Calculus

Problem 32001

Evaluate the integral: $\int_{1}^{5} \sqrt{\frac{5}{x}} \, dx$

See Solution

Problem 32002

Evaluate the integral: $\int \sqrt[3]{5-8 t} \, dt$ and find the result.

See Solution

Problem 32003

Find the tangent line to $y=2 \sin x$ at $(\pi / 6,1)$ in the form $y=m x+b$ . What are $m$ and $b$ ?

See Solution

Problem 32004

Prove that wavefunctions for different energy levels in a particle in a box are orthogonal: $\int \psi_n^* \psi_m \, dx = 0$ for $n \neq m$ .

See Solution

Problem 32005

Find the tangent and normal line equations for $y=(3+6x)^{2}$ at the point $(-4,441)$ . Use $y=mx+b$ form.

See Solution

Problem 32006

Differentiate the integral $\int_{-x}^{\sin x}(1-t^{2}) dt$ with respect to $x$ .

See Solution

Problem 32007

Evaluate the integral $\int_{2}^{16} \frac{d x}{4 x \sqrt{\ln x}}$ and select the correct answer.

See Solution

Problem 32008

Differentiate the integral: $\frac{d}{d x} \int_{-\frac{\pi}{2}}^{x} \sec ^{2} t \, dt$

See Solution

Problem 32009

Find the first and second derivatives of $f(x)=4 \sin x+3 \cos x$ . What are $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ ?

See Solution

Problem 32010

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

See Solution

Problem 32011

Calculate the integrals: (a) $\int x^{2} \sin (x) \, dx$ and (b) $\int \tan^{3}(x) \, dx$ .

See Solution

Problem 32012

Find the value(s) of $x$ where the tangent line to $f(x)=8x^{3}-36x^{2}+x-72$ is parallel to $y=x-1.1$ .

See Solution

Problem 32013

Find the $x$ -values where the tangent to $f(x)=2 x^{3}+3 x^{2}-120 x+7$ is horizontal.

See Solution

Problem 32014

Find the average rate of change for $f(x)=x^{2}-2x$ from $x=-1$ to $x=1$ . Options: $1/2$ , $-1/2$ , $-1$ , $-2$ , $1$ , $2$ .

See Solution

Problem 32015

What does $\lim _{x \rightarrow a} f(x)=\infty$ mean? Check all true interpretations from these options.

See Solution

Problem 32016

Calculate the integral $g \int \frac{8}{4 x-3} \, dx$ .

See Solution

Problem 32017

How do we calculate the average rate of change of a function $h(x)$ from $a$ to $b$ ? Choose the correct option.

See Solution

Problem 32018

Given a function $y=f(x)$ with limits at $x=5$ and $x=-2$ , determine which statements about asymptotes and limits are true.

See Solution

Problem 32019

Calculate the integral: $\int \frac{12 h^{2}}{h^{3}+8} d h$

See Solution

Problem 32020

Evaluate the integral: $\int \frac{(4-\sqrt{u})^{7}}{\sqrt{u}} d u$ .

See Solution

Problem 32021

Find the general solutions to the linear homogeneous equation $u_{xxy}=0$ .

See Solution

Problem 32022

Find velocity and acceleration for $s=(1/3)t^3 - 5t^2 + 25t + 4$ . Then, find acceleration after 1 second and when velocity is 0.

See Solution

Problem 32023

Given the function $y=f(x)$ with limits at $x=5$ and $x=-2$ , which statements about asymptotes could be true?

See Solution

Problem 32024

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

See Solution

Problem 32025

Solve the equation: $y \int\left(\frac{-5}{x-1}+\frac{7}{(4-x)^{2}}\right) dx = \int \frac{-5}{x-1} dx + \int \frac{7}{(4-x)^{2}} dx$

See Solution

Problem 32026

Find local maxima and minima for $f(x) = \log x + \sin x + 4$ using its graph and the sign of the first derivative.

See Solution

Problem 32027

Find (A) the derivative of $\frac{T(x)}{B(x)}$ without the quotient rule, and (B) $\frac{T^{\prime}(x)}{B^{\prime}(x)}$ for $T(x)=x^{8}, B(x)=x^{3}$ .

See Solution

Problem 32028

Find the derivative $f^{\prime}(x)$ for $f(x)=5 x^{4}(x^{3}-2)$ and identify the correct product rule application.

See Solution

Problem 32029

Find $f^{\prime}(x)$ for $f(x)=\frac{x}{x-16}$ . Which option shows the correct quotient rule application?

See Solution

Problem 32030

Find $f^{\prime}(x)$ for $f(x)=5 x^{4}(x^{3}-2)$ and identify the correct product rule application from the options.

See Solution

Problem 32031

Find the derivative of the function $f(x)=7 x e^{x}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 32032

Find the derivative using the product rule for $y=(4 x^{2}+5)(5 x-4)$ . What is $y^{\prime}=?$

See Solution

Problem 32033

Find the derivative $f^{\prime}(x)$ of $f(x)=4 x^{5} \ln x$ and identify the correct product rule application.

See Solution

Problem 32034

Find $f^{\prime}(x)$ for $f(x)=4 x^{5} \ln x$ . Identify the correct product rule application.

See Solution

Problem 32035

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{e^{x}}{8 x^{2}+9}$ .

See Solution

Problem 32036

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{\ln x}{8+x}$ .

See Solution

Problem 32037

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{4-e^{x}}{9+e^{x}}$ .

See Solution

Problem 32038

Find the derivative $h^{\prime}(x)$ for $h(x)=\frac{\ln x^{9}}{f(x)}$ , where $f(x)$ is differentiable.

See Solution

Problem 32039

Find the domain, range, and horizontal asymptotes of $f(x)=\frac{8 \log 2 x}{\sin x+2}$ .

See Solution

Problem 32040

Find the derivative $y^{\prime}$ for $y=\frac{3x-4}{x^{2}+9x}$ . Simplify your answer.

See Solution

Problem 32041

Find the derivative $h^{\prime}(x)$ for $h(x)=10 e^{x} f(x)$ . Choose the correct expression for $h^{\prime}(x)$ .

See Solution

Problem 32042

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{2-e^{x}}{9+e^{x}}$ .

See Solution

Problem 32043

Find the velocity $v(t)$ of the particle at $t=3$ , when $s(t)=t^{4}-9t+17$ . Also, find when it's at rest and its distance in 8 sec.

See Solution

Problem 32044

Find $f^{\prime}(x)$ and the tangent line at $x=2$ for $f(x)=(2+3 x)(5-2 x)$ .

See Solution

Problem 32045

Find the Riemann sum $R_{N}$ for the function $f(x)=\frac{1}{2} x+2$ on the interval $[2,4]$ .

See Solution

Problem 32046

Find the derivative $\frac{dy}{dt}$ for $y=(2.2t-2t^{2})(4t+1.8)$ . Calculate $\frac{dy}{dt}=\square$ .

See Solution

Problem 32047

A ball is thrown from a 48 ft building at $96 \mathrm{ft/sec}$ . Find its max height and velocity when it hits the ground.

See Solution

Problem 32048

Find $h'(x)$ for $h(x)=\frac{f(x)}{x^{12}}$ . Choose the correct option for $h'(x)$ .

See Solution

Problem 32049

Find the derivative $f^{\prime}(x)$ and the tangent line equation at $x=2$ for $f(x)=(2+3x)(5-2x)$ .

See Solution

Problem 32050

Find $f^{\prime}(x)$ and the tangent line at $x=1$ for $f(x)=(2+4 x)(3-2 x)$ .

See Solution

Problem 32051

Find the tangent line equation for $f(x)=(x^{2}+9)(x-2)$ at the point (0,-18). $y=$

See Solution

Problem 32052

Find the derivative $f^{\prime}(x)$ and the values of $x$ where $f^{\prime}(x)=0$ for $f(x)=(2x-45)(x^{2}+168)$ .

See Solution

Problem 32053

Find $f^{\prime}(x)$ and the tangent line equation for $f$ at $x=2$ , where $f(x)=\frac{x-5}{2 x-7}$ .

See Solution

Problem 32054

Find the derivative $f'(x)$ of the function $f(x)=2 \sqrt{x}(x^{3}-2 \sqrt{x}+6)$ and evaluate $f'(4)$ .

See Solution

Problem 32055

Find the limit: $R_{N}=\lim _{N \rightarrow \infty}\left(6+\frac{N+1}{N}\right)$ .

See Solution

Problem 32056

Find the derivative of $2 x \log x^{8}$ and simplify it. What is $\frac{d}{d x}\left[2 x \log x^{8}\right]=\square$ ?

See Solution

Problem 32057

Find the derivative and simplify: $\frac{d}{d x}\left[9 x \log x^{6}\right] = \square$ .

See Solution

Problem 32058

Find the tangent line equation for $f(x)=\frac{x+7}{x-8}$ at the point $\left(0,-\frac{7}{8}\right)$ .

See Solution

Problem 32059

Find $f^{\prime}(x)$ using the product rule for $f(x)=x^{9}(x^{7}-4)$ . Which option shows the product rule correctly?

See Solution

Problem 32060

Find $f^{\prime}(x)$ using the product rule for $f(x)=x^{9}(x^{7}-4)$ . Which option shows the correct derivative?
Also, simplify $f(x)$ completely: $f(x)=\square$ .

See Solution

Problem 32061

Is the equation true or false? $\frac{d}{d x} \int_{x}^{5} \frac{5 t}{\sin t} d t=\frac{5 x}{\sin x}-\frac{25}{\sin 5}$

See Solution

Problem 32062

Find the derivative $S^{\prime}(t)$ of the sales function $S(t)=\frac{80 t^{2}}{t^{2}+150}$ .

See Solution

Problem 32063

Calculate the indefinite integral and use $C$ for the constant: $\int \sin^{3} 2 \theta \sqrt{\cos 2 \theta} d \theta$

See Solution

Problem 32064

The sales formula for a movie is $S(t)=\frac{80 t^{2}}{t^{2}+150}$ . Find $S^{\prime}(t)$ , $S(10)$ , $S^{\prime}(10)$ , and estimate sales after 11 months.

See Solution

Problem 32065

Find the sales $S(t)=\frac{80 t^{2}}{t^{2}+150}$ for $t$ months. Compute $S^{\prime}(t)$ and evaluate $S(10)$ , $S^{\prime}(10)$ .

See Solution

Problem 32066

Find the Instantaneous Rate of Change at $x=A$ for $f(x)=2 \sin \left(\frac{200000}{x+200}\right)-4$ . Is it increasing? Show calculations.

See Solution

Problem 32067

Find $f^{\prime}(x)$ using the product rule for $f(x)=x^{9}(x^{7}-4)$ . Which option shows the product rule result? Simplify $f(x)$ to $f(x)=x^{16}-4x^{9}$ . What is $f^{\prime}(x)=\square$ ?

See Solution

Problem 32068

Given the function $S(t)=\frac{80 t^{2}}{t^{2}+150}$ , find $S^{\prime}(t)$ and evaluate $S(10)$ and $S^{\prime}(10)$ . What do these values indicate?

See Solution

Problem 32069

Calculate the indefinite integral: $\int \frac{\cos ^{5} t}{\sqrt{\sin t}} d t$ (use $C$ for the constant).

See Solution

Problem 32070

Find the derivative $f^{\prime}(x)$ of $f(x)=\frac{x}{x^{2}+100}$ and where $f^{\prime}(x)=0$ .

See Solution

Problem 32071

Calculate the integral of $(\sin(x))^{3}$ and include the constant of integration $C$ .

See Solution

Problem 32072

Find the sales $S(t)$ of DVDs using $S(t)=\frac{80 t^{2}}{t^{2}+150}$ . Calculate $S^{\prime}(t)$ and $S(10)$ .

See Solution

Problem 32073

Find the derivative of the function $f(x)=\frac{4 x^{2}+8 x+29}{\sqrt{x}}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 32074

Given $f^{\prime}(x)=\int_{7}^{x} 5\left(-2 t^{-4}+4\right)^{2} d t$ , determine the behavior of $f$ for $7<x<13$ .

See Solution

Problem 32075

Find $f^{\prime}(x)$ for $f(x)=x^{9}(x^{7}-4)$ using the product rule. Which option shows this correctly?

See Solution

Problem 32076

Calculate the indefinite integral: $\int \cos^{4} x \sin x \, dx$ (use $C$ for the constant).

See Solution

Problem 32077

Find the tangent and normal line equations for $y=(3+6x)^{2}$ at the point $(-4,441)$ . Use $y=mx+b$ form.

See Solution

Problem 32078

Find the derivative $y'$ for $y=\frac{7x-2}{x^2+8x}$ . Simplify your answer.

See Solution

Problem 32079

Find $f(a)$ and $f(a+h)$ for $f(x)=6+8x^{2}$ , then compute $\frac{f(a+h)-f(a)}{h}$ .

See Solution

Problem 32080

Find the derivative of the function $y=6 \pi^{2}$ . What is $y^{\prime}$ ?

See Solution

Problem 32081

Find the instantaneous rate of change of $f(x)=2 \sin \left(\frac{200000}{x+200}\right)-4$ at $x=0$ . Show your work and determine if the function is increasing.

See Solution

Problem 32082

Find the rate of increase of the surface area $S = 4 \pi r^{2}$ of a balloon when $r = 1$ inch, in sq. inches/sec.

See Solution

Problem 32083

Calculate the integral $\int_{0}^{3}\left(2+\sqrt{9-x^{2}}\right) d x$ .

See Solution

Problem 32084

Find the instantaneous rate of change of $f(x) = 2 \sin \left(\frac{200000}{x+200}\right) - 4$ at $x = A$ . Is it increasing?

See Solution

Problem 32085

Solve the initial value problem: $x^{2} y' + x y = 1$ , $x > 0$ , $y(1) = 2$ .

See Solution

Problem 32086

Find $f(a)$ and $f(a+h)$ for $f(x)=4-1 x+10 x^{2}$ , then compute $\frac{f(a+h)-f(a)}{h}$ for $h \neq 0$ .

See Solution

Problem 32087

Find the derivative of the function $f(u)=\sqrt{5} u+\sqrt{8 u}$ , i.e., calculate $f'(u)$ .

See Solution

Problem 32088

Find the derivative of the function $f(u)=\sqrt{5} u+\sqrt{8 u}$ , i.e., calculate $f^{\prime}(u)$ .

See Solution

Problem 32089

Find the rate of increase of the surface area $S=4 \pi r^{2}$ of a balloon for $r=1$ , $r=3$ , and $r=6$ inches.

See Solution

Problem 32090

Sketch the slope field for $y' = x$ at $x = -1, 0, 1$ .

See Solution

Problem 32091

Use Euler's method to find the first three approximations for $y^{\prime}=x+y$ , $y(0)=1$ , with $d x=0.1$ .

See Solution

Problem 32092

Determine if the sequences converge or diverge; if they converge, find the limit.
a. $a_{n}=\sqrt{\frac{3+5 n^{2}}{n+n^{2}}}$ b. $a_{n}=\frac{3^{n}}{n^{3}}$ c. $a_{n}=\frac{2}{(0.8)^{n}}$

See Solution

Problem 32093

Check if the series converges or diverges, and find the sum if it converges. a. $\sum_{n=0}^{\infty}(-1)^{n}\left(\frac{3}{5^{n}}\right)$ b. $\frac{9}{4}-\frac{27}{8}+\frac{81}{16}-\frac{243}{32}+\ldots$

See Solution

Problem 32094

Find the sum of the series: a. $\sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)}$ b. $\sum_{n=0}^{\infty} 2\left(\frac{e}{\pi}\right)^{n}$

See Solution

Problem 32095

Show why the series diverges: $\sum_{n=1}^{\infty} \frac{n(n+1)}{(n+2)(n+3)}$

See Solution

Problem 32096

Find average velocities for $s(t)=-16 t^{2}+100 t$ over intervals $[2,3]$ , $[2.9,3]$ , $[2.99,3]$ , $[2.999,3]$ , and $[2.9999,3]$ , then conjecture instantaneous velocity at $t=3$ .

See Solution

Problem 32097

Determine if the series $\sum_{n=1}^{\infty} \frac{\ln n}{n}$ converges or diverges using the integral test. Show your work.

See Solution

Problem 32098

Determine convergence or divergence of the series $\sum_{n=1}^{\infty} \frac{2 n+1}{(n+1)^{2}}$ using the limit comparison test.

See Solution

Problem 32099

Find the missing expression in the equation: $\frac{d}{d x}(4 x+7)^{5}=5(4 x+7)^{4} ?$

See Solution

Problem 32100

Determine if the series $\sum_{n=1}^{\infty} \frac{1}{2^{n}+\sqrt{n}}$ converges or diverges using the direct comparison test.

See Solution

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Calculus

Problem 32001

Evaluate the integral: ∫155x dx\int_{1}^{5} \sqrt{\frac{5}{x}} \, dx∫15​x5​​dx

Problem 32002

Evaluate the integral: ∫5−8t3 dt\int \sqrt[3]{5-8 t} \, dt∫35−8t​dt and find the result.

Problem 32003

Find the tangent line to y=2sin⁡xy=2 \sin xy=2sinx at (π/6,1)(\pi / 6,1)(π/6,1) in the form y=mx+by=m x+by=mx+b. What are mmm and bbb?

Problem 32004

Prove that wavefunctions for different energy levels in a particle in a box are orthogonal: ∫ψn∗ψm dx=0\int \psi_n^* \psi_m \, dx = 0∫ψn∗​ψm​dx=0 for n≠mn \neq mn=m.

Problem 32005

Find the tangent and normal line equations for y=(3+6x)2y=(3+6x)^{2}y=(3+6x)2 at the point (−4,441)(-4,441)(−4,441). Use y=mx+by=mx+by=mx+b form.

Problem 32006

Differentiate the integral ∫−xsin⁡x(1−t2)dt\int_{-x}^{\sin x}(1-t^{2}) dt∫−xsinx​(1−t2)dt with respect to xxx.

Problem 32007

Evaluate the integral ∫216dx4xln⁡x\int_{2}^{16} \frac{d x}{4 x \sqrt{\ln x}}∫216​4xlnx​dx​ and select the correct answer.

Problem 32008

Differentiate the integral: ddx∫−π2xsec⁡2t dt\frac{d}{d x} \int_{-\frac{\pi}{2}}^{x} \sec ^{2} t \, dtdxd​∫−2π​x​sec2tdt

Problem 32009

Find the first and second derivatives of f(x)=4sin⁡x+3cos⁡xf(x)=4 \sin x+3 \cos xf(x)=4sinx+3cosx. What are f′(x)f^{\prime}(x)f′(x) and f′′(x)f^{\prime \prime}(x)f′′(x)?

Problem 32010

Calculate the integral: ∫4x(5−3x2)5 dx\int 4 x(5-3 x^{2})^{5} \, dx∫4x(5−3x2)5dx

Problem 32011

Calculate the integrals: (a) ∫x2sin⁡(x) dx\int x^{2} \sin (x) \, dx∫x2sin(x)dx and (b) ∫tan⁡3(x) dx\int \tan^{3}(x) \, dx∫tan3(x)dx.

Problem 32012

Find the value(s) of xxx where the tangent line to f(x)=8x3−36x2+x−72f(x)=8x^{3}-36x^{2}+x-72f(x)=8x3−36x2+x−72 is parallel to y=x−1.1y=x-1.1y=x−1.1.

Problem 32013

Find the xxx-values where the tangent to f(x)=2x3+3x2−120x+7f(x)=2 x^{3}+3 x^{2}-120 x+7f(x)=2x3+3x2−120x+7 is horizontal.

Problem 32014

Find the average rate of change for f(x)=x2−2xf(x)=x^{2}-2xf(x)=x2−2x from x=−1x=-1x=−1 to x=1x=1x=1. Options: 1/21/21/2, −1/2-1/2−1/2, −1-1−1, −2-2−2, 111, 222.

Problem 32015

What does lim⁡x→af(x)=∞\lim _{x \rightarrow a} f(x)=\inftylimx→a​f(x)=∞ mean? Check all true interpretations from these options.

Problem 32016

Calculate the integral g∫84x−3 dxg \int \frac{8}{4 x-3} \, dxg∫4x−38​dx.

Problem 32017

How do we calculate the average rate of change of a function h(x)h(x)h(x) from aaa to bbb? Choose the correct option.

Problem 32018

Given a function y=f(x)y=f(x)y=f(x) with limits at x=5x=5x=5 and x=−2x=-2x=−2, determine which statements about asymptotes and limits are true.

Problem 32019

Calculate the integral: ∫12h2h3+8dh\int \frac{12 h^{2}}{h^{3}+8} d h∫h3+812h2​dh

Problem 32020

Evaluate the integral: ∫(4−u)7udu\int \frac{(4-\sqrt{u})^{7}}{\sqrt{u}} d u∫u​(4−u​)7​du.

Problem 32021

Find the general solutions to the linear homogeneous equation uxxy=0u_{xxy}=0uxxy​=0.

Problem 32022

Find velocity and acceleration for s=(1/3)t3−5t2+25t+4s=(1/3)t^3 - 5t^2 + 25t + 4s=(1/3)t3−5t2+25t+4. Then, find acceleration after 1 second and when velocity is 0.

Problem 32023

Given the function y=f(x)y=f(x)y=f(x) with limits at x=5x=5x=5 and x=−2x=-2x=−2, which statements about asymptotes could be true?

Problem 32024

Find the limit: lim⁡x→−∞x2−3x−3x2+2x2x+x2+1\lim _{x \rightarrow-\infty} \frac{\sqrt{x^{2}-3 x}-\sqrt{3 x^{2}+2 x}}{2 x+\sqrt{x^{2}+1}}limx→−∞​2x+x2+1​x2−3x​−3x2+2x​​.

Problem 32025

Solve the equation: y∫(−5x−1+7(4−x)2)dx=∫−5x−1dx+∫7(4−x)2dxy \int\left(\frac{-5}{x-1}+\frac{7}{(4-x)^{2}}\right) dx = \int \frac{-5}{x-1} dx + \int \frac{7}{(4-x)^{2}} dxy∫(x−1−5​+(4−x)27​)dx=∫x−1−5​dx+∫(4−x)27​dx

Problem 32026

Find local maxima and minima for f(x)=log⁡x+sin⁡x+4f(x) = \log x + \sin x + 4f(x)=logx+sinx+4 using its graph and the sign of the first derivative.

Problem 32027

Find (A) the derivative of T(x)B(x)\frac{T(x)}{B(x)}B(x)T(x)​ without the quotient rule, and (B) T′(x)B′(x)\frac{T^{\prime}(x)}{B^{\prime}(x)}B′(x)T′(x)​ for T(x)=x8,B(x)=x3T(x)=x^{8}, B(x)=x^{3}T(x)=x8,B(x)=x3.

Problem 32028

Find the derivative f′(x)f^{\prime}(x)f′(x) for f(x)=5x4(x3−2)f(x)=5 x^{4}(x^{3}-2)f(x)=5x4(x3−2) and identify the correct product rule application.

Problem 32029

Find f′(x)f^{\prime}(x)f′(x) for f(x)=xx−16f(x)=\frac{x}{x-16}f(x)=x−16x​. Which option shows the correct quotient rule application?

Problem 32030

Find f′(x)f^{\prime}(x)f′(x) for f(x)=5x4(x3−2)f(x)=5 x^{4}(x^{3}-2)f(x)=5x4(x3−2) and identify the correct product rule application from the options.

Problem 32031

Find the derivative of the function f(x)=7xexf(x)=7 x e^{x}f(x)=7xex. What is f′(x)f^{\prime}(x)f′(x)?

Problem 32032

Find the derivative using the product rule for y=(4x2+5)(5x−4)y=(4 x^{2}+5)(5 x-4)y=(4x2+5)(5x−4). What is y′=?y^{\prime}=?y′=?

Problem 32033

Find the derivative f′(x)f^{\prime}(x)f′(x) of f(x)=4x5ln⁡xf(x)=4 x^{5} \ln xf(x)=4x5lnx and identify the correct product rule application.

Problem 32034

Find f′(x)f^{\prime}(x)f′(x) for f(x)=4x5ln⁡xf(x)=4 x^{5} \ln xf(x)=4x5lnx. Identify the correct product rule application.

Problem 32035

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=ex8x2+9f(x)=\frac{e^{x}}{8 x^{2}+9}f(x)=8x2+9ex​.

Problem 32036

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=ln⁡x8+xf(x)=\frac{\ln x}{8+x}f(x)=8+xlnx​.

Problem 32037

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=4−ex9+exf(x)=\frac{4-e^{x}}{9+e^{x}}f(x)=9+ex4−ex​.

Problem 32038

Find the derivative h′(x)h^{\prime}(x)h′(x) for h(x)=ln⁡x9f(x)h(x)=\frac{\ln x^{9}}{f(x)}h(x)=f(x)lnx9​, where f(x)f(x)f(x) is differentiable.

Problem 32039

Find the domain, range, and horizontal asymptotes of f(x)=8log⁡2xsin⁡x+2f(x)=\frac{8 \log 2 x}{\sin x+2}f(x)=sinx+28log2x​.

Problem 32040

Evaluate the integral: $\int_{1}^{5} \sqrt{\frac{5}{x}} \, dx$

Evaluate the integral: $\int \sqrt[3]{5-8 t} \, dt$ and find the result.

Find the tangent line to $y=2 \sin x$ at $(\pi / 6,1)$ in the form $y=m x+b$ . What are $m$ and $b$ ?

Prove that wavefunctions for different energy levels in a particle in a box are orthogonal: $\int \psi_n^* \psi_m \, dx = 0$ for $n \neq m$ .

Find the tangent and normal line equations for $y=(3+6x)^{2}$ at the point $(-4,441)$ . Use $y=mx+b$ form.

Differentiate the integral $\int_{-x}^{\sin x}(1-t^{2}) dt$ with respect to $x$ .

Evaluate the integral $\int_{2}^{16} \frac{d x}{4 x \sqrt{\ln x}}$ and select the correct answer.

Differentiate the integral: $\frac{d}{d x} \int_{-\frac{\pi}{2}}^{x} \sec ^{2} t \, dt$

Find the first and second derivatives of $f(x)=4 \sin x+3 \cos x$ . What are $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ ?

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

Calculate the integrals: (a) $\int x^{2} \sin (x) \, dx$ and (b) $\int \tan^{3}(x) \, dx$ .

Find the value(s) of $x$ where the tangent line to $f(x)=8x^{3}-36x^{2}+x-72$ is parallel to $y=x-1.1$ .

Find the $x$ -values where the tangent to $f(x)=2 x^{3}+3 x^{2}-120 x+7$ is horizontal.

Find the average rate of change for $f(x)=x^{2}-2x$ from $x=-1$ to $x=1$ . Options: $1/2$ , $-1/2$ , $-1$ , $-2$ , $1$ , $2$ .

What does $\lim _{x \rightarrow a} f(x)=\infty$ mean? Check all true interpretations from these options.

Calculate the integral $g \int \frac{8}{4 x-3} \, dx$ .

How do we calculate the average rate of change of a function $h(x)$ from $a$ to $b$ ? Choose the correct option.

Given a function $y=f(x)$ with limits at $x=5$ and $x=-2$ , determine which statements about asymptotes and limits are true.

Calculate the integral: $\int \frac{12 h^{2}}{h^{3}+8} d h$

Evaluate the integral: $\int \frac{(4-\sqrt{u})^{7}}{\sqrt{u}} d u$ .

Find the general solutions to the linear homogeneous equation $u_{xxy}=0$ .

Find velocity and acceleration for $s=(1/3)t^3 - 5t^2 + 25t + 4$ . Then, find acceleration after 1 second and when velocity is 0.

Given the function $y=f(x)$ with limits at $x=5$ and $x=-2$ , which statements about asymptotes could be true?

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

Solve the equation: $y \int\left(\frac{-5}{x-1}+\frac{7}{(4-x)^{2}}\right) dx = \int \frac{-5}{x-1} dx + \int \frac{7}{(4-x)^{2}} dx$

Find local maxima and minima for $f(x) = \log x + \sin x + 4$ using its graph and the sign of the first derivative.

Find (A) the derivative of $\frac{T(x)}{B(x)}$ without the quotient rule, and (B) $\frac{T^{\prime}(x)}{B^{\prime}(x)}$ for $T(x)=x^{8}, B(x)=x^{3}$ .

Find the derivative $f^{\prime}(x)$ for $f(x)=5 x^{4}(x^{3}-2)$ and identify the correct product rule application.

Find $f^{\prime}(x)$ for $f(x)=\frac{x}{x-16}$ . Which option shows the correct quotient rule application?

Find $f^{\prime}(x)$ for $f(x)=5 x^{4}(x^{3}-2)$ and identify the correct product rule application from the options.

Find the derivative of the function $f(x)=7 x e^{x}$ . What is $f^{\prime}(x)$ ?

Find the derivative using the product rule for $y=(4 x^{2}+5)(5 x-4)$ . What is $y^{\prime}=?$

Find the derivative $f^{\prime}(x)$ of $f(x)=4 x^{5} \ln x$ and identify the correct product rule application.

Find $f^{\prime}(x)$ for $f(x)=4 x^{5} \ln x$ . Identify the correct product rule application.

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{e^{x}}{8 x^{2}+9}$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{\ln x}{8+x}$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{4-e^{x}}{9+e^{x}}$ .

Find the derivative $h^{\prime}(x)$ for $h(x)=\frac{\ln x^{9}}{f(x)}$ , where $f(x)$ is differentiable.

Find the domain, range, and horizontal asymptotes of $f(x)=\frac{8 \log 2 x}{\sin x+2}$ .

Find the derivative $y^{\prime}$ for $y=\frac{3x-4}{x^{2}+9x}$ . Simplify your answer.

Find the derivative $h^{\prime}(x)$ for $h(x)=10 e^{x} f(x)$ . Choose the correct expression for $h^{\prime}(x)$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{2-e^{x}}{9+e^{x}}$ .

Find the velocity $v(t)$ of the particle at $t=3$ , when $s(t)=t^{4}-9t+17$ . Also, find when it's at rest and its distance in 8 sec.

Find $f^{\prime}(x)$ and the tangent line at $x=2$ for $f(x)=(2+3 x)(5-2 x)$ .

Find the Riemann sum $R_{N}$ for the function $f(x)=\frac{1}{2} x+2$ on the interval $[2,4]$ .

Find the derivative $\frac{dy}{dt}$ for $y=(2.2t-2t^{2})(4t+1.8)$ . Calculate $\frac{dy}{dt}=\square$ .

A ball is thrown from a 48 ft building at $96 \mathrm{ft/sec}$ . Find its max height and velocity when it hits the ground.

Find $h'(x)$ for $h(x)=\frac{f(x)}{x^{12}}$ . Choose the correct option for $h'(x)$ .

Find the derivative $f^{\prime}(x)$ and the tangent line equation at $x=2$ for $f(x)=(2+3x)(5-2x)$ .

Find $f^{\prime}(x)$ and the tangent line at $x=1$ for $f(x)=(2+4 x)(3-2 x)$ .

Find the tangent line equation for $f(x)=(x^{2}+9)(x-2)$ at the point (0,-18). $y=$

Find the derivative $f^{\prime}(x)$ and the values of $x$ where $f^{\prime}(x)=0$ for $f(x)=(2x-45)(x^{2}+168)$ .

Find $f^{\prime}(x)$ and the tangent line equation for $f$ at $x=2$ , where $f(x)=\frac{x-5}{2 x-7}$ .

Find the derivative $f'(x)$ of the function $f(x)=2 \sqrt{x}(x^{3}-2 \sqrt{x}+6)$ and evaluate $f'(4)$ .

Find the limit: $R_{N}=\lim _{N \rightarrow \infty}\left(6+\frac{N+1}{N}\right)$ .

Find the derivative of $2 x \log x^{8}$ and simplify it. What is $\frac{d}{d x}\left[2 x \log x^{8}\right]=\square$ ?

Find the derivative and simplify: $\frac{d}{d x}\left[9 x \log x^{6}\right] = \square$ .

Find the tangent line equation for $f(x)=\frac{x+7}{x-8}$ at the point $\left(0,-\frac{7}{8}\right)$ .

Find $f^{\prime}(x)$ using the product rule for $f(x)=x^{9}(x^{7}-4)$ . Which option shows the product rule correctly?

Find $f^{\prime}(x)$ using the product rule for $f(x)=x^{9}(x^{7}-4)$ . Which option shows the correct derivative?
Also, simplify $f(x)$ completely: $f(x)=\square$ .

Is the equation true or false? $\frac{d}{d x} \int_{x}^{5} \frac{5 t}{\sin t} d t=\frac{5 x}{\sin x}-\frac{25}{\sin 5}$

Find the derivative $S^{\prime}(t)$ of the sales function $S(t)=\frac{80 t^{2}}{t^{2}+150}$ .

Calculate the indefinite integral and use $C$ for the constant: $\int \sin^{3} 2 \theta \sqrt{\cos 2 \theta} d \theta$

The sales formula for a movie is $S(t)=\frac{80 t^{2}}{t^{2}+150}$ . Find $S^{\prime}(t)$ , $S(10)$ , $S^{\prime}(10)$ , and estimate sales after 11 months.

Find the sales $S(t)=\frac{80 t^{2}}{t^{2}+150}$ for $t$ months. Compute $S^{\prime}(t)$ and evaluate $S(10)$ , $S^{\prime}(10)$ .

Find the Instantaneous Rate of Change at $x=A$ for $f(x)=2 \sin \left(\frac{200000}{x+200}\right)-4$ . Is it increasing? Show calculations.

Find $f^{\prime}(x)$ using the product rule for $f(x)=x^{9}(x^{7}-4)$ . Which option shows the product rule result? Simplify $f(x)$ to $f(x)=x^{16}-4x^{9}$ . What is $f^{\prime}(x)=\square$ ?

Given the function $S(t)=\frac{80 t^{2}}{t^{2}+150}$ , find $S^{\prime}(t)$ and evaluate $S(10)$ and $S^{\prime}(10)$ . What do these values indicate?

Calculate the indefinite integral: $\int \frac{\cos ^{5} t}{\sqrt{\sin t}} d t$ (use $C$ for the constant).

Find the derivative $f^{\prime}(x)$ of $f(x)=\frac{x}{x^{2}+100}$ and where $f^{\prime}(x)=0$ .

Calculate the integral of $(\sin(x))^{3}$ and include the constant of integration $C$ .

Find the sales $S(t)$ of DVDs using $S(t)=\frac{80 t^{2}}{t^{2}+150}$ . Calculate $S^{\prime}(t)$ and $S(10)$ .

Find the derivative of the function $f(x)=\frac{4 x^{2}+8 x+29}{\sqrt{x}}$ . What is $f^{\prime}(x)$ ?

Given $f^{\prime}(x)=\int_{7}^{x} 5\left(-2 t^{-4}+4\right)^{2} d t$ , determine the behavior of $f$ for $7<x<13$ .