Calculus

Problem 23901

Find the absolute max and min of the function $f(x)=e^{x} \sin x$ on the interval $[0, 2 \pi]$ .

See Solution

Problem 23902

Evaluate the integral $\int x \cos(4 x^{2}) \, dx$ with the substitution $u=4 x^{2}$ .

See Solution

Problem 23903

Evaluate the integral $\int x \cos(4 x^{2}) dx$ with the substitution $u=4 x^{2}$ . What is the result?

See Solution

Problem 23904

Find the work done in stretching a spring 70 cm with a force of 800 N. (Remember to consider the units.)

See Solution

Problem 23905

Find relative extrema of $g$ where $g^{\prime}(x)=(5-x) x^{-3}$ for $x>0$ and justify your answers.

See Solution

Problem 23906

Find the derivative of the function $f(t)=\frac{\ln (8 t)}{8 t-2}$ : $f^{\prime}(t)=$

See Solution

Problem 23907

Find the length and width of a rectangle on the $x$ -axis under the curve $y=36-x^{2}$ for maximum area.

See Solution

Problem 23908

Evaluate the integral using substitution: $\int_{0}^{1} \frac{d x}{\sqrt{9-x^{2}}}$ .

See Solution

Problem 23909

Find $f'(x)$ and $f'(-3)$ if $f(x)=\int_{0}^{x} t^{6} dt$ .

See Solution

Problem 23910

Determine if the mutual fund amount $m(t)=\sin \left(\frac{e}{3}\right)^{t}$ is increasing or decreasing at $t=4$ years. Justify.

See Solution

Problem 23911

Evaluate the integral using substitution: $\int_{\pi / 6}^{\pi / 2} \cot x \csc^{4} x \, dx$ .

See Solution

Problem 23912

Find $f^{\prime \prime}(x)$ if $f(x)=\int_{0}^{x}(t^{3}+5 t^{2}+2) dt$ .

See Solution

Problem 23913

A 2 kg ball is dropped from 15 meters. What is its impact speed? Ignore air resistance. A. 12.1 m/s B. 17.1 m/s C. 147 m/s D. 294 m/s

See Solution

Problem 23914

Evaluate the integral: $\int e^{t} \cot \left(e^{t}-2\right) d t$

See Solution

Problem 23915

Find the horizontal and vertical asymptotes of $f(x) = \frac{2x^2 - 1}{x^2 + 3}$ using limits.

See Solution

Problem 23916

Find tangent line equations to $y=\frac{x-1}{x+1}$ parallel to $x-2y=3$ . List answers as comma-separated equations.

See Solution

Problem 23917

Maximize volume of a file from a 10-in by 18-in sheet. What height maximizes the volume?

See Solution

Problem 23918

Evaluate the integral: $\int x^{2} \sqrt{x^{3}+9} \, dx$

See Solution

Problem 23919

Show that the equation $-2 x^{5}+9 x^{4}-7 x^{3}-13 x=0$ has a solution in $[3,4]$ using the Intermediate Value Theorem. Then, use Newton's method to find the solution to six decimal places.

See Solution

Problem 23920

Find the horizontal and vertical asymptotes of $f(x) = \frac{2x + 1}{x^2 - x}$ using limits to describe the behavior.

See Solution

Problem 23921

A rock is thrown on Mars with height $H=13t-1.86t^{2}$ . Find velocities after 1s and at $t=a$ , time to hit ground, and impact velocity.

See Solution

Problem 23922

Evaluate the integral $\int 2 x\left(x^{2}+8\right)^{6} dx$ using the substitution $u=x^{2}+8$ .

See Solution

Problem 23923

Evaluate the integral using symmetry: $\int_{-x / 2}^{\pi / 2} \sin ^{5} 3 x \, dx$ .

See Solution

Problem 23924

Calculate the work done in moving an electron from $(-2,4)$ to $(1,4)$ with one fixed at $(2,4)$ under inverse square force.

See Solution

Problem 23925

Find the integral $\int -16 x \sin(8 x^{2}+1) \, dx$ using the substitution $u=8 x^{2}+1$ .

See Solution

Problem 23926

Find the integral $\int\left(7 x^{6}+2\right) \sqrt{x^{7}+2 x} \, dx$ using $u=x^{7}+2 x$ and verify by differentiation.

See Solution

Problem 23927

Find an antiderivative of $e^{8 x-3}$ and verify by differentiating your answer. $\int e^{8 x-3} \mathrm{dx}=\square$

See Solution

Problem 23928

Does the function $f(x)=\frac{x}{x+2}$ meet the Mean Value Theorem conditions on $[1,4]$ ? Find $c$ if it does. $c=$

See Solution

Problem 23929

Determine the horizontal asymptote of the function $f(x)=\frac{6 x}{2 x^{2}+1}$ .

See Solution

Problem 23930

Differentiate implicitly to find $\frac{d x}{d y}$ for the equation $x^{5} y^{2}-x^{3} y+3 x y^{3}=0$ .

See Solution

Problem 23931

Find the average value of $f(x)=-2x+6$ over the interval $[-6,3]$ .

See Solution

Problem 23932

Find the rate of change $\frac{d A}{d t}$ when $A=5$ and $\frac{dB}{dt}=3$ for $A^{3}+B^{3}=133$ .

See Solution

Problem 23933

Evaluate the integral $\int \sqrt{7 x+5} \, dx$ using the substitution $u=ax+b$ .

See Solution

Problem 23934

A 10 ft ladder leans against a wall. If it slides away at 0.7 ft/s, find the angle's rate of change (in rad/s) when 6 ft from the wall.

See Solution

Problem 23935

Evaluate the integral of $\csc^{2}(8\theta + 4) \, d\theta$ .

See Solution

Problem 23936

Evaluate the integral using a change of variables: $\int x^{15} e^{x^{16}} d x$ .

See Solution

Problem 23937

A circle's radius increases by $4 \mathrm{~mm}$ /s. Find the area change rate when the radius is $25 \mathrm{~mm}$ . Area change: $\square$ (round to nearest thousandth).

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Problem 23938

Evaluate the integral: $\int \frac{4 x^{4}}{\sqrt{7-8 x^{5}}} d x$ using a change of variables.

See Solution

Problem 23939

Explain why the function is discontinuous at $x = 4$ . Possible reasons: $f(4)$ defined, $\lim_{x \to 4} f(x)$ finite but not equal; $f(4)$ undefined; $\lim_{x \to 4} f(x)$ does not exist; $\lim_{x \to 4^{+}} f(x)$ and $\lim_{x \to 4^{-}} f(x)$ finite but not equal; none of the above.

See Solution

Problem 23940

Evaluate the integral $\int\left(x^{2}+x\right)^{9}(2 x+1) dx$ using a change of variables or a table.

See Solution

Problem 23941

Evaluate the integral: $\int_{0}^{6} 9 \sqrt{4 x+1} \, dx$ .

See Solution

Problem 23942

Evaluate the integral: $\int \frac{9 d x}{\sqrt{36-81 x^{2}}}$ .

See Solution

Problem 23943

Evaluate the integral: $\int \frac{d x}{\sqrt{7 x+8}}$ with the substitution $u=7 x+8$ .

See Solution

Problem 23944

Evaluate the integral using a change of variables: $\int \frac{1}{13 x-4} d x$ .

See Solution

Problem 23945

Find $\frac{d y}{d t}$ when $y=x^{3}+1$ , $\frac{d x}{d t}=4$ , and $x=5$ . Simplify $\frac{d y}{d t}=\square$ .

See Solution

Problem 23946

Evaluate the integral $\int_{0}^{2} \frac{2 x}{(x^{2}+4)^{3}} dx$ using a variable change or a table.

See Solution

Problem 23947

Evaluate the integral: $\int \frac{9 d x}{\sqrt{36-81 x^{2}}}$ .

See Solution

Problem 23948

Evaluate the integral $\int \frac{d x}{\sqrt{7 x+8}}$ using the substitution $u=7 x+8$ .

See Solution

Problem 23949

Calculate the integral: $\int \frac{d x}{\sqrt{7 x+8}}$

See Solution

Problem 23950

Evaluate the integral $\int_{0}^{8} \frac{p}{\sqrt{36+p^{2}}} d p$ using a change of variables or a table.

See Solution

Problem 23951

Calculate the area between the curve $f(x)=x \sin \left(x^{2}\right)$ and the $x$ -axis from $x=0$ to $x=\sqrt{\pi}$ .

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Problem 23952

Trouvez les extrêmes relatifs et absolus des fonctions suivantes sur les intervalles donnés :
1. $f(x)=x^{3}-9 x^{2}+24 x-2,[0,5]$
2. $f(x)=x \sqrt{1-x},[-1,1]$
3. $f(x)=\frac{3 x-4}{x^{2}+1},[-2,2]$
4. $f(x)=\sqrt{x^{2}+x+1},[-2,1]$
5. $f(x)=x+2 \cos x,[-\pi, \pi]$
6. $f(x)=x^{2} e^{-x},[-1,3]$

See Solution

Problem 23953

Evaluate the integral using substitution: $\int_{0}^{1} x^{7} \cdot e^{x^{8}} \, dx$ .

See Solution

Problem 23954

Find the rate of change of average cost when producing 177 belts, given $C(x)=720+37x-0.067x^{2}$ . Calculate $\overline{\mathrm{C}}^{\prime}(177)=\square$ .

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Problem 23955

Evaluate the integral $\int_{0}^{\pi / 3} \frac{3 \sin (3 t)}{6-\cos (3 t)} dt$ .

See Solution

Problem 23956

Calculate the integral: $\int_{1}^{e^{7}} \frac{(\ln x)^{2}}{x} d x$ .

See Solution

Problem 23957

Find the function $f$ given that $f'''(x) = \cos(x)$ , with conditions $f(0)=8$ , $f'(0)=3$ , $f''(0)=3$ .

See Solution

Problem 23958

Evaluate the integral: $\int\left(e^{4 x}+2 e^{-x}\right) d x$

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Problem 23959

Find the limits as $x$ approaches 2: (a) $\lim [f(x)+4 g(x)]$ (b) $\lim [g(x)]^{3}$ (c) $\lim \sqrt{f(x)}$ (d) $\lim \frac{4 f(x)}{g(x)}$ (e) $\lim \frac{g(x)}{h(x)}$ (f) $\lim \frac{g(x) h(x)}{f(x)}$ If DNE, state that.

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Problem 23960

Given the limits, find the following:
(a) $\lim _{x \rightarrow 2}[f(x)+4 g(x)]$ (b) $\lim _{x \rightarrow 2}[g(x)]^{3}$ (c) $\lim _{x \rightarrow 2} \sqrt{f(x)}$ (d) $\lim _{x \rightarrow 2} \frac{4 f(x)}{g(x)}$ (e) $\lim _{x \rightarrow 2} \frac{g(x)}{h(x)}$

See Solution

Problem 23961

Find the max and min of $f(x)=4+81 x-3 x^{3}$ on $[0,4]$ . What are the values?

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Problem 23962

Calculate the value of $e^{-\infty}$ .

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Problem 23963

Given the function $h(x) = \frac{x-1}{x^2-x-12}$ , find intercepts, asymptotes, and behavior at vertical asymptotes.

See Solution

Problem 23964

Find the limit: $\lim _{x \rightarrow \infty} \ln (x+5)$ .

See Solution

Problem 23965

Find the intercepts and asymptotes of $h(x) = \frac{x-1}{x^{2}-x-12}$ . Use limits for vertical asymptote behavior.

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Problem 23966

A $110$ -kg roller coaster starts at rest at $15.5 \, \mathrm{m}$ and goes over hill C at $6.8 \, \mathrm{m}$ . Find its speed.

See Solution

Problem 23967

Use the Mean Value Theorem for $y = \sin(3x)$ on the interval $[0, \pi]$ .

See Solution

Problem 23968

Find the limit: $\lim _{x \rightarrow-2} \sin (x \pi)$ .

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Problem 23969

What is the average amount in your account after investing \$20000 at 12\% interest compounded continuously for 2 years?

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Problem 23970

Find $f^{(2022)}(x)$ given that $f^{(2020)}(x)=\cot^{-1}(x)+x^{2}$ .

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Problem 23971

Find the function for the power series: $\sum_{k=0}^{\infty}(-1)^{k} \frac{x^{k}}{13^{k}}$ . What is $f(x)=\square$ ?

See Solution

Problem 23972

Find the limit: $\lim _{x \rightarrow 8}(3+\sqrt[3]{x})(2-5 x^{2}+x^{3})$ . If none, enter DNE.

See Solution

Problem 23973

Find $\lim _{x \rightarrow 8} \frac{f(x)}{g(x)-h(x)}$ given $\lim _{x \rightarrow 8} f(x)=18$ , $\lim _{x \rightarrow 8} g(x)=6$ , and $\lim _{x \rightarrow 8} h(x)=4$ .

See Solution

Problem 23974

Find the limit: $\lim _{x \rightarrow 1} \frac{x^{2}-4 x+3}{x-1}$ . If it doesn't exist, write DNE.

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Problem 23975

Evaluate the limit: $\lim _{x \rightarrow 3} \frac{x^{2}-3 x}{x^{2}-2 x-3}$ . If it doesn't exist, write DNE.

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Problem 23976

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{9+h}-3}{h}$ . If it doesn't exist, write DNE.

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Problem 23977

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{(-4+h)^{2}-16}{h}$ . If it doesn't exist, enter DNE.

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Problem 23978

Find $\lim _{x \rightarrow 8} \frac{f(x)}{g(x)-h(x)}$ given $\lim _{x \rightarrow 8} f(x)=18$ , $\lim _{x \rightarrow 8} g(x)=6$ , $\lim _{x \rightarrow 8} h(x)=4$ . It equals 9. Justify using limit laws. Select all that apply: A, B, C, D, E, F, G.

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Problem 23979

Determine if the sequence $\{(-0.015)^{n}\}$ converges or diverges, and if so, describe its behavior. What is the limit?

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Problem 23980

Find the limit of the sequence $\left\{\frac{n^{4}}{n^{5}+1}\right\}$ or state if it diverges.

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Problem 23981

For which values of $r$ does the sequence $\{r^n\}$ converge or diverge? Select the correct option.

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Problem 23982

Find the limit as $x$ approaches -7: $\lim _{x \rightarrow-7} \frac{8 x+56}{|x+7|}$ .

See Solution

Problem 23983

Evaluate the integral: $\int 13 \tan^{5} x \sec^{4} x \, dx = \square$

See Solution

Problem 23984

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin 6 x}{x}$ . If no answer, enter DNE.

See Solution

Problem 23985

Evaluate the integral: $\int 5 \tan^{5} x \sec^{4} x \, dx = \square$

See Solution

Problem 23986

Calculate the integral: $\int \frac{2 x^{3}-4 x-8}{(x^{2}-x)(x^{2}+4)} \, dx$

See Solution

Problem 23987

Find the infinite limit: $\lim _{x \rightarrow-4^{-}} \frac{x+5}{x+4}$ . Is it $\infty$ or $-\infty$ ?

See Solution

Problem 23988

Find the limit of the sequence $-\frac{\sin 3 n}{4 n}$ or state if it diverges.
A. The limit is $\square$ . B. The sequence diverges.

See Solution

Problem 23989

Find the half-life of a substance that decays as $A=A_{0} e^{-0.021 t}$ . Round to two decimal places.

See Solution

Problem 23990

Estimate Walmart's total revenue from January 2004 to January 2014 using $R(t)=172 e^{0.08 t}$ in billion dollars.

See Solution

Problem 23991

How long for \$ 1625 to double at 6.5\% continuous compounding? Round to two decimal places.

See Solution

Problem 23992

Find the intercepts and asymptotes of the function $f(x)=\frac{x^{2}+x-2}{x^{2}-9}$ and describe behavior at vertical asymptotes using limits.

See Solution

Problem 23993

Find the infinite limit: $\lim _{x \rightarrow 8^{-}} \frac{e^{x}}{(x-8)^{3}}$ .

See Solution

Problem 23994

Find $f^{\prime}(x)$ for the following functions: (i) $f(x)=\left(\frac{2 x-1}{3+\sqrt{x}}\right)^{5}+\tan (\cos x+\sin x)$ (ii) $f(x)=\frac{5}{(\cot x+5)}$ (iii) $f(x)=\sin^{-1}(2 x+3)+\tan^{-1}(\sqrt{x})+\ln (x^{2}+1)+2^{x}$ (iv) $f(x)=\sec (4 \ln x)-\csc^{-1}(1-2 x)$ (v) $f(x)=\left(4 x^{5}-6 x\right)\left(3-\frac{1}{x}\right)^{3}(\cos x-\cot x)$ (vi) $f(x)=\left(x^{2}+2\right)^{(x+1)}$

See Solution

Problem 23995

Evaluate the limit using continuity: $\lim _{x \rightarrow 9} \frac{16+\sqrt{x}}{\sqrt{16+x}}$

See Solution

Problem 23996

Find the left and right limits of the function $f(x)$ at $a=0$ , where $f(x)=e^{x}$ for $x<0$ and $f(x)=x^{2}$ for $x \geq 0$ .

See Solution

Problem 23997

Determine the intervals where the function $f^{\prime}(x)=-x^{3}-12 x^{2}-36 x$ is decreasing.

See Solution

Problem 23998

Find the discontinuities of the function $y=\frac{4}{1+e^{7/x}}$ for $x=\square$ .

See Solution

Problem 23999

Find the approximation of $f(81.1)$ using the tangent line to $f(x)=-x^{\frac{3}{4}}$ at $x=81$ .

See Solution

Problem 24000

Find the area of the shaded region under the curve $y=4(\sin x) \sqrt{1+\cos x}$ over the interval $[0, 2\pi]$ .

See Solution

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Calculus

Problem 23901

Find the absolute max and min of the function f(x)=exsin⁡xf(x)=e^{x} \sin xf(x)=exsinx on the interval [0,2π][0, 2 \pi][0,2π].

Problem 23902

Evaluate the integral ∫xcos⁡(4x2) dx\int x \cos(4 x^{2}) \, dx∫xcos(4x2)dx with the substitution u=4x2u=4 x^{2}u=4x2.

Problem 23903

Evaluate the integral ∫xcos⁡(4x2)dx\int x \cos(4 x^{2}) dx∫xcos(4x2)dx with the substitution u=4x2u=4 x^{2}u=4x2. What is the result?

Problem 23904

Find the work done in stretching a spring 70 cm with a force of 800 N. (Remember to consider the units.)

Problem 23905

Find relative extrema of ggg where g′(x)=(5−x)x−3g^{\prime}(x)=(5-x) x^{-3}g′(x)=(5−x)x−3 for x>0x>0x>0 and justify your answers.

Problem 23906

Find the derivative of the function f(t)=ln⁡(8t)8t−2f(t)=\frac{\ln (8 t)}{8 t-2}f(t)=8t−2ln(8t)​: f′(t)=f^{\prime}(t)=f′(t)=

Problem 23907

Find the length and width of a rectangle on the xxx-axis under the curve y=36−x2y=36-x^{2}y=36−x2 for maximum area.

Problem 23908

Evaluate the integral using substitution: ∫01dx9−x2\int_{0}^{1} \frac{d x}{\sqrt{9-x^{2}}}∫01​9−x2​dx​.

Problem 23909

Find f′(x)f'(x)f′(x) and f′(−3)f'(-3)f′(−3) if f(x)=∫0xt6dtf(x)=\int_{0}^{x} t^{6} dtf(x)=∫0x​t6dt.

Problem 23910

Determine if the mutual fund amount m(t)=sin⁡(e3)tm(t)=\sin \left(\frac{e}{3}\right)^{t}m(t)=sin(3e​)t is increasing or decreasing at t=4t=4t=4 years. Justify.

Problem 23911

Evaluate the integral using substitution: ∫π/6π/2cot⁡xcsc⁡4x dx\int_{\pi / 6}^{\pi / 2} \cot x \csc^{4} x \, dx∫π/6π/2​cotxcsc4xdx.

Problem 23912

Find f′′(x)f^{\prime \prime}(x)f′′(x) if f(x)=∫0x(t3+5t2+2)dtf(x)=\int_{0}^{x}(t^{3}+5 t^{2}+2) dtf(x)=∫0x​(t3+5t2+2)dt.

Problem 23913

A 2 kg ball is dropped from 15 meters. What is its impact speed? Ignore air resistance. A. 12.1 m/s B. 17.1 m/s C. 147 m/s D. 294 m/s

Problem 23914

Evaluate the integral: ∫etcot⁡(et−2)dt\int e^{t} \cot \left(e^{t}-2\right) d t∫etcot(et−2)dt

Problem 23915

Find the horizontal and vertical asymptotes of f(x)=2x2−1x2+3f(x) = \frac{2x^2 - 1}{x^2 + 3}f(x)=x2+32x2−1​ using limits.

Problem 23916

Find tangent line equations to y=x−1x+1y=\frac{x-1}{x+1}y=x+1x−1​ parallel to x−2y=3x-2y=3x−2y=3. List answers as comma-separated equations.

Problem 23917

Maximize volume of a file from a 10-in by 18-in sheet. What height maximizes the volume?

Problem 23918

Evaluate the integral: ∫x2x3+9 dx\int x^{2} \sqrt{x^{3}+9} \, dx∫x2x3+9​dx

Problem 23919

Show that the equation −2x5+9x4−7x3−13x=0-2 x^{5}+9 x^{4}-7 x^{3}-13 x=0−2x5+9x4−7x3−13x=0 has a solution in [3,4][3,4][3,4] using the Intermediate Value Theorem. Then, use Newton's method to find the solution to six decimal places.

Problem 23920

Find the horizontal and vertical asymptotes of f(x)=2x+1x2−xf(x) = \frac{2x + 1}{x^2 - x}f(x)=x2−x2x+1​ using limits to describe the behavior.

Problem 23921

A rock is thrown on Mars with height H=13t−1.86t2H=13t-1.86t^{2}H=13t−1.86t2. Find velocities after 1s and at t=at=at=a, time to hit ground, and impact velocity.

Problem 23922

Evaluate the integral ∫2x(x2+8)6dx\int 2 x\left(x^{2}+8\right)^{6} dx∫2x(x2+8)6dx using the substitution u=x2+8u=x^{2}+8u=x2+8.

Problem 23923

Evaluate the integral using symmetry: ∫−x/2π/2sin⁡53x dx\int_{-x / 2}^{\pi / 2} \sin ^{5} 3 x \, dx∫−x/2π/2​sin53xdx.

Problem 23924

Calculate the work done in moving an electron from (−2,4)(-2,4)(−2,4) to (1,4)(1,4)(1,4) with one fixed at (2,4)(2,4)(2,4) under inverse square force.

Problem 23925

Find the integral ∫−16xsin⁡(8x2+1) dx\int -16 x \sin(8 x^{2}+1) \, dx∫−16xsin(8x2+1)dx using the substitution u=8x2+1u=8 x^{2}+1u=8x2+1.

Problem 23926

Find the integral ∫(7x6+2)x7+2x dx\int\left(7 x^{6}+2\right) \sqrt{x^{7}+2 x} \, dx∫(7x6+2)x7+2x​dx using u=x7+2xu=x^{7}+2 xu=x7+2x and verify by differentiation.

Problem 23927

Find an antiderivative of e8x−3e^{8 x-3}e8x−3 and verify by differentiating your answer. ∫e8x−3dx=□\int e^{8 x-3} \mathrm{dx}=\square∫e8x−3dx=□

Problem 23928

Does the function f(x)=xx+2f(x)=\frac{x}{x+2}f(x)=x+2x​ meet the Mean Value Theorem conditions on [1,4][1,4][1,4]? Find ccc if it does. c=c=c=

Problem 23929

Determine the horizontal asymptote of the function f(x)=6x2x2+1f(x)=\frac{6 x}{2 x^{2}+1}f(x)=2x2+16x​.

Problem 23930

Differentiate implicitly to find dxdy\frac{d x}{d y}dydx​ for the equation x5y2−x3y+3xy3=0x^{5} y^{2}-x^{3} y+3 x y^{3}=0x5y2−x3y+3xy3=0.

Problem 23931

Find the average value of f(x)=−2x+6f(x)=-2x+6f(x)=−2x+6 over the interval [−6,3][-6,3][−6,3].

Problem 23932

Find the rate of change dAdt\frac{d A}{d t}dtdA​ when A=5A=5A=5 and dBdt=3\frac{dB}{dt}=3dtdB​=3 for A3+B3=133A^{3}+B^{3}=133A3+B3=133.

Problem 23933

Evaluate the integral ∫7x+5 dx\int \sqrt{7 x+5} \, dx∫7x+5​dx using the substitution u=ax+bu=ax+bu=ax+b.

Problem 23934

A 10 ft ladder leans against a wall. If it slides away at 0.7 ft/s, find the angle's rate of change (in rad/s) when 6 ft from the wall.

Problem 23935

Evaluate the integral of csc⁡2(8θ+4) dθ\csc^{2}(8\theta + 4) \, d\thetacsc2(8θ+4)dθ.

Problem 23936

Evaluate the integral using a change of variables: ∫x15ex16dx\int x^{15} e^{x^{16}} d x∫x15ex16dx.

Problem 23937

A circle's radius increases by 4 mm4 \mathrm{~mm}4 mm/s. Find the area change rate when the radius is 25 mm25 \mathrm{~mm}25 mm. Area change: □\square□ (round to nearest thousandth).

Problem 23938

Evaluate the integral: ∫4x47−8x5dx\int \frac{4 x^{4}}{\sqrt{7-8 x^{5}}} d x∫7−8x5​4x4​dx using a change of variables.

Problem 23939

Problem 23940

Evaluate the integral ∫(x2+x)9(2x+1)dx\int\left(x^{2}+x\right)^{9}(2 x+1) dx∫(x2+x)9(2x+1)dx using a change of variables or a table.

Find the absolute max and min of the function $f(x)=e^{x} \sin x$ on the interval $[0, 2 \pi]$ .

Evaluate the integral $\int x \cos(4 x^{2}) \, dx$ with the substitution $u=4 x^{2}$ .

Evaluate the integral $\int x \cos(4 x^{2}) dx$ with the substitution $u=4 x^{2}$ . What is the result?

Find relative extrema of $g$ where $g^{\prime}(x)=(5-x) x^{-3}$ for $x>0$ and justify your answers.

Find the derivative of the function $f(t)=\frac{\ln (8 t)}{8 t-2}$ : $f^{\prime}(t)=$

Find the length and width of a rectangle on the $x$ -axis under the curve $y=36-x^{2}$ for maximum area.

Evaluate the integral using substitution: $\int_{0}^{1} \frac{d x}{\sqrt{9-x^{2}}}$ .

Find $f'(x)$ and $f'(-3)$ if $f(x)=\int_{0}^{x} t^{6} dt$ .

Determine if the mutual fund amount $m(t)=\sin \left(\frac{e}{3}\right)^{t}$ is increasing or decreasing at $t=4$ years. Justify.

Evaluate the integral using substitution: $\int_{\pi / 6}^{\pi / 2} \cot x \csc^{4} x \, dx$ .

Find $f^{\prime \prime}(x)$ if $f(x)=\int_{0}^{x}(t^{3}+5 t^{2}+2) dt$ .

Evaluate the integral: $\int e^{t} \cot \left(e^{t}-2\right) d t$

Find the horizontal and vertical asymptotes of $f(x) = \frac{2x^2 - 1}{x^2 + 3}$ using limits.

Find tangent line equations to $y=\frac{x-1}{x+1}$ parallel to $x-2y=3$ . List answers as comma-separated equations.

Evaluate the integral: $\int x^{2} \sqrt{x^{3}+9} \, dx$

Show that the equation $-2 x^{5}+9 x^{4}-7 x^{3}-13 x=0$ has a solution in $[3,4]$ using the Intermediate Value Theorem. Then, use Newton's method to find the solution to six decimal places.

Find the horizontal and vertical asymptotes of $f(x) = \frac{2x + 1}{x^2 - x}$ using limits to describe the behavior.

A rock is thrown on Mars with height $H=13t-1.86t^{2}$ . Find velocities after 1s and at $t=a$ , time to hit ground, and impact velocity.

Evaluate the integral $\int 2 x\left(x^{2}+8\right)^{6} dx$ using the substitution $u=x^{2}+8$ .

Evaluate the integral using symmetry: $\int_{-x / 2}^{\pi / 2} \sin ^{5} 3 x \, dx$ .

Calculate the work done in moving an electron from $(-2,4)$ to $(1,4)$ with one fixed at $(2,4)$ under inverse square force.

Find the integral $\int -16 x \sin(8 x^{2}+1) \, dx$ using the substitution $u=8 x^{2}+1$ .

Find the integral $\int\left(7 x^{6}+2\right) \sqrt{x^{7}+2 x} \, dx$ using $u=x^{7}+2 x$ and verify by differentiation.

Find an antiderivative of $e^{8 x-3}$ and verify by differentiating your answer. $\int e^{8 x-3} \mathrm{dx}=\square$

Does the function $f(x)=\frac{x}{x+2}$ meet the Mean Value Theorem conditions on $[1,4]$ ? Find $c$ if it does. $c=$

Determine the horizontal asymptote of the function $f(x)=\frac{6 x}{2 x^{2}+1}$ .

Differentiate implicitly to find $\frac{d x}{d y}$ for the equation $x^{5} y^{2}-x^{3} y+3 x y^{3}=0$ .

Find the average value of $f(x)=-2x+6$ over the interval $[-6,3]$ .

Find the rate of change $\frac{d A}{d t}$ when $A=5$ and $\frac{dB}{dt}=3$ for $A^{3}+B^{3}=133$ .

Evaluate the integral $\int \sqrt{7 x+5} \, dx$ using the substitution $u=ax+b$ .

Evaluate the integral of $\csc^{2}(8\theta + 4) \, d\theta$ .

Evaluate the integral using a change of variables: $\int x^{15} e^{x^{16}} d x$ .

A circle's radius increases by $4 \mathrm{~mm}$ /s. Find the area change rate when the radius is $25 \mathrm{~mm}$ . Area change: $\square$ (round to nearest thousandth).

Evaluate the integral: $\int \frac{4 x^{4}}{\sqrt{7-8 x^{5}}} d x$ using a change of variables.

Evaluate the integral $\int\left(x^{2}+x\right)^{9}(2 x+1) dx$ using a change of variables or a table.

Evaluate the integral: $\int_{0}^{6} 9 \sqrt{4 x+1} \, dx$ .

Evaluate the integral: $\int \frac{9 d x}{\sqrt{36-81 x^{2}}}$ .

Evaluate the integral: $\int \frac{d x}{\sqrt{7 x+8}}$ with the substitution $u=7 x+8$ .

Evaluate the integral using a change of variables: $\int \frac{1}{13 x-4} d x$ .

Find $\frac{d y}{d t}$ when $y=x^{3}+1$ , $\frac{d x}{d t}=4$ , and $x=5$ . Simplify $\frac{d y}{d t}=\square$ .

Evaluate the integral $\int_{0}^{2} \frac{2 x}{(x^{2}+4)^{3}} dx$ using a variable change or a table.

Evaluate the integral: $\int \frac{9 d x}{\sqrt{36-81 x^{2}}}$ .

Evaluate the integral $\int \frac{d x}{\sqrt{7 x+8}}$ using the substitution $u=7 x+8$ .

Calculate the integral: $\int \frac{d x}{\sqrt{7 x+8}}$

Evaluate the integral $\int_{0}^{8} \frac{p}{\sqrt{36+p^{2}}} d p$ using a change of variables or a table.

Calculate the area between the curve $f(x)=x \sin \left(x^{2}\right)$ and the $x$ -axis from $x=0$ to $x=\sqrt{\pi}$ .

Evaluate the integral using substitution: $\int_{0}^{1} x^{7} \cdot e^{x^{8}} \, dx$ .

Find the rate of change of average cost when producing 177 belts, given $C(x)=720+37x-0.067x^{2}$ . Calculate $\overline{\mathrm{C}}^{\prime}(177)=\square$ .

Evaluate the integral $\int_{0}^{\pi / 3} \frac{3 \sin (3 t)}{6-\cos (3 t)} dt$ .

Calculate the integral: $\int_{1}^{e^{7}} \frac{(\ln x)^{2}}{x} d x$ .

Find the function $f$ given that $f'''(x) = \cos(x)$ , with conditions $f(0)=8$ , $f'(0)=3$ , $f''(0)=3$ .

Evaluate the integral: $\int\left(e^{4 x}+2 e^{-x}\right) d x$

Find the max and min of $f(x)=4+81 x-3 x^{3}$ on $[0,4]$ . What are the values?

Calculate the value of $e^{-\infty}$ .

Given the function $h(x) = \frac{x-1}{x^2-x-12}$ , find intercepts, asymptotes, and behavior at vertical asymptotes.

Find the limit: $\lim _{x \rightarrow \infty} \ln (x+5)$ .

Find the intercepts and asymptotes of $h(x) = \frac{x-1}{x^{2}-x-12}$ . Use limits for vertical asymptote behavior.

A $110$ -kg roller coaster starts at rest at $15.5 \, \mathrm{m}$ and goes over hill C at $6.8 \, \mathrm{m}$ . Find its speed.

Use the Mean Value Theorem for $y = \sin(3x)$ on the interval $[0, \pi]$ .

Find the limit: $\lim _{x \rightarrow-2} \sin (x \pi)$ .

Find $f^{(2022)}(x)$ given that $f^{(2020)}(x)=\cot^{-1}(x)+x^{2}$ .

Find the function for the power series: $\sum_{k=0}^{\infty}(-1)^{k} \frac{x^{k}}{13^{k}}$ . What is $f(x)=\square$ ?

Find the limit: $\lim _{x \rightarrow 8}(3+\sqrt[3]{x})(2-5 x^{2}+x^{3})$ . If none, enter DNE.

Find the limit: $\lim _{x \rightarrow 1} \frac{x^{2}-4 x+3}{x-1}$ . If it doesn't exist, write DNE.