Calculus

Problem 9701

Find the derivative of the function $g(x)=\int_{1}^{x} \ln(\varpi+t^{2}) dt$ .

See Solution

Problem 9702

Find the indefinite integral and evaluate the definite integral from 1 to 3:
$\int\left(x^{2}+2 x-3\right) d x$ and $\int_{1}^{3}\left(x^{2}+2 x-3\right) d x$ .

See Solution

Problem 9703

Find the integral $\int_{0}^{4} f(x) \, dx$ for the given piecewise linear function defined by 4 segments.

See Solution

Problem 9704

Calculate the integral from 0 to 9 of the function $4 t^{2}+t^{3/2}$ .

See Solution

Problem 9705

Find the maximum blood pressure $B$ from the function $B(x)=375 x^{2}-3750 x^{3}$ for $0 \leq x \leq 0.10$ .

See Solution

Problem 9706

Find the correct Newton's method formula for $f(x)=1-\tan 2x$ and compute $x_{1}$ and $x_{2}$ from $x_{0}=1$ .

See Solution

Problem 9707

Find the derivative of the function $g(x)=\int_{0}^{x} \sqrt{t^{3}+t^{5}} dt$ .

See Solution

Problem 9708

Rewrite the integral $\int_{1}^{9} \frac{4+x^{2}}{\sqrt{x}} d x$ using rational exponents and evaluate it.

See Solution

Problem 9709

Find the absolute max and min of the function $f(x)=x^{2}+\frac{480}{x}$ for $x \in (0, \infty)$ .

See Solution

Problem 9710

Find the derivative of $y=\frac{-4}{(4x^{5}+9)^{3}}$ . What is $\frac{dy}{dx}=?$

See Solution

Problem 9711

Calculate the Riemann sum $M_{5}=\int_{1}^{3} \frac{x}{x^{2}+3} d x$ with $n=5$ .

See Solution

Problem 9712

Evaluate the integral from 1 to 9 of the function $2x + \frac{1}{x}$ .

See Solution

Problem 9713

Find the derivative of $f(t)=4 \sqrt{6 t^{2}+9}$ . What is $u=g(t)$ where $u$ is a function of $t$ ?

See Solution

Problem 9714

Find the derivative of $f(t)=4 \sqrt{6 t^{2}+9}$ . What is $u=g(t)$ for $f=h(u)$ ?

See Solution

Problem 9715

Find the derivative of $g(w)=\int_{0}^{w} \sin(8+t^{9}) \, dt$ using the fundamental theorem of calculus.

See Solution

Problem 9716

Find $x_{2}$ using Newton's method for $f(x)=5-\tan(6x)$ with $x_{0}=1$ . Compute $x_{1}$ first.

See Solution

Problem 9717

Find the derivative of the function $m(t)=-5 t(3 t^{8}-1)^{4}$ . What is $m'(t)$ ?

See Solution

Problem 9718

Evaluate the integral from 0 to 1 of (3x^{e} + 5e^{x}) dx.

See Solution

Problem 9719

Find $F'(x)$ for the function defined by $F(x)=\int_{x}^{x^{2}} e^{t^{7}} d t$ .

See Solution

Problem 9720

Find the slope of the tangent line of $f(x)=\ln \left(\frac{x}{5}\right)$ at $x=7$ . Choices: A. 14 B. $\frac{1}{7}$ C. 7 D. -7 E. $-\frac{1}{7}$

See Solution

Problem 9721

Find the limit: $\lim _{h \rightarrow 0} \frac{\sec \left(\frac{\pi}{6}+h\right)-\sec \left(\frac{\pi}{6}\right)}{h}$ .

See Solution

Problem 9722

Find the derivative of the product: $(f(x) \cdot g(x))' = f'(x) \cdot g(x) + g'(x) \cdot f(x)$ for $(x^{2} \cdot e^{x})'$ .

See Solution

Problem 9723

Find the average rate of change of the cost $C(x)=4500+1530 x-0.04 x^{3}$ from year 4 to year 7. Include units.

See Solution

Problem 9724

Find the average rate of change of cost $C(x)=4500+1530 x-0.04 x^{3}$ from year 4 to year 7. Include units.

See Solution

Problem 9725

Find the second derivative $g^{\prime \prime}(20)$ for the function $g(x)=\left(3-\frac{x}{5}\right)^{5}$ .

See Solution

Problem 9726

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

See Solution

Problem 9727

Calculate the indefinite integral and include the constant $C$ . $\int(4 x^{3}+8 x+9) dx$

See Solution

Problem 9728

Calculate the indefinite integral: $\int\left(\sqrt[4]{x}+\frac{1}{\sqrt[4]{x}}\right) d x$ (use $C$ for the constant).

See Solution

Problem 9729

Find the indefinite integral: $\int(u+4)(5u+1) \, du$ (use $C$ for the constant).

See Solution

Problem 9730

Calculate the indefinite integral and include the constant $C$ for integration: $\int \frac{1+\sqrt{x}+x}{x} d x$

See Solution

Problem 9731

Find the derivative of the function $y = \frac{e^{17x} + e^{-17x}}{x}$ .

See Solution

Problem 9732

Compute the indefinite integral and include the constant $C$ :
$\int\left(4+5^{x}\right) d x$

See Solution

Problem 9733

Find the maximum value of $f(x)=\frac{18}{x^{2}-9}$ for $-3<x<3$ .

See Solution

Problem 9734

Find the indefinite integral and evaluate the definite integral from 1 to 6 for $8 t^{3}-6 t^{-2}$ . Use $C$ for the constant.

See Solution

Problem 9735

Find the volume of a solid with disc cross-sections of radius $e^{-x}$ for $1 \leq x < \infty$ . Round to four decimal places.

See Solution

Problem 9736

Cactus wrens' weight fits $G(t) = \frac{31.8}{1 + \frac{31.8 - 2.37}{2.37} e^{-0.0125t}}$ .
Find $G(1)$ , $G(5)$ , $G(12)$ and describe growth rate changes.

See Solution

Problem 9737

Find the total distance traveled by a particle with velocity $v(t) = 3t - 8$ from $t = 0$ to $t = 5$ .

See Solution

Problem 9738

Given the velocity $v(t)=3t-8$ for $0 \leq t \leq 5$ , find displacement and total distance traveled in meters.

See Solution

Problem 9739

Find the volume of the solid in polar coordinates: $0 \leq r \leq R$ , $0 \leq \theta \leq 2\pi$ , $0 \leq z \leq \theta$ .

See Solution

Problem 9740

Find the integral to compute the volume of region $\boldsymbol{D}$ revolved around $x=5$ :
1. $\int_{x=0}^{1} 2 \pi(5-x)(1-x^{4}) d x$
2. Other options include $\int_{x=0}^{1} \pi(1-x^{4})^{2} d x$ and others.

See Solution

Problem 9741

Find the derivative of $y=(2x^{4}-x+1)(-x^{5}+1)$ .

See Solution

Problem 9742

Find the price elasticity of demand $E$ for tissues at $p = \$32$ . Interpret $E$ .
Also, find the price for max revenue and demand at that price.

See Solution

Problem 9743

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

See Solution

Problem 9744

Find the derivative of $y=\frac{a x+b}{c x+d}$ .

See Solution

Problem 9745

Approximate $\sqrt{81.1}$ using the tangent line of $f(x)=\sqrt{x}$ at $x=81$ : $L(x)=9+\frac{1}{18}(x-81)$ . Answer to 9 sig. figs.

See Solution

Problem 9746

Does $x$ approach infinity as $2f(x)$ approaches 0 for the function $f(x)=\frac{(x+3)^{2}}{(x+3)(x+1)}$ ?

See Solution

Problem 9747

Does $f(x)=\frac{(x+3)^{2}}{(x+3)(x+1)}$ approach 0 as $x$ approaches infinity?

See Solution

Problem 9748

Approximate $\sqrt{81.1}$ using linear approximation with $f(x)=\sqrt{x}$ at $x=81$ . Find the tangent line $L(x)=$ . Give answer to 9 significant figures.

See Solution

Problem 9749

Find the max and min of $f(x)=2x^{3}+12x^{2}-192x+4$ on $[-8, 5]$ and check its continuity in that interval.

See Solution

Problem 9750

Find the tangent line equation for $y=\frac{x+1}{x-1}$ at the point $(2,3)$ .

See Solution

Problem 9751

Find the derivatives at $x=2$ : 57. $\frac{d}{dx}[f(x)g(x)]$ and 58. $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]$ .

See Solution

Problem 9752

Find the local minimum and maximum of the function $f(x)=2 x^{3}-24 x^{2}+42 x+2$ .

See Solution

Problem 9753

Given $f(2)=3$ , $f'(2)=3$ , $g(2)=3$ , and $g'(2)=\frac{1}{3}$ , find the derivatives at $x=2$ for the given functions.

See Solution

Problem 9754

Evaluate $f(x)=5 x^{2}-10 x+3$ at $x=-1$ and $x=3$ . Does Rolle's Theorem apply? Find $c$ where $f'(c)=0$ .

See Solution

Problem 9755

Find $K^{\prime}(2)$ for $K(x)=[f(x)]^{3}$ given $f(2)=-4$ and $f^{\prime}(2)=8$ .

See Solution

Problem 9756

Find $H^{\prime}(2)$ for $H(x) = f(g(x))$ using the given values of $f$ and $g$ .

See Solution

Problem 9757

Find the approximate slope of $f(x)=\frac{1}{x-1}$ at $x=1.2$ . Choices: $-\frac{1}{25}$ , $\frac{1}{25}$ , 25, $-25$ .

See Solution

Problem 9758

Differentiate $y=5 x^{3}(2-x)^{4}$ using differentiation rules.

See Solution

Problem 9759

An object falls from $188 \mathrm{ft}$ with height $s=188-16 t^{2}$ . Find velocity, time to hit ground, and impact velocity.

See Solution

Problem 9760

Analyze the behavior of $f(x)=\frac{2 x+5}{x+3}$ as $x \rightarrow+\infty$ . What does it approach?

See Solution

Problem 9761

Find local extrema as ordered pairs for: a. $f(x)=x^{3}-6 x^{2}+5$ and b. $g(x)=x \sqrt{9-x^{2}}$ .

See Solution

Problem 9762

What happens to $f(x)=\frac{1}{3 x+5}$ as $x \rightarrow-\frac{5}{3}^{+}$ ? Options: undefined, $-\infty$ , $\infty$ , or $0$ .

See Solution

Problem 9763

Find the derivative of $y=2(2 x-1)^{5/4}(2 x+1)^{3/4}$ using differentiation rules.

See Solution

Problem 9764

Find the limit: $\lim _{x \rightarrow \infty} \frac{x^{2}-6 x-4}{13-7 x^{2}}$ .

See Solution

Problem 9765

Find the first derivative of $g(x)=4x^{3}-12x^{2}-180x$ and then the second derivative. Evaluate $g^{\prime \prime}(5)$ to determine concavity.

See Solution

Problem 9766

Given $f(x)$ and $g(x)$ values, find $P^{\prime}(2)$ where $P(x) = f(x) \times g(x)$ .

See Solution

Problem 9767

Find $R^{\prime}(4)$ for $R(x)=(f(x)-g(x))^{2}$ using the values of $f$ and $g$ given in the table.

See Solution

Problem 9768

Differentiate $f+g$ and evaluate at $x=1$ . Use $f'(1)$ and $g'(1)$ values from the table.

See Solution

Problem 9769

Determine the horizontal asymptotes by finding these limits:
1. $\lim _{x \rightarrow \infty} \frac{-6 x}{6+2 x}$
2. $\lim _{x \rightarrow-\infty} \frac{7 x-3}{x^{3}+11 x-4}$
3. $\lim _{x \rightarrow \infty} \frac{x^{2}-6 x-4}{13-7 x^{2}}$
4. $\lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}+12 x}}{2-8 x}$
5. $\lim _{x \rightarrow-\infty} \frac{\sqrt{x^{2}+12 x}}{2-8 x}$

See Solution

Problem 9770

Find the slope of the secant line for $f(x)=x^{3}-6x$ on $[2,6]$ and values of $c$ where $f'(c)=m$ .

See Solution

Problem 9771

Find all points $(x, y)$ on the graph of $f(x)=x^{3}+9 x^{2}+20 x+13$ where the tangent slope is -4.

See Solution

Problem 9772

Find the second derivative $f^{\prime \prime}(x)$ of the function $f(x)=-2 x^{3}-\frac{2}{3} x^{4}-6 x-\frac{3}{2} x^{-2}$ .

See Solution

Problem 9773

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the function $y=-\frac{5}{3} x^{3}-\frac{1}{2} x^{-1}-2 x^{2}$ .

See Solution

Problem 9774

Which integrals equal zero by symmetry? Explain: (a) $\int_{-1}^{2} \frac{x}{\sqrt{9+x^{2}}} dx$ (b) $\int_{-2}^{2} \frac{x^{2}}{\sqrt{9+x^{2}}} dx$ (c) $\int_{-1}^{1} \frac{x^{3}}{\sqrt{9+x^{2}}} dx$
Calculate the following: (a) $\int_{\sin (-\sqrt{2})}^{\sin (\sqrt{2})} \frac{x^{1234567}}{\sqrt{1+x^{8}}} dx$ (b) $\lim_{N \to \infty} \sum_{n=1}^{N} \frac{1}{N+n}$
Estimate $\int_{0}^{2} x^{2} dx$ using a Riemann sum with 4 equal intervals.
Find $F'(\sqrt{\pi})$ where $F(x)=\int_{1+\cos(x^{2})}^{3} e^{3(t-1)^{2}} dt$ .
Find the Taylor series of $f$ about $x=0$ up to quadratic term, where $f(x)=\int_{0}^{x} e^{\sin t} dt$ .

See Solution

Problem 9775

Find the third derivative of the function $f(x)=\frac{1}{6} x^{5}-\frac{1}{2} x^{4}+\frac{1}{6} x^{2}$ .

See Solution

Problem 9776

Find the derivative of $y=(2x^{3}-2x^{2}+8x-6)e^{x^{3}}$ .

See Solution

Problem 9777

Find the third derivative of the function $y=-\frac{1}{12} x^{6}-x^{-1}-\frac{1}{4} x^{2}+\frac{1}{8} x^{4}$ .

See Solution

Problem 9778

Find the derivative of the function $f(x)=5x-x^{5}$ using the limit definition of a derivative. No need to simplify.

See Solution

Problem 9779

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the function $y=\frac{5}{3}+x+x^{3}$ .

See Solution

Problem 9780

Find the second derivative of the function $f(x)=-x^{-1}-\frac{2}{3} x^{4}-\frac{1}{9}$ .

See Solution

Problem 9781

Find the instantaneous rate of change of $f(x)=\sqrt{2 x}$ at $x=2$ using the limit definition.

See Solution

Problem 9782

Find the derivative of the function $f(x)=5 x^{5}-3 x^{2}$ using the limit definition, without simplifying.

See Solution

Problem 9783

Calculate total hours of daylight for Miami: $f(x)=\frac{-2}{x} \cos \left(\frac{1}{2} x\right)+12$ and Fairbanks: $f(x)=-9 \cdot \cos (.5 x)+12$ in 2016.

See Solution

Problem 9784

Find the derivative of $f(x)=3 x^{2}-x^{3}$ at $x=5$ using the limit method. No need to simplify your answer.

See Solution

Problem 9785

Find the slope of the tangent line for $f(x)=4 \ln (x+1)$ at $x=8$ using the limit definition. No simplification needed.

See Solution

Problem 9786

Find the limits $\lim _{x \rightarrow 3^{+}} f(x)$ , $\lim _{x \rightarrow 3^{-}} f(x)$ , and values for $f(3)$ and $f(1)$ .

See Solution

Problem 9787

Find the slope of the curve $f(x)=\frac{1}{x-1}$ at $x=1.2$ .

See Solution

Problem 9788

Find the volume of the region under $y=\ln \sqrt{x}$ from $x=1$ to $x=e$ when revolved around the $x$ -axis.

See Solution

Problem 9789

Find the derivative of -5 cos(x) with respect to x.

See Solution

Problem 9790

Find the derivative of the function: $\frac{d}{d x}(6 \sin (-6 x-5))$ .

See Solution

Problem 9791

Find the integral to compute the volume of the region $D$ revolved around $x=5$ : $y=1-x^{4}$ , $x,y \geq 0$ .

See Solution

Problem 9792

Find the derivative of $-6 \sin (-x)$ with respect to $x$ .

See Solution

Problem 9793

Find the derivative $f^{\prime}(x)$ of the function $f(x)=2\left(-4 x^{2}-4 x-3\right)^{3}$ .

See Solution

Problem 9794

Find the derivative of the function $f(x)=(8 x+9)^{4}$ , denoted as $f^{\prime}(x)$ .

See Solution

Problem 9795

Find the derivative $\frac{d y}{d x}$ for the function $y=-5(2 x^{2}-x-5)^{4}$ .

See Solution

Problem 9796

Find the derivative of the function $y=\frac{3}{\sqrt[5]{-3 x^{2}+8 x-7}}$ , represented as $\frac{d y}{d x}$ .

See Solution

Problem 9797

Find the derivative of the function $y=\frac{1}{\sqrt[3]{2 x^{2}-8 x-3}}$ , i.e., compute $\frac{d y}{d x}$ .

See Solution

Problem 9798

Find the derivative of the function $y=3 x e^{-6 x}-e^{x^{2}}$ .

See Solution

Problem 9799

Find the derivative of the function $y=3\left(7 x^{2}+2\right)^{-\frac{5}{4}}$ , i.e., compute $\frac{d y}{d x}$ .

See Solution

Problem 9800

Find the intervals where the function $f(x)=\frac{2 x}{x-7}$ is increasing.

See Solution

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Calculus

Problem 9701

Find the derivative of the function g(x)=∫1xln⁡(ϖ+t2)dtg(x)=\int_{1}^{x} \ln(\varpi+t^{2}) dtg(x)=∫1x​ln(ϖ+t2)dt.

Problem 9702

Find the indefinite integral and evaluate the definite integral from 1 to 3: ∫(x2+2x−3)dx\int\left(x^{2}+2 x-3\right) d x∫(x2+2x−3)dx and ∫13(x2+2x−3)dx\int_{1}^{3}\left(x^{2}+2 x-3\right) d x∫13​(x2+2x−3)dx.

Problem 9703

Find the integral ∫04f(x) dx\int_{0}^{4} f(x) \, dx∫04​f(x)dx for the given piecewise linear function defined by 4 segments.

Problem 9704

Calculate the integral from 0 to 9 of the function 4t2+t3/24 t^{2}+t^{3/2}4t2+t3/2.

Problem 9705

Find the maximum blood pressure BBB from the function B(x)=375x2−3750x3B(x)=375 x^{2}-3750 x^{3}B(x)=375x2−3750x3 for 0≤x≤0.100 \leq x \leq 0.100≤x≤0.10.

Problem 9706

Find the correct Newton's method formula for f(x)=1−tan⁡2xf(x)=1-\tan 2xf(x)=1−tan2x and compute x1x_{1}x1​ and x2x_{2}x2​ from x0=1x_{0}=1x0​=1.

Problem 9707

Find the derivative of the function g(x)=∫0xt3+t5dtg(x)=\int_{0}^{x} \sqrt{t^{3}+t^{5}} dtg(x)=∫0x​t3+t5​dt.

Problem 9708

Rewrite the integral ∫194+x2xdx\int_{1}^{9} \frac{4+x^{2}}{\sqrt{x}} d x∫19​x​4+x2​dx using rational exponents and evaluate it.

Problem 9709

Find the absolute max and min of the function f(x)=x2+480xf(x)=x^{2}+\frac{480}{x}f(x)=x2+x480​ for x∈(0,∞)x \in (0, \infty)x∈(0,∞).

Problem 9710

Find the derivative of y=−4(4x5+9)3y=\frac{-4}{(4x^{5}+9)^{3}}y=(4x5+9)3−4​. What is dydx=?\frac{dy}{dx}=?dxdy​=?

Problem 9711

Calculate the Riemann sum M5=∫13xx2+3dxM_{5}=\int_{1}^{3} \frac{x}{x^{2}+3} d xM5​=∫13​x2+3x​dx with n=5n=5n=5.

Problem 9712

Evaluate the integral from 1 to 9 of the function 2x+1x2x + \frac{1}{x}2x+x1​.

Problem 9713

Find the derivative of f(t)=46t2+9f(t)=4 \sqrt{6 t^{2}+9}f(t)=46t2+9​. What is u=g(t)u=g(t)u=g(t) where uuu is a function of ttt?

Problem 9714

Find the derivative of f(t)=46t2+9f(t)=4 \sqrt{6 t^{2}+9}f(t)=46t2+9​. What is u=g(t)u=g(t)u=g(t) for f=h(u)f=h(u)f=h(u)?

Problem 9715

Find the derivative of g(w)=∫0wsin⁡(8+t9) dtg(w)=\int_{0}^{w} \sin(8+t^{9}) \, dtg(w)=∫0w​sin(8+t9)dt using the fundamental theorem of calculus.

Problem 9716

Find x2x_{2}x2​ using Newton's method for f(x)=5−tan⁡(6x)f(x)=5-\tan(6x)f(x)=5−tan(6x) with x0=1x_{0}=1x0​=1. Compute x1x_{1}x1​ first.

Problem 9717

Find the derivative of the function m(t)=−5t(3t8−1)4m(t)=-5 t(3 t^{8}-1)^{4}m(t)=−5t(3t8−1)4. What is m′(t)m'(t)m′(t)?

Problem 9718

Evaluate the integral from 0 to 1 of (3x^{e} + 5e^{x}) dx.

Problem 9719

Find F′(x)F'(x)F′(x) for the function defined by F(x)=∫xx2et7dtF(x)=\int_{x}^{x^{2}} e^{t^{7}} d tF(x)=∫xx2​et7dt.

Problem 9720

Find the slope of the tangent line of f(x)=ln⁡(x5)f(x)=\ln \left(\frac{x}{5}\right)f(x)=ln(5x​) at x=7x=7x=7. Choices: A. 14 B. 17\frac{1}{7}71​ C. 7 D. -7 E. −17-\frac{1}{7}−71​

Problem 9721

Find the limit: lim⁡h→0sec⁡(π6+h)−sec⁡(π6)h\lim _{h \rightarrow 0} \frac{\sec \left(\frac{\pi}{6}+h\right)-\sec \left(\frac{\pi}{6}\right)}{h}limh→0​hsec(6π​+h)−sec(6π​)​.

Problem 9722

Find the derivative of the product: (f(x)⋅g(x))′=f′(x)⋅g(x)+g′(x)⋅f(x)(f(x) \cdot g(x))' = f'(x) \cdot g(x) + g'(x) \cdot f(x)(f(x)⋅g(x))′=f′(x)⋅g(x)+g′(x)⋅f(x) for (x2⋅ex)′(x^{2} \cdot e^{x})'(x2⋅ex)′.

Problem 9723

Find the average rate of change of the cost C(x)=4500+1530x−0.04x3C(x)=4500+1530 x-0.04 x^{3}C(x)=4500+1530x−0.04x3 from year 4 to year 7. Include units.

Problem 9724

Find the average rate of change of cost C(x)=4500+1530x−0.04x3C(x)=4500+1530 x-0.04 x^{3}C(x)=4500+1530x−0.04x3 from year 4 to year 7. Include units.

Problem 9725

Find the second derivative g′′(20)g^{\prime \prime}(20)g′′(20) for the function g(x)=(3−x5)5g(x)=\left(3-\frac{x}{5}\right)^{5}g(x)=(3−5x​)5.

Problem 9726

Find the derivative of y=tan⁡2xy=\tan ^{2} xy=tan2x.

Problem 9727

Calculate the indefinite integral and include the constant CCC. ∫(4x3+8x+9)dx\int(4 x^{3}+8 x+9) dx∫(4x3+8x+9)dx

Problem 9728

Calculate the indefinite integral: ∫(x4+1x4)dx\int\left(\sqrt[4]{x}+\frac{1}{\sqrt[4]{x}}\right) d x∫(4x​+4x​1​)dx (use CCC for the constant).

Problem 9729

Find the indefinite integral: ∫(u+4)(5u+1) du\int(u+4)(5u+1) \, du∫(u+4)(5u+1)du (use CCC for the constant).

Problem 9730

Calculate the indefinite integral and include the constant CCC for integration: ∫1+x+xxdx\int \frac{1+\sqrt{x}+x}{x} d x∫x1+x​+x​dx

Problem 9731

Find the derivative of the function y=e17x+e−17xx y = \frac{e^{17x} + e^{-17x}}{x} y=xe17x+e−17x​.

Problem 9732

Compute the indefinite integral and include the constant CCC: ∫(4+5x)dx \int\left(4+5^{x}\right) d x ∫(4+5x)dx

Problem 9733

Find the maximum value of f(x)=18x2−9f(x)=\frac{18}{x^{2}-9}f(x)=x2−918​ for −3<x<3-3<x<3−3<x<3.

Problem 9734

Find the indefinite integral and evaluate the definite integral from 1 to 6 for 8t3−6t−28 t^{3}-6 t^{-2}8t3−6t−2. Use CCC for the constant.

Problem 9735

Find the volume of a solid with disc cross-sections of radius e−xe^{-x}e−x for 1≤x<∞1 \leq x < \infty1≤x<∞. Round to four decimal places.

Problem 9736

Cactus wrens' weight fits G(t)=31.81+31.8−2.372.37e−0.0125tG(t) = \frac{31.8}{1 + \frac{31.8 - 2.37}{2.37} e^{-0.0125t}}G(t)=1+2.3731.8−2.37​e−0.0125t31.8​. Find G(1)G(1)G(1), G(5)G(5)G(5), G(12)G(12)G(12) and describe growth rate changes.

Problem 9737

Find the total distance traveled by a particle with velocity v(t)=3t−8v(t) = 3t - 8v(t)=3t−8 from t=0t = 0t=0 to t=5t = 5t=5.

Problem 9738

Given the velocity v(t)=3t−8v(t)=3t-8v(t)=3t−8 for 0≤t≤50 \leq t \leq 50≤t≤5, find displacement and total distance traveled in meters.

Problem 9739

Find the volume of the solid in polar coordinates: 0≤r≤R0 \leq r \leq R0≤r≤R, 0≤θ≤2π0 \leq \theta \leq 2\pi0≤θ≤2π, 0≤z≤θ0 \leq z \leq \theta0≤z≤θ.

Problem 9740

Find the derivative of the function $g(x)=\int_{1}^{x} \ln(\varpi+t^{2}) dt$ .

Find the indefinite integral and evaluate the definite integral from 1 to 3:
$\int\left(x^{2}+2 x-3\right) d x$ and $\int_{1}^{3}\left(x^{2}+2 x-3\right) d x$ .

Find the integral $\int_{0}^{4} f(x) \, dx$ for the given piecewise linear function defined by 4 segments.

Calculate the integral from 0 to 9 of the function $4 t^{2}+t^{3/2}$ .

Find the maximum blood pressure $B$ from the function $B(x)=375 x^{2}-3750 x^{3}$ for $0 \leq x \leq 0.10$ .

Find the correct Newton's method formula for $f(x)=1-\tan 2x$ and compute $x_{1}$ and $x_{2}$ from $x_{0}=1$ .

Find the derivative of the function $g(x)=\int_{0}^{x} \sqrt{t^{3}+t^{5}} dt$ .

Rewrite the integral $\int_{1}^{9} \frac{4+x^{2}}{\sqrt{x}} d x$ using rational exponents and evaluate it.

Find the absolute max and min of the function $f(x)=x^{2}+\frac{480}{x}$ for $x \in (0, \infty)$ .

Find the derivative of $y=\frac{-4}{(4x^{5}+9)^{3}}$ . What is $\frac{dy}{dx}=?$

Calculate the Riemann sum $M_{5}=\int_{1}^{3} \frac{x}{x^{2}+3} d x$ with $n=5$ .

Evaluate the integral from 1 to 9 of the function $2x + \frac{1}{x}$ .

Find the derivative of $f(t)=4 \sqrt{6 t^{2}+9}$ . What is $u=g(t)$ where $u$ is a function of $t$ ?

Find the derivative of $f(t)=4 \sqrt{6 t^{2}+9}$ . What is $u=g(t)$ for $f=h(u)$ ?

Find the derivative of $g(w)=\int_{0}^{w} \sin(8+t^{9}) \, dt$ using the fundamental theorem of calculus.

Find $x_{2}$ using Newton's method for $f(x)=5-\tan(6x)$ with $x_{0}=1$ . Compute $x_{1}$ first.

Find the derivative of the function $m(t)=-5 t(3 t^{8}-1)^{4}$ . What is $m'(t)$ ?

Find $F'(x)$ for the function defined by $F(x)=\int_{x}^{x^{2}} e^{t^{7}} d t$ .

Find the slope of the tangent line of $f(x)=\ln \left(\frac{x}{5}\right)$ at $x=7$ . Choices: A. 14 B. $\frac{1}{7}$ C. 7 D. -7 E. $-\frac{1}{7}$

Find the limit: $\lim _{h \rightarrow 0} \frac{\sec \left(\frac{\pi}{6}+h\right)-\sec \left(\frac{\pi}{6}\right)}{h}$ .

Find the derivative of the product: $(f(x) \cdot g(x))' = f'(x) \cdot g(x) + g'(x) \cdot f(x)$ for $(x^{2} \cdot e^{x})'$ .

Find the average rate of change of the cost $C(x)=4500+1530 x-0.04 x^{3}$ from year 4 to year 7. Include units.

Find the average rate of change of cost $C(x)=4500+1530 x-0.04 x^{3}$ from year 4 to year 7. Include units.

Find the second derivative $g^{\prime \prime}(20)$ for the function $g(x)=\left(3-\frac{x}{5}\right)^{5}$ .

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

Calculate the indefinite integral and include the constant $C$ . $\int(4 x^{3}+8 x+9) dx$

Calculate the indefinite integral: $\int\left(\sqrt[4]{x}+\frac{1}{\sqrt[4]{x}}\right) d x$ (use $C$ for the constant).

Find the indefinite integral: $\int(u+4)(5u+1) \, du$ (use $C$ for the constant).

Calculate the indefinite integral and include the constant $C$ for integration: $\int \frac{1+\sqrt{x}+x}{x} d x$

Find the derivative of the function $y = \frac{e^{17x} + e^{-17x}}{x}$ .

Compute the indefinite integral and include the constant $C$ :
$\int\left(4+5^{x}\right) d x$

Find the maximum value of $f(x)=\frac{18}{x^{2}-9}$ for $-3<x<3$ .

Find the indefinite integral and evaluate the definite integral from 1 to 6 for $8 t^{3}-6 t^{-2}$ . Use $C$ for the constant.

Find the volume of a solid with disc cross-sections of radius $e^{-x}$ for $1 \leq x < \infty$ . Round to four decimal places.

Cactus wrens' weight fits $G(t) = \frac{31.8}{1 + \frac{31.8 - 2.37}{2.37} e^{-0.0125t}}$ .
Find $G(1)$ , $G(5)$ , $G(12)$ and describe growth rate changes.

Find the total distance traveled by a particle with velocity $v(t) = 3t - 8$ from $t = 0$ to $t = 5$ .

Given the velocity $v(t)=3t-8$ for $0 \leq t \leq 5$ , find displacement and total distance traveled in meters.

Find the volume of the solid in polar coordinates: $0 \leq r \leq R$ , $0 \leq \theta \leq 2\pi$ , $0 \leq z \leq \theta$ .

Find the derivative of $y=(2x^{4}-x+1)(-x^{5}+1)$ .

Find the price elasticity of demand $E$ for tissues at $p = \$32$ . Interpret $E$ .
Also, find the price for max revenue and demand at that price.

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

Find the derivative of $y=\frac{a x+b}{c x+d}$ .

Approximate $\sqrt{81.1}$ using the tangent line of $f(x)=\sqrt{x}$ at $x=81$ : $L(x)=9+\frac{1}{18}(x-81)$ . Answer to 9 sig. figs.

Does $x$ approach infinity as $2f(x)$ approaches 0 for the function $f(x)=\frac{(x+3)^{2}}{(x+3)(x+1)}$ ?

Does $f(x)=\frac{(x+3)^{2}}{(x+3)(x+1)}$ approach 0 as $x$ approaches infinity?

Approximate $\sqrt{81.1}$ using linear approximation with $f(x)=\sqrt{x}$ at $x=81$ . Find the tangent line $L(x)=$ . Give answer to 9 significant figures.

Find the max and min of $f(x)=2x^{3}+12x^{2}-192x+4$ on $[-8, 5]$ and check its continuity in that interval.

Find the tangent line equation for $y=\frac{x+1}{x-1}$ at the point $(2,3)$ .

Find the derivatives at $x=2$ : 57. $\frac{d}{dx}[f(x)g(x)]$ and 58. $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]$ .

Find the local minimum and maximum of the function $f(x)=2 x^{3}-24 x^{2}+42 x+2$ .

Given $f(2)=3$ , $f'(2)=3$ , $g(2)=3$ , and $g'(2)=\frac{1}{3}$ , find the derivatives at $x=2$ for the given functions.

Evaluate $f(x)=5 x^{2}-10 x+3$ at $x=-1$ and $x=3$ . Does Rolle's Theorem apply? Find $c$ where $f'(c)=0$ .

Find $K^{\prime}(2)$ for $K(x)=[f(x)]^{3}$ given $f(2)=-4$ and $f^{\prime}(2)=8$ .

Find $H^{\prime}(2)$ for $H(x) = f(g(x))$ using the given values of $f$ and $g$ .

Find the approximate slope of $f(x)=\frac{1}{x-1}$ at $x=1.2$ . Choices: $-\frac{1}{25}$ , $\frac{1}{25}$ , 25, $-25$ .

Differentiate $y=5 x^{3}(2-x)^{4}$ using differentiation rules.

An object falls from $188 \mathrm{ft}$ with height $s=188-16 t^{2}$ . Find velocity, time to hit ground, and impact velocity.

Analyze the behavior of $f(x)=\frac{2 x+5}{x+3}$ as $x \rightarrow+\infty$ . What does it approach?

Find local extrema as ordered pairs for: a. $f(x)=x^{3}-6 x^{2}+5$ and b. $g(x)=x \sqrt{9-x^{2}}$ .

What happens to $f(x)=\frac{1}{3 x+5}$ as $x \rightarrow-\frac{5}{3}^{+}$ ? Options: undefined, $-\infty$ , $\infty$ , or $0$ .

Find the derivative of $y=2(2 x-1)^{5/4}(2 x+1)^{3/4}$ using differentiation rules.

Find the limit: $\lim _{x \rightarrow \infty} \frac{x^{2}-6 x-4}{13-7 x^{2}}$ .

Find the first derivative of $g(x)=4x^{3}-12x^{2}-180x$ and then the second derivative. Evaluate $g^{\prime \prime}(5)$ to determine concavity.

Given $f(x)$ and $g(x)$ values, find $P^{\prime}(2)$ where $P(x) = f(x) \times g(x)$ .

Find $R^{\prime}(4)$ for $R(x)=(f(x)-g(x))^{2}$ using the values of $f$ and $g$ given in the table.

Differentiate $f+g$ and evaluate at $x=1$ . Use $f'(1)$ and $g'(1)$ values from the table.

Find the slope of the secant line for $f(x)=x^{3}-6x$ on $[2,6]$ and values of $c$ where $f'(c)=m$ .

Find all points $(x, y)$ on the graph of $f(x)=x^{3}+9 x^{2}+20 x+13$ where the tangent slope is -4.

Find the second derivative $f^{\prime \prime}(x)$ of the function $f(x)=-2 x^{3}-\frac{2}{3} x^{4}-6 x-\frac{3}{2} x^{-2}$ .

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the function $y=-\frac{5}{3} x^{3}-\frac{1}{2} x^{-1}-2 x^{2}$ .