \rightarrow -\infty$ ?

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Problem 19604

Find the tangent line equation to $y(x)=9^{x-8}$ at the point $(8,1)$ .

See Solution

Problem 19605

Estimate the area under $f(x)=\frac{2}{x+1}$ from $x=1$ to $x=7$ using 2 rectangles (upper sum). Round to the nearest tenth.

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Problem 19606

Analyze the graph of $f^{\prime}(x)$ on $[0,6]$ to answer:
a) When is $f(x)$ increasing or decreasing? b) Find local extrema $x$ -coordinates. c) When is $f$ concave up or down? d) Identify points of inflection $x$ -coordinates.

See Solution

Problem 19607

Given a function $f$ on $[-3,4]$ with $f(0)=3$ and $f'$ as a line segment and semicircle:
a) When is $f(x)$ increasing or decreasing? Justify.
b) Find $x$ -coordinates of relative maxima. Justify.
c) When is $f$ concave up or down? Identify points of inflection. Justify.
d) Find the tangent line equation at $(0,3)$ .

See Solution

Problem 19608

Determine if the series $\sum_{n=1}^{\infty} \frac{(n !)^{2} m^{n}}{e^{n^{2}}}$ converges.

See Solution

Problem 19609

Examine the convergence of these series:
1) $\sum_{n=1}^{\infty} \frac{(n !)^{2} m^{n}}{e^{n^{2}}}$
2) $\sum_{n=1}^{\infty}(-1)^{n} \frac{\sqrt{n}}{n+5 m}$
3) $\sum_{n=1}^{\infty}\left(\frac{n-m}{n}\right)^{n}$

See Solution

Problem 19610

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

See Solution

Problem 19611

Find the expression for the slope of the tangent line to $f(x)=-2 x^{3} e^{x}$ . Choices include: -6 $x^{2}$ , -6 $x^{2}$ e $x$ , 6 $x^{2}$ e $x$ -6 $x^{3}$ e $x$ , -6 $x^{2}$ e $x$ -2 $x^{3}$ e $x$ .

See Solution

Problem 19612

Find the expression for the slope of the tangent line to the graph of $f(x)=-2 x^{3} e^{x}$ .

See Solution

Problem 19613

Investigate the convergence of the series: $\sum_{n=1}^{\infty}\left(\frac{n-m}{n}\right)^{n}$ .

See Solution

Problem 19614

Find the volume of the solid formed by rotating the region $Q$ between $u(y)=y^{\frac{1}{2}}$ and $v(y)=\frac{1}{y}$ from $y=1$ to $y=4$ around the $y$ -axis.

See Solution

Problem 19615

Find the area between $y=9-x^{2}$ (from $0$ to $5$ ) and $y=\frac{1}{x}$ (from $2$ to $6$ ), counting below $x$ -axis as negative.

See Solution

Problem 19616

Evaluate the definite integral $\int_{0}^{6}\left(x^{2}-2\right) d x$ . What is the result?

See Solution

Problem 19617

Évaluez l'intégrale suivante: $\int x \sqrt{1+4 x} \, dx$

See Solution

Problem 19618

Find the derivative $\mathrm{dy} / \mathrm{dx}$ using implicit differentiation for the equation $2xy - y^2 = 1$ .

See Solution

Problem 19619

A particle moves with $z=k r$ , $r=r_{0} e^{C \theta}$ . Find $r(t)$ at $t=0$ , compute velocity/acceleration, and curvature.

See Solution

Problem 19620

Find $\frac{d y}{d x}$ for $y=\int_{-1}^{2 x} \sin ^{2}(t) \sqrt[3]{t} d t$ .

See Solution

Problem 19621

Find the velocity, acceleration magnitudes of point M, curvature in terms of $z$ , and polar equation of projection H.

See Solution

Problem 19622

Find the average velocity of an object moving at $30 \mathrm{~m/s}$ from $t_1=10s$ to $t_2=30s$ .

See Solution

Problem 19623

Evaluate the integral of $2x \cos(x^2) \, dx$ .

See Solution

Problem 19624

An apple drops from a building. What is its speed after 5 seconds? Use $g=9.8 \mathrm{~m} / \mathrm{s}^{2}$ . Options: a) 14.8 b) 49 c) 2.0 d) 9.8

See Solution

Problem 19625

Find $\frac{d y}{d x}$ for $y=\int_{2}^{\sin (x)} t^{2} e^{t} d t$ .

See Solution

Problem 19626

Evaluate the integral $\int \sec^2(\theta) \tan^2(\theta) d\theta$ .

See Solution

Problem 19627

Berechne die Steigung der Funktionen an den angegebenen Stellen: a) $f(x)=(6-2 x)^{3}$ bei $a=5$ b) $f(x)=3 \cdot(0,5 x^{2}-1)^{5}$ bei $a=-2$ c) $f(x)=\frac{5}{x^{2}+3 x}$ bei $a=-1$ d) $f(x)=(\sqrt{x}+2)^{3}$ bei $a=4$

See Solution

Problem 19628

Evaluate the integral: $\int \frac{1+2 x}{1+16 x^{2}} d x$

See Solution

Problem 19629

Evaluate the integral $\int_{0}^{3} \frac{e^{x}}{1+e^{2 x}} d x$ .

See Solution

Problem 19630

Evaluate the integral: $\int 4 x^{2 / 3} dx$

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Problem 19631

Find the derivative $\frac{d y}{d x}$ for the equation $x y^{2}=4$ at the point (4,1).

See Solution

Problem 19632

Find the cost function given that the marginal cost is $C^{\prime}(x)=8x-2$ and fixed cost is \$7.

See Solution

Problem 19633

Calculate the integral: $\int 9 z \sqrt{3 z^{2}-7} \, dz$

See Solution

Problem 19634

Find $\frac{dy}{dt}$ given $x^{3}=16y^{5}-8$ , $\frac{dx}{dt}=8$ , and $y=1$ .

See Solution

Problem 19635

Calculate the integral: $\int 4(2 x+5)^{3} \, dx$

See Solution

Problem 19636

Evaluate the integral: $\int 14 e^{-0.2 x} d x$

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Problem 19637

Evaluate the integral of the function: $\int(2 x^{5}-7 x^{3}+8) \, dx$

See Solution

Problem 19638

Calculate the integral: $\int \frac{19}{2+5 y} d y$

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Problem 19639

Find the limit $L = \lim _{x \rightarrow c} \frac{f(x)}{g(x)}$ if $\lim _{x \rightarrow c} f(x)=-6$ and $\lim _{x \rightarrow c} g(x)=\frac{1}{3}$ .

See Solution

Problem 19640

Find the derivative $d q / d p$ for the demand equation $9 p^{2}+q^{2}=1,400$ .

See Solution

Problem 19641

Find the limit $L$ as $t$ approaches 1 for the piecewise function:
$h(t) = \begin{cases} t^{3}+1 & \text{if } t<1 \\ \frac{1}{2}(t+1) & \text{if } t \geq 1 \end{cases}$
Does the limit exist? Explain.

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Problem 19642

Find the derivative of $F(t)=e^{6 t \sin (2 t)}$ . What is $F^{\prime}(t)$ ?

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Problem 19643

Evaluate the limit using L'Hospital's Rule: $\lim _{x \rightarrow 0^{+}}(1+3 x)^{1 /(2 x)}$ . State the indeterminate form.

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Problem 19644

A 6-ft man walks away from a 22-ft lamppost at 5 ft/sec. Find the rate of change of his shadow length when 60 ft away.

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Problem 19645

Evaluate the integral: $\int t^{5} e^{-t^{6}} d t$ (use $C$ for the constant of integration).

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Problem 19646

Check if $f(x)=x^{3}-8x-5$ meets the Mean Value Theorem on $[1,4]$ and find guaranteed point(s).

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Problem 19647

Find the volume change rate of a cylinder (radius 4 cm, height 10 cm) when height increases at 2 cm/min.

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Problem 19648

Find the derivative of $y=(x^2+4)^3$ .

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Problem 19649

Find the derivative of $f(x)=2 x^{2}-6 x-3$ . Which option is correct: a) $2 x^{2}-6$ , b) $4 x^{2}-6$ , c) $4 x-6$ , d) $2 x-6$ ?

See Solution

Problem 19650

Find the average rate of change for $f(x)=x^{3}-2 x^{2}+x+1$ over $[0,2]$ and $[2,4]$ .

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Problem 19651

Find the slope of the graph $y=\sqrt{2x^{2}+1}$ at the point $(2,3)$ . Options: a. $\frac{1}{4}$ b. $\frac{4}{3}$ c. 12 d. $\frac{3}{2}$ e. None.

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Problem 19652

Find the slope of the graph $y = \sqrt{2x^{2}+1}$ at the point $(2,3)$ . Choose the correct answer: a. $\frac{1}{4}$ b. $\frac{4}{3}$ c. 12 d. $\frac{3}{2}$ e. None of these.

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Problem 19653

Find the $x$ -values where $f(x)=(x-8)^{2/5}$ is differentiable. Use interval notation for your answer.

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Problem 19654

Find the second derivative of the function $f(x)=4 x^{4}-6 x^{2}+5$ .

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Problem 19655

Determine if the series $\sum_{n=2}^{\infty} \frac{1}{\sqrt{n}-1}$ converges or diverges using the Comparison Test.

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Problem 19656

Given $f(x)=(x-1)^{\frac{2}{3}}+2$ , find critical points, apply the First Derivative Test, and determine extremes on $[0,9]$ .

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Problem 19657

Find $h(9)$ and $h^{\prime}(9)$ given $f(9)=8$ , $f^{\prime}(9)=3$ , $g(9)=3$ , $g^{\prime}(9)=7$ , and $h(x)=2 f(x)+3 g(x)$ .

See Solution

Problem 19658

Find the marginal revenue from the function $R(x)=x^{2/3}+2x^{1/2}+6x+2$ at $x=10$ . Choose the correct option.

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Problem 19659

A ball is thrown from a 700-ft building with initial velocity -30 ft/s.
(a) Find $s(t)$ and $v(t)$ . (b) Average velocity on $[1,3]$ . (c) Instantaneous velocities at $t=1$ and $t=3$ . (d) Time to reach ground level (to 3 decimals). (e) Impact velocity (to 1 decimal).

See Solution

Problem 19660

Find the derivative $\frac{d V}{d r}$ of the volume formula $V=\frac{4}{3} \pi r^{3}$ at $r=2$ .

See Solution

Problem 19661

Find the average rate of change for $f(x)=-2x^{2}+5x-2$ on the interval $[-1,1]$ .

See Solution

Problem 19662

If \$5000 is invested at 4% annual interest compounded continuously, find the amount in the account after 8 years.

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Problem 19663

Find the tangent line equation for $y=\frac{-3}{5 x+2}$ at $x=3$ .

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Problem 19664

Find the tangent line equation for $f(x)=\frac{x+6}{x-1}$ at the point $\left(\frac{1}{2},-13\right)$ . $y=\square$

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Problem 19665

Differentiate $y=\left(\sec ^{2}\left(x^{2}\right)\right)^{7}$ .

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Problem 19666

Differentiate $y=\csc (-4 x)$ .

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Problem 19667

Find the derivative of $y=3^{2x}$ .

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Problem 19668

Determine if the series $\sum_{n=1}^{\infty} \frac{n^{2}+n+1}{n^{4}+n^{2}}$ converges or diverges using the Limit Comparison Test.

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Problem 19669

Analyze the graph of $f(x)$ : True/False for asymptotes, intercepts, symmetry, and behavior of $f'$ , $f''$ . Create a sign chart.

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Problem 19670

Find the limit as $x$ approaches 0 from the left for the expression $\frac{1}{x}$ .

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Problem 19671

Calculate the population after 10 years using $p(t)=3430 e^{0.007 t}$ . Options: A) 11,036 B) 3679 C) 7357 D) 3945

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Problem 19672

Check if the Mean Value Theorem applies to $f(x)=x^{2/3}$ on $[1,27]$ . Options: Yes, No (continuity), No (differentiability), NA.

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Problem 19673

Estimate $f(801)$ , $f(800.5)$ , and $f(799.75)$ given $f(800)=6000$ and $f'(800)=10$ .

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Problem 19674

Determine if the sequence $a_{n}=\ln(n+5)-\ln(n)$ converges or diverges. Find the limit if it converges.

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Problem 19675

Determine if the series $\sum_{n=1}^{\infty} \frac{24}{n(n+3)}$ converges or diverges. If convergent, find the sum.

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Problem 19676

Determine if the series $\sum_{n=1}^{\infty} \frac{3^{n}+n}{n+5^{n}}$ converges or diverges using the Comparison Test.

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Problem 19677

Cut a 60-inch wire into two pieces for a circle and a square. Find $x$ to maximize their combined area.
a) Find the radius $r$ of the circle in terms of $x$ .
b) Write the objective function and its domain.
c) Differentiate the function, find critical points, and provide exact and approximate values (rounded to two decimals).

See Solution

Problem 19678

Cut a 60-inch wire into two pieces to form a circle and a square. Maximize the total area.
a) Find the radius $r$ of the circle in terms of $x$ .
b) Write the objective function to maximize and its domain.
c) Differentiate the function and find critical points (exact and approximate).
d) Evaluate the function at endpoints and critical points to find min/max values. Round to two decimals with units.

See Solution

Problem 19679

Is the series $\sum_{n=3}^{\infty} \frac{5 n-6}{n^{2}-2 n}$ convergent or divergent?

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Problem 19680

Find the values of $p$ for which the series $\sum_{n=1}^{\infty} n^{4}(1+n^{5})^{p}$ converges (as an inequality).

See Solution

Problem 19681

Determine the values of $p$ for which the series $\sum_{n=1}^{\infty} n^{3}(1+n^{4})^{p}$ converges.

See Solution

Problem 19682

Determine the values of $p$ for which the series $\sum_{n=1}^{\infty} n^{9}(1+n^{10})^{p}$ converges.

See Solution

Problem 19683

Find the derivative of $f(x)=(3 x+4)^{3}$ and evaluate $f^{\prime}(3)$ .

See Solution

Problem 19684

Find the derivative of the function $y=x^{7x+4}$ .

See Solution

Problem 19685

A cube's edges expand at 8 cm/s. Find volume change rate at edges 4 cm and 15 cm: $\mathrm{cm}^{3} / \mathrm{sec}$ .

See Solution

Problem 19686

Is the series $10 - 6 + 3.6 - 2.16 + \cdots$ convergent or divergent? If convergent, find the sum.

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Problem 19687

Find the position function for an object with acceleration $a(t)=5 \cos t+2 \sin t$ , initial velocity $v(0)=4$ , and position $s(0)=3$ .

See Solution

Problem 19688

Determine if the sequence $a_{n}=\ln (n+4)-\ln (n)$ converges or diverges. If it converges, find the limit.

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Problem 19689

Find the general solution for the differential equation $y^{(4)}+12 y^{\prime \prime}+36 y=0$ . What is $y(x)=\square$ ?

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Problem 19690

Find the derivative of $f(x)=9^{x-1}$ . What is $f'(x)$ ?

See Solution

Problem 19691

Calculate the indefinite integral and use $C$ for the constant of integration: $\int \frac{e^{4 x}-e^{2 x}+1}{e^{x}} d x$

See Solution

Problem 19692

Determine the values of $p$ that make the series $\sum_{n=1}^{\infty} n^{9}(1+n^{10})^{p}$ convergent.

See Solution

Problem 19693

Find $x$ in $[0,2]$ where the instantaneous rate of change of $f(x)=x^{3}-e^{x}$ equals the average rate of change.

See Solution

Problem 19694

Check if the series $\sum_{n=1}^{\infty} \frac{6}{n(n+3)}$ converges or diverges and find the sum if it converges.

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Problem 19695

Find intervals where the function $f$ is increasing given $f^{\prime}(x)=\frac{(x-2)^{3}}{x}$ . Choose: (A) $(-\infty, 0)$ and $(2, \infty)$ (B) $(-\infty, 0)$ only (C) $(-\infty, 0)$ and $(0,2)$ (D) $(0,2)$ and $(2, \infty)$

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Problem 19696

Find the tangent line equation for $h(x)=-x e^{9-x}$ at the point $(9,-9)$ . $y=$

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Problem 19697

Calculate the sum: $\sum_{n=1}^{\infty} \frac{21}{n(n+3)}$ .

See Solution

Problem 19698

Zeichne den Graphen von $f(x)=x+1$ und finde die Fläche zwischen Graph und $x$ -Achse für $I=[a;b]$ mit: a) $a=0, b=2$ , b) $a=0, b=4$ , c) $a=0, b=8$ , d) $a=0, b$ (variabel).

See Solution

Problem 19699

Is the series $\sum_{n=3}^{\infty} \frac{5 n-4}{n^{2}-2 n}$ convergent or divergent?

See Solution

Problem 19700

Find the function $f(x)$ and number $c$ such that $\lim _{\Delta x \rightarrow 0} \frac{\frac{4}{2-(-1+\Delta x)^{2}}-4}{\Delta x} = f^{\prime}(c)$ . Options: (A) $f(x)=2-x^{2}, c=-1$ ; (B) $f(x)=\frac{4}{2-x^{2}}, c=-1$ ; (C) $f(x)=\frac{4}{2-x^{2}}, c=4$ ; (D) $f(x)=2-x^{2}, c=4$ .

See Solution

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Calculus

Problem 19601

Gegeben ist die Funktion f(x)=0,1x2f(x)=0,1 x^{2}f(x)=0,1x2. Berechne die Ableitung an x0=3x_{0}=3x0​=3, die allgemeine Ableitung und die Tangentengleichung an x0=5x_{0}=5x0​=5.

Problem 19602

A ball is thrown up at 30ft/s30 \mathrm{ft/s}30ft/s from a 145ft145 \mathrm{ft}145ft cliff. Find its speed after 3 seconds. Gravity is −32ft/s2-32 \mathrm{ft/s}^2−32ft/s2.

Problem 19603

Which one indicates an upward direction: y→∞y \rightarrow \inftyy→∞ or y→−∞y \rightarrow -\inftyy→−∞?

Problem 19604

Find the tangent line equation to y(x)=9x−8y(x)=9^{x-8}y(x)=9x−8 at the point (8,1)(8,1)(8,1).

Problem 19605

Estimate the area under f(x)=2x+1f(x)=\frac{2}{x+1}f(x)=x+12​ from x=1x=1x=1 to x=7x=7x=7 using 2 rectangles (upper sum). Round to the nearest tenth.

Problem 19606

Analyze the graph of f′(x)f^{\prime}(x)f′(x) on [0,6][0,6][0,6] to answer: a) When is f(x)f(x)f(x) increasing or decreasing? b) Find local extrema xxx-coordinates. c) When is fff concave up or down? d) Identify points of inflection xxx-coordinates.

Problem 19607

Problem 19608

Determine if the series ∑n=1∞(n!)2mnen2\sum_{n=1}^{\infty} \frac{(n !)^{2} m^{n}}{e^{n^{2}}}∑n=1∞​en2(n!)2mn​ converges.

Problem 19609

Problem 19610

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=exxy=\frac{e^{x}}{x}y=xex​.

Problem 19611

Find the expression for the slope of the tangent line to f(x)=−2x3exf(x)=-2 x^{3} e^{x}f(x)=−2x3ex. Choices include: -6x2x^{2}x2, -6x2x^{2}x2exxx, 6x2x^{2}x2exxx-6x3x^{3}x3exxx, -6x2x^{2}x2exxx-2x3x^{3}x3exxx.

Problem 19612

Find the expression for the slope of the tangent line to the graph of f(x)=−2x3exf(x)=-2 x^{3} e^{x}f(x)=−2x3ex.

Problem 19613

Investigate the convergence of the series: ∑n=1∞(n−mn)n\sum_{n=1}^{\infty}\left(\frac{n-m}{n}\right)^{n}n=1∑∞​(nn−m​)n.

Problem 19614

Find the volume of the solid formed by rotating the region QQQ between u(y)=y12u(y)=y^{\frac{1}{2}}u(y)=y21​ and v(y)=1yv(y)=\frac{1}{y}v(y)=y1​ from y=1y=1y=1 to y=4y=4y=4 around the yyy-axis.

Problem 19615

Find the area between y=9−x2y=9-x^{2}y=9−x2 (from 000 to 555) and y=1xy=\frac{1}{x}y=x1​ (from 222 to 666), counting below xxx-axis as negative.

Problem 19616

Evaluate the definite integral ∫06(x2−2)dx\int_{0}^{6}\left(x^{2}-2\right) d x∫06​(x2−2)dx. What is the result?

Problem 19617

Évaluez l'intégrale suivante: ∫x1+4x dx\int x \sqrt{1+4 x} \, dx∫x1+4x​dx

Problem 19618

Find the derivative dy/dx\mathrm{dy} / \mathrm{dx}dy/dx using implicit differentiation for the equation 2xy−y2=12xy - y^2 = 12xy−y2=1.

Problem 19619

A particle moves with z=krz=k rz=kr, r=r0eCθr=r_{0} e^{C \theta}r=r0​eCθ. Find r(t)r(t)r(t) at t=0t=0t=0, compute velocity/acceleration, and curvature.

Problem 19620

Find dydx\frac{d y}{d x}dxdy​ for y=∫−12xsin⁡2(t)t3dty=\int_{-1}^{2 x} \sin ^{2}(t) \sqrt[3]{t} d ty=∫−12x​sin2(t)3t​dt.

Problem 19621

Find the velocity, acceleration magnitudes of point M, curvature in terms of zzz, and polar equation of projection H.

Problem 19622

Find the average velocity of an object moving at 30 m/s30 \mathrm{~m/s}30 m/s from t1=10st_1=10st1​=10s to t2=30st_2=30st2​=30s.

Problem 19623

Evaluate the integral of 2xcos⁡(x2) dx2x \cos(x^2) \, dx2xcos(x2)dx.

Problem 19624

An apple drops from a building. What is its speed after 5 seconds? Use g=9.8 m/s2g=9.8 \mathrm{~m} / \mathrm{s}^{2}g=9.8 m/s2. Options: a) 14.8 b) 49 c) 2.0 d) 9.8

Problem 19625

Find dydx\frac{d y}{d x}dxdy​ for y=∫2sin⁡(x)t2etdty=\int_{2}^{\sin (x)} t^{2} e^{t} d ty=∫2sin(x)​t2etdt.

Problem 19626

Evaluate the integral ∫sec⁡2(θ)tan⁡2(θ)dθ\int \sec^2(\theta) \tan^2(\theta) d\theta∫sec2(θ)tan2(θ)dθ.

Problem 19627

Problem 19628

Evaluate the integral: ∫1+2x1+16x2dx\int \frac{1+2 x}{1+16 x^{2}} d x∫1+16x21+2x​dx

Problem 19629

Evaluate the integral ∫03ex1+e2xdx\int_{0}^{3} \frac{e^{x}}{1+e^{2 x}} d x∫03​1+e2xex​dx.

Problem 19630

Evaluate the integral: ∫4x2/3dx\int 4 x^{2 / 3} dx∫4x2/3dx

Problem 19631

Find the derivative dydx\frac{d y}{d x}dxdy​ for the equation xy2=4x y^{2}=4xy2=4 at the point (4,1).

Problem 19632

Find the cost function given that the marginal cost is C′(x)=8x−2C^{\prime}(x)=8x-2C′(x)=8x−2 and fixed cost is \$7.

Problem 19633

Calculate the integral: ∫9z3z2−7 dz\int 9 z \sqrt{3 z^{2}-7} \, dz∫9z3z2−7​dz

Problem 19634

Find dydt\frac{dy}{dt}dtdy​ given x3=16y5−8x^{3}=16y^{5}-8x3=16y5−8, dxdt=8\frac{dx}{dt}=8dtdx​=8, and y=1y=1y=1.

Problem 19635

Calculate the integral: ∫4(2x+5)3 dx\int 4(2 x+5)^{3} \, dx∫4(2x+5)3dx

Problem 19636

Evaluate the integral: ∫14e−0.2xdx\int 14 e^{-0.2 x} d x∫14e−0.2xdx

Problem 19637

Evaluate the integral of the function: ∫(2x5−7x3+8) dx\int(2 x^{5}-7 x^{3}+8) \, dx∫(2x5−7x3+8)dx

Problem 19638

Calculate the integral: ∫192+5ydy\int \frac{19}{2+5 y} d y∫2+5y19​dy

Problem 19639

Find the limit L=lim⁡x→cf(x)g(x)L = \lim _{x \rightarrow c} \frac{f(x)}{g(x)}L=limx→c​g(x)f(x)​ if lim⁡x→cf(x)=−6\lim _{x \rightarrow c} f(x)=-6limx→c​f(x)=−6 and lim⁡x→cg(x)=13\lim _{x \rightarrow c} g(x)=\frac{1}{3}limx→c​g(x)=31​.

Problem 19640

Find the derivative dq/dpd q / d pdq/dp for the demand equation 9p2+q2=1,4009 p^{2}+q^{2}=1,4009p2+q2=1,400.

Problem 19641

Find the limit LLL as ttt approaches 1 for the piecewise function: h(t)={t3+1if t<112(t+1)if t≥1 h(t) = \begin{cases} t^{3}+1 & \text{if } t<1 \\ \frac{1}{2}(t+1) & \text{if } t \geq 1 \end{cases} h(t)={t3+121​(t+1)​if t<1if t≥1​ Does the limit exist? Explain.

Gegeben ist die Funktion $f(x)=0,1 x^{2}$ . Berechne die Ableitung an $x_{0}=3$ , die allgemeine Ableitung und die Tangentengleichung an $x_{0}=5$ .

A ball is thrown up at $30 \mathrm{ft/s}$ from a $145 \mathrm{ft}$ cliff. Find its speed after 3 seconds. Gravity is $-32 \mathrm{ft/s}^2$ .

Which one indicates an upward direction: $y \rightarrow \infty$ or $y \rightarrow -\infty$ ?

Find the tangent line equation to $y(x)=9^{x-8}$ at the point $(8,1)$ .

Estimate the area under $f(x)=\frac{2}{x+1}$ from $x=1$ to $x=7$ using 2 rectangles (upper sum). Round to the nearest tenth.

Analyze the graph of $f^{\prime}(x)$ on $[0,6]$ to answer:
a) When is $f(x)$ increasing or decreasing? b) Find local extrema $x$ -coordinates. c) When is $f$ concave up or down? d) Identify points of inflection $x$ -coordinates.

Determine if the series $\sum_{n=1}^{\infty} \frac{(n !)^{2} m^{n}}{e^{n^{2}}}$ converges.

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

Find the expression for the slope of the tangent line to $f(x)=-2 x^{3} e^{x}$ . Choices include: -6 $x^{2}$ , -6 $x^{2}$ e $x$ , 6 $x^{2}$ e $x$ -6 $x^{3}$ e $x$ , -6 $x^{2}$ e $x$ -2 $x^{3}$ e $x$ .

Find the expression for the slope of the tangent line to the graph of $f(x)=-2 x^{3} e^{x}$ .

Investigate the convergence of the series: $\sum_{n=1}^{\infty}\left(\frac{n-m}{n}\right)^{n}$ .

Find the volume of the solid formed by rotating the region $Q$ between $u(y)=y^{\frac{1}{2}}$ and $v(y)=\frac{1}{y}$ from $y=1$ to $y=4$ around the $y$ -axis.

Find the area between $y=9-x^{2}$ (from $0$ to $5$ ) and $y=\frac{1}{x}$ (from $2$ to $6$ ), counting below $x$ -axis as negative.

Evaluate the definite integral $\int_{0}^{6}\left(x^{2}-2\right) d x$ . What is the result?

Évaluez l'intégrale suivante: $\int x \sqrt{1+4 x} \, dx$

Find the derivative $\mathrm{dy} / \mathrm{dx}$ using implicit differentiation for the equation $2xy - y^2 = 1$ .

A particle moves with $z=k r$ , $r=r_{0} e^{C \theta}$ . Find $r(t)$ at $t=0$ , compute velocity/acceleration, and curvature.

Find $\frac{d y}{d x}$ for $y=\int_{-1}^{2 x} \sin ^{2}(t) \sqrt[3]{t} d t$ .

Find the velocity, acceleration magnitudes of point M, curvature in terms of $z$ , and polar equation of projection H.

Find the average velocity of an object moving at $30 \mathrm{~m/s}$ from $t_1=10s$ to $t_2=30s$ .

Evaluate the integral of $2x \cos(x^2) \, dx$ .

An apple drops from a building. What is its speed after 5 seconds? Use $g=9.8 \mathrm{~m} / \mathrm{s}^{2}$ . Options: a) 14.8 b) 49 c) 2.0 d) 9.8

Find $\frac{d y}{d x}$ for $y=\int_{2}^{\sin (x)} t^{2} e^{t} d t$ .

Evaluate the integral $\int \sec^2(\theta) \tan^2(\theta) d\theta$ .

Evaluate the integral: $\int \frac{1+2 x}{1+16 x^{2}} d x$

Evaluate the integral $\int_{0}^{3} \frac{e^{x}}{1+e^{2 x}} d x$ .

Evaluate the integral: $\int 4 x^{2 / 3} dx$

Find the derivative $\frac{d y}{d x}$ for the equation $x y^{2}=4$ at the point (4,1).

Find the cost function given that the marginal cost is $C^{\prime}(x)=8x-2$ and fixed cost is \$7.

Calculate the integral: $\int 9 z \sqrt{3 z^{2}-7} \, dz$

Find $\frac{dy}{dt}$ given $x^{3}=16y^{5}-8$ , $\frac{dx}{dt}=8$ , and $y=1$ .

Calculate the integral: $\int 4(2 x+5)^{3} \, dx$

Evaluate the integral: $\int 14 e^{-0.2 x} d x$

Evaluate the integral of the function: $\int(2 x^{5}-7 x^{3}+8) \, dx$

Calculate the integral: $\int \frac{19}{2+5 y} d y$

Find the limit $L = \lim _{x \rightarrow c} \frac{f(x)}{g(x)}$ if $\lim _{x \rightarrow c} f(x)=-6$ and $\lim _{x \rightarrow c} g(x)=\frac{1}{3}$ .

Find the derivative $d q / d p$ for the demand equation $9 p^{2}+q^{2}=1,400$ .

Find the limit $L$ as $t$ approaches 1 for the piecewise function:
$h(t) = \begin{cases} t^{3}+1 & \text{if } t<1 \\ \frac{1}{2}(t+1) & \text{if } t \geq 1 \end{cases}$
Does the limit exist? Explain.

Find the derivative of $F(t)=e^{6 t \sin (2 t)}$ . What is $F^{\prime}(t)$ ?

Evaluate the limit using L'Hospital's Rule: $\lim _{x \rightarrow 0^{+}}(1+3 x)^{1 /(2 x)}$ . State the indeterminate form.

Evaluate the integral: $\int t^{5} e^{-t^{6}} d t$ (use $C$ for the constant of integration).

Check if $f(x)=x^{3}-8x-5$ meets the Mean Value Theorem on $[1,4]$ and find guaranteed point(s).

Find the derivative of $y=(x^2+4)^3$ .

Find the derivative of $f(x)=2 x^{2}-6 x-3$ . Which option is correct: a) $2 x^{2}-6$ , b) $4 x^{2}-6$ , c) $4 x-6$ , d) $2 x-6$ ?

Find the average rate of change for $f(x)=x^{3}-2 x^{2}+x+1$ over $[0,2]$ and $[2,4]$ .

Find the slope of the graph $y=\sqrt{2x^{2}+1}$ at the point $(2,3)$ . Options: a. $\frac{1}{4}$ b. $\frac{4}{3}$ c. 12 d. $\frac{3}{2}$ e. None.

Find the slope of the graph $y = \sqrt{2x^{2}+1}$ at the point $(2,3)$ . Choose the correct answer: a. $\frac{1}{4}$ b. $\frac{4}{3}$ c. 12 d. $\frac{3}{2}$ e. None of these.

Find the $x$ -values where $f(x)=(x-8)^{2/5}$ is differentiable. Use interval notation for your answer.

Find the second derivative of the function $f(x)=4 x^{4}-6 x^{2}+5$ .

Determine if the series $\sum_{n=2}^{\infty} \frac{1}{\sqrt{n}-1}$ converges or diverges using the Comparison Test.

Given $f(x)=(x-1)^{\frac{2}{3}}+2$ , find critical points, apply the First Derivative Test, and determine extremes on $[0,9]$ .

Find $h(9)$ and $h^{\prime}(9)$ given $f(9)=8$ , $f^{\prime}(9)=3$ , $g(9)=3$ , $g^{\prime}(9)=7$ , and $h(x)=2 f(x)+3 g(x)$ .

Find the marginal revenue from the function $R(x)=x^{2/3}+2x^{1/2}+6x+2$ at $x=10$ . Choose the correct option.

A ball is thrown from a 700-ft building with initial velocity -30 ft/s.
(a) Find $s(t)$ and $v(t)$ . (b) Average velocity on $[1,3]$ . (c) Instantaneous velocities at $t=1$ and $t=3$ . (d) Time to reach ground level (to 3 decimals). (e) Impact velocity (to 1 decimal).

Find the derivative $\frac{d V}{d r}$ of the volume formula $V=\frac{4}{3} \pi r^{3}$ at $r=2$ .

Find the average rate of change for $f(x)=-2x^{2}+5x-2$ on the interval $[-1,1]$ .

Find the tangent line equation for $y=\frac{-3}{5 x+2}$ at $x=3$ .

Find the tangent line equation for $f(x)=\frac{x+6}{x-1}$ at the point $\left(\frac{1}{2},-13\right)$ . $y=\square$

Differentiate $y=\left(\sec ^{2}\left(x^{2}\right)\right)^{7}$ .

Differentiate $y=\csc (-4 x)$ .

Find the derivative of $y=3^{2x}$ .

Determine if the series $\sum_{n=1}^{\infty} \frac{n^{2}+n+1}{n^{4}+n^{2}}$ converges or diverges using the Limit Comparison Test.

Analyze the graph of $f(x)$ : True/False for asymptotes, intercepts, symmetry, and behavior of $f'$ , $f''$ . Create a sign chart.

Find the limit as $x$ approaches 0 from the left for the expression $\frac{1}{x}$ .

Calculate the population after 10 years using $p(t)=3430 e^{0.007 t}$ . Options: A) 11,036 B) 3679 C) 7357 D) 3945

Check if the Mean Value Theorem applies to $f(x)=x^{2/3}$ on $[1,27]$ . Options: Yes, No (continuity), No (differentiability), NA.

Estimate $f(801)$ , $f(800.5)$ , and $f(799.75)$ given $f(800)=6000$ and $f'(800)=10$ .

Determine if the sequence $a_{n}=\ln(n+5)-\ln(n)$ converges or diverges. Find the limit if it converges.