Calculus

Problem 23801

Find the surface area from revolving $y=4-x^{2}$ around the $y$ -axis.

See Solution

Problem 23802

Evaluate the integral $\int(14 x+5) \sqrt{7 x^{2}+5 x} dx$ using the substitution $u=7 x^{2}+5 x$ .

See Solution

Problem 23803

Find the horizontal asymptotes by calculating these limits:
1. $\lim _{x \rightarrow \infty} \frac{-13 x}{14+2 x}=\square$
2. $\lim _{x \rightarrow-\infty} \frac{14 x-14}{x^{3}+13 x-7}=$
3. $\lim _{x \rightarrow \infty} \frac{x^{2}-4 x-3}{5-4 x^{2}}=\square$
4. $\lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}+12 x}}{8-3 x}=\square$
5. $\lim _{x \rightarrow-\infty} \frac{\sqrt{x^{2}+12 x}}{8-3 x}=\square$

See Solution

Problem 23804

Find an antiderivative of $e^{4x+7}$ and verify by differentiating your result.

See Solution

Problem 23805

Calculate the average rate of change of $f(x)=\sqrt{x}$ from $x=2$ to $x=9$ .

See Solution

Problem 23806

Evaluate the integral $\int \sqrt{6 x+2} \, dx$ using the substitution $u = ax + b$ .

See Solution

Problem 23807

Evaluate the integral using a change of variables: $\int 2 x\left(x^{2}-3\right)^{101} d x$ .

See Solution

Problem 23808

Evaluate the integral: $\int \frac{2 x^{3}}{\sqrt{7-8 x^{4}}} d x$ using a change of variables or a table.

See Solution

Problem 23809

Evaluate the integral: $\int \frac{1}{15 x-3} d x$ using a change of variables.

See Solution

Problem 23810

Find the tangent line to $f(x)=\sqrt{x}$ at $(4,2)$ and use it to approximate $f(5)$ . Give your answer as a decimal.

See Solution

Problem 23811

Find $\frac{d y}{d x}$ if $y=\ln (3 x) \csc x$ .

See Solution

Problem 23812

Evaluate the integral $\int x^{2}\left(x^{3}+13\right)^{5} d x$ using a change of variables or a table.

See Solution

Problem 23813

Find the average rate of change of $f(x)=\frac{1}{x+2}$ on $[9, 9+h]$ . Express your answer in terms of $h$ .

See Solution

Problem 23814

Find the derivative $\frac{d y}{d x}$ if $y=\tan^{-1}(2 x^{3})$ .

See Solution

Problem 23815

Evaluate the integral: $\int \frac{1}{12 x-5} d x$ using a change of variables.

See Solution

Problem 23816

Find the average rate of change of $f(x)=x^{2}+3$ from $f(4)$ to $f(10)$ .

See Solution

Problem 23817

Find the acceleration of the particle at $t=4$ seconds given its position $s(t)=\frac{t^{3}}{3}-2 t^{2}+3 t$ .

See Solution

Problem 23818

Evaluate the integral using a change of variables: $\int \sec 17 w \tan 17 w \, dw$ .

See Solution

Problem 23819

Find the average rate of change of $f(x) = 4x^3 + 16$ from $f(6)$ to $f(16)$ .

See Solution

Problem 23820

Evaluate the integral: $\int \cos x \csc^{10} x \, dx$ using a change of variables or a table.

See Solution

Problem 23821

Find the tangent line equation for $f(x)=\frac{3}{x}$ at the point $\left(-6,-\frac{1}{2}\right)$ . The equation is $y=\square$ .

See Solution

Problem 23822

Find the general solution of the differential equation $\frac{d y}{d x}=\frac{x^{2}+2}{3 y^{2}}$ .

See Solution

Problem 23823

Is there an $a$ in $[-4, 6)$ such that the Mean Value Theorem gives a $c$ where $f^{\prime}(c)=\frac{1}{3}$ ? Justify.

See Solution

Problem 23824

Find the absolute extreme values of $f(x)=5 \csc x$ on the interval $\left[-\frac{5 \pi}{6},-\frac{\pi}{6}\right]$ .

See Solution

Problem 23825

Find the rate of change of $f(x)=x^{3}+x+1$ from $x=1$ to $x=3$ . Options: 14, 8, 20, 28.

See Solution

Problem 23826

Let $f$ be a twice differentiable function with $f(2) = 5$ and $f(5) = 2$ . Define $g(x) = f(f(x))$ .
(a) Show there exists $c$ such that $2 < c < 5$ and $f'(c) = -1$ . (b) Prove $g'(2) = g'(5)$ and find $k$ where $2 < k < 5$ and $g''(k) = 0$ . (c) Let $h(x) = f(x) - x$ . Show there exists $r$ such that $2 < r < 5$ and $h(r) = 0$ .

See Solution

Problem 23827

Evaluate the integral: $\int \frac{e^{2 x}}{e^{2 x}+3} d x$ using a change of variables or a table.

See Solution

Problem 23828

Solve $\frac{d y}{d t}=-\frac{3}{4} \sqrt{t}$ with the initial condition $(0,10)$ .

See Solution

Problem 23829

Evaluate the integral using the substitution $u=x^{2}+2$ : $\int 2 x\left(x^{2}+2\right)^{-3} d x=\square$

See Solution

Problem 23830

Find the values of $t \geq 0$ where the acceleration of the particle, given $v(t)=(t-5)(t-2)^{2}$ , is $0$ .

See Solution

Problem 23831

Find the horizontal and vertical asymptotes of the function $f(x)=\frac{x+3}{x+2}$ .

See Solution

Problem 23832

Evaluate the integral $\int_{0}^{\pi / 15} \cos 5 x \, dx$ using a change of variables or a table.

See Solution

Problem 23833

Find the sign of $f^{\prime}(x)$ for $f(x)=\sin^2(x)+\cos(x)$ in intervals around critical points: $\left(\frac{-\pi}{2}, \frac{-\pi}{3}\right)$ , $\left(\frac{-\pi}{3}, 0\right)$ , $\left(0, \frac{\pi}{3}\right)$ , $\left(\frac{\pi}{3}, \frac{\pi}{2}\right)$ .

See Solution

Problem 23834

Determine the concavity of $f(x)$ where $f^{\prime \prime}(x)=x(x-1)(x+1)e^{x(x-1)}$ in the intervals $(-\infty,-1)$ , $(-1,0)$ , $(0,1)$ , $(1,+\infty)$ .

See Solution

Problem 23835

Find the sign of $f'(x)$ for $f(x)=\sin^2(x)+\cos(x)$ in the intervals around its critical points in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ .

See Solution

Problem 23836

Evaluate the integral using the substitution $u=4x$ :
$\int 12 \csc(4x) \cot(4x) dx = \int(\square) du$

See Solution

Problem 23837

Find values of $x$ in $-4 < x < 3$ where $f$ is continuous but not differentiable, given vertical tangent at $x=-2$ .

See Solution

Problem 23838

Evaluate the integral using $u=9-x^{8}$ : $\int \frac{x^{7}}{\left(9-x^{8}\right)^{2}} d x=\square$

See Solution

Problem 23839

A train's location is $s(t)=\frac{110}{t}$ for $3 \leq t \leq 9$ . Graph $s(t)$ and find average velocity from $t=3$ to $t=9$ .

See Solution

Problem 23840

Evaluate the integral: $\int 12 \csc (4 x) \cot (4 x) \, dx = \square$

See Solution

Problem 23841

Evaluate the integral $\int_{0}^{5} 6 e^{2 x} d x$ using a change of variables or a table.

See Solution

Problem 23842

Find and simplify: a. $\frac{f(x)-f(a)}{x-a}$ , b. $\frac{f(x+h)-f(x)}{h}$ for $f(x)=8x^{2}$ .

See Solution

Problem 23843

Evaluate the integral: $\int -\csc(9x-4) \cot(9x-4) \, dx$ . Choose the correct variable change from $x$ to $u$ .

See Solution

Problem 23844

Evaluate the integral using a change of variables: $\int_{0}^{2} \frac{2 x}{\left(x^{2}+2\right)^{3}} d x$ .

See Solution

Problem 23845

Find the limits: (i) $\lim _{x \rightarrow \infty} \frac{4 x^{3}-x+5}{2+3 x^{3}}$ , (ii) $\lim _{x \rightarrow 3} \frac{(x+5)(x-4)}{(x-3)(x+2)}$ (show sign chart).

See Solution

Problem 23846

Rewrite the integral as $u$ substitution: $\int -\csc(9x-4) \cot(9x-4) dx = \int (\square) du$

See Solution

Problem 23847

Evaluate the integral: $\int -\csc(9x-4) \cot(9x-4) \, dx = \square$

See Solution

Problem 23848

Evaluate the integral: $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^{2} x \cos x \, dx$ using a change of variables.

See Solution

Problem 23849

Use the product rule to find the derivative of (5 x^{7}+9 x^{6})(6 e^{\wedge} x-2). No need to expand.

See Solution

Problem 23850

Evaluate the integral $\int r^{2}\left(\frac{r^{3}}{12}+1\right)^{3} d r$ and find the correct substitution for $u$ . Options: A, B, C, D.

See Solution

Problem 23851

Evaluate the integral $\int r^{2}\left(\frac{r^{3}}{12}+1\right)^{3} d r$ and find the correct substitution for $u$ .

See Solution

Problem 23852

Evaluate the integral using geometry: $\int_{1}^{4}|3 x-6| d x$ .

See Solution

Problem 23853

Evaluate the integral: $\int r^{2}\left(\frac{r^{3}}{12}+1\right)^{3} d r=\square$

See Solution

Problem 23854

Invest $\$ 20000$ at $12\%$ continuous interest. Find the average balance over 2 years. Average balance $=$ dollars.

See Solution

Problem 23855

If you invest \$20000 at 12% interest compounded continuously, what is the average balance over 2 years?

See Solution

Problem 23856

Evaluate the integral $\int \frac{x}{\sqrt{x-7}} d x$ and find the correct change of variables from $x$ to $u$ . Choose A, B, C, or D.

See Solution

Problem 23857

A circle's radius grows at $2 \mathrm{~mm/s}$ . Find the area change rate when the radius is $10 \mathrm{~mm}$ .

See Solution

Problem 23858

Determine the continuity of the function $f$ at $x=1$ based on the given piecewise definition.

See Solution

Problem 23859

Rewrite the integral using the substitution $u = \sqrt{x - 7}$ . Find $\int \frac{x}{u} du$ .

See Solution

Problem 23860

Find the area between $y=14 x$ and $y=x^{3}-2 x$ from $x=1$ to $x=2$ . Area =

See Solution

Problem 23861

Find the area between $y=0$ and $y=x^{2}-49$ from $x=0$ to $x=7$ .

See Solution

Problem 23862

Evaluate the integral: $\int \sqrt{\frac{x^{2}-4}{x^{8}}} d x = \square$

See Solution

Problem 23863

Evaluate the integral using substitution: $\int e^{(-\sin x)}(-\cos x) d x=\square$

See Solution

Problem 23864

Evaluate the integral using geometry: $\int_{1}^{4}|4 x-12| d x$ .

See Solution

Problem 23865

Find the area between $y=x^{2}$ and $y=-5$ from $x=-1$ to $x=1$ . Calculate the area: $\text{Area} =$

See Solution

Problem 23866

Evaluate the integral $\int \frac{4}{(4 x-5) \ln (4 x-5)} d x = \square$

See Solution

Problem 23867

Differentiate $x^{6}+y^{4}=e^{y}$ to find $\frac{d y}{d x}$ . Then find the slope $m$ at point $P(-1,0)$ .

See Solution

Problem 23868

Evaluate the integral $\int_{1/5}^{\sqrt{2}/5} \frac{dx}{x \sqrt{25x^2 - 1}}$ using a change of variables.

See Solution

Problem 23869

Find the area between the curves $y=e^{x}$ and $y=x$ from $x=0$ to $x=2$ . Area =

See Solution

Problem 23870

Evaluate the integral $\int \frac{(\arcsin x)^{2} d x}{\sqrt{1-x^{2}}}$ . What is the result?

See Solution

Problem 23871

A circle's radius grows at $3 \mathrm{~mm/s}$ . Find the area change rate when the radius is $13 \mathrm{~mm}$ . Area change: $\mathfrak{\square}$ (Round to nearest thousandth.)

See Solution

Problem 23872

Evaluate the integral: $\int \frac{(2 r-1) \cos \sqrt{3(2 r-1)^{2}+6}}{\sqrt{3(2 r-1)^{2}+6}} d r=\square$

See Solution

Problem 23873

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

See Solution

Problem 23874

Find the integral of $\cot x = \frac{\cos x}{\sin x}$ ; what is the integral of $\cot x$ ? Answer: $\square$ .

See Solution

Problem 23875

Find $f^{\prime}(-1)$ for $f(x)=x^{9} h(x)$ , given $h(-1)=2$ and $h^{\prime}(-1)=5$ .

See Solution

Problem 23876

Evaluate the integral using the Substitution Formula: $\int_{-\pi}^{\pi}-3 \cos ^{2} x \sin x d x=\square$

See Solution

Problem 23877

Express the limit of the Riemann sum $R_{n}=\frac{3}{n} \sum_{i=1}^{n}\left(2+3 \frac{i}{n}\right) \ln\left(2+3 \frac{i}{n}\right)$ as a definite integral as $n \to \infty$ .

See Solution

Problem 23878

Evaluate the integrals using the Substitution Formula:
a. $\int_{0}^{1} t \sqrt{25+24 t} d t$
b. $\int_{1}^{34} t \sqrt{25+24 t} d t$

See Solution

Problem 23879

Evaluate the integrals using the Substitution Formula:
a. $\int_{0}^{1} t \sqrt{25+24 t} d t=\square$
b. $\int_{1}^{34} t \sqrt{25+24 t} d t=\square$

See Solution

Problem 23880

Consider the function $f(x)=\left\{\begin{array}{ll} e^{2 x}, & x \geq 1 \\ e^{3-x}, & x<1 \end{array}\right.$ . Which statement is true?
1. $f$ is decreasing on both sides of $x=1$ .
2. $x=1$ is not a critical point of $f$ .
3. $f(1)$ is a local maximum of $f$ .
4. $f(1)$ is a local minimum of $f$ .
5. $f$ is increasing on both sides of $x=1$ .

See Solution

Problem 23881

Which statements are true? A continuous function is differentiable; a continuous function may not be differentiable; a differentiable function is continuous; a differentiable function may not be continuous.

See Solution

Problem 23882

Find the integral using substitutions: $\int \frac{d x}{\sqrt{2+\sqrt{5+x}}}$ (Hint: Start with $u=\sqrt{5+x}$ ).

See Solution

Problem 23883

Find the area between $R(t)=90+20t$ and $C(t)=70+10t$ from $t=0$ to $t=5$ .

See Solution

Problem 23884

Evaluate the integral using substitution: $\int_{0}^{5 \pi / 4} \tan \frac{x}{5} \, dx$ .

See Solution

Problem 23885

A particle's position on the $y$ -axis is given by $y(t)=6 t-2 t^{3}+10$ .
a. Find when the particle is highest for $0 \leq t \leq 2$ . b. Determine the maximum speed in the same interval.

See Solution

Problem 23886

Find the absolute max and min of $g(x)=3 x^{3} e^{-x}$ on $[-1, 6]$ . Graph it and identify extrema points.

See Solution

Problem 23887

Calculate the area between the curves $h(t)=-30 t^{2}+250 t+800$ and $s(t)=-33 t^{2}+250 t+700$ for $2 \leq t \leq 6$ . What does this area signify?

See Solution

Problem 23888

Find the following limits, if they exist: (i) $\lim _{x \rightarrow 0} 2 \sin (4 x)-\cos 3 x+5$ (ii) $\lim _{x \rightarrow 0} \frac{\tan (3 x)}{x^{2}-3 x}$ (No L'Hospital's Rule) (iii) $\lim _{x \rightarrow 1} \frac{1-x+\ln x}{1+\cos \pi x}$ (Check L'Hospital's Rule applicability first.)

See Solution

Problem 23889

Rewrite the integral using $u$ : $\int_{1}^{4} \frac{3(\ln x)^{2}}{x} d x=\int_{0}^{\square} \square d u$

See Solution

Problem 23890

Differentiate $\sin(x^{2} y^{2})$ and set it equal to the derivative of $x$ . What is the result?

See Solution

Problem 23891

Find the global minimum coordinates of $f(x)=x^{3}-6x+4$ on the interval $[0,3]$ .

See Solution

Problem 23892

Evaluate the integral using substitution: $\int_{0}^{1} \sqrt{x+1} \, dx$ .

See Solution

Problem 23893

Find the derivative of $\frac{\ln x}{x^{2}+1}$ .

See Solution

Problem 23894

Find the derivative of $\frac{\ln x}{x^{2}+1}$ .

See Solution

Problem 23895

Evaluate the integral: $\int_{0}^{\ln(\sqrt{3})} \frac{e^{x} d x}{1+e^{2 x}}$

See Solution

Problem 23896

Use the Substitution Formula to evaluate $\int_{1}^{4} \frac{3(\ln x)^{2}}{x} dx$ . Choose $u$ from options A-D.

See Solution

Problem 23897

Find the volume of the solid formed by revolving the area under $y=\sqrt{9-x^{2}}$ in the first quadrant around the $x$ -axis.

See Solution

Problem 23898

Calculate the area between the curve $y=3^{2-x}$ and the $x$ -axis from $x=0$ to $x=2$ .

See Solution

Problem 23899

Find $f'(2)$ for $f(x)=(g(x) \cdot h(x))^{3}$ given $g(2)=2, g'(2)=4, h(2)=4, h'(2)=4$ . Round to two decimal places.

See Solution

Problem 23900

Find $f^{\prime}(1)$ given that $f(x+h)-f(x)=-5 h x^{2}-2 h x+5 h^{2} x+4 h^{2}-h^{3}$ .

See Solution

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Calculus

Problem 23801

Find the surface area from revolving y=4−x2y=4-x^{2}y=4−x2 around the yyy-axis.

Problem 23802

Evaluate the integral ∫(14x+5)7x2+5xdx\int(14 x+5) \sqrt{7 x^{2}+5 x} dx∫(14x+5)7x2+5x​dx using the substitution u=7x2+5xu=7 x^{2}+5 xu=7x2+5x.

Problem 23803

Problem 23804

Find an antiderivative of e4x+7e^{4x+7}e4x+7 and verify by differentiating your result.

Problem 23805

Calculate the average rate of change of f(x)=xf(x)=\sqrt{x}f(x)=x​ from x=2x=2x=2 to x=9x=9x=9.

Problem 23806

Evaluate the integral ∫6x+2 dx\int \sqrt{6 x+2} \, dx∫6x+2​dx using the substitution u=ax+bu = ax + bu=ax+b.

Problem 23807

Evaluate the integral using a change of variables: ∫2x(x2−3)101dx\int 2 x\left(x^{2}-3\right)^{101} d x∫2x(x2−3)101dx.

Problem 23808

Evaluate the integral: ∫2x37−8x4dx\int \frac{2 x^{3}}{\sqrt{7-8 x^{4}}} d x∫7−8x4​2x3​dx using a change of variables or a table.

Problem 23809

Evaluate the integral: ∫115x−3dx\int \frac{1}{15 x-3} d x∫15x−31​dx using a change of variables.

Problem 23810

Find the tangent line to f(x)=xf(x)=\sqrt{x}f(x)=x​ at (4,2)(4,2)(4,2) and use it to approximate f(5)f(5)f(5). Give your answer as a decimal.

Problem 23811

Find dydx\frac{d y}{d x}dxdy​ if y=ln⁡(3x)csc⁡xy=\ln (3 x) \csc xy=ln(3x)cscx.

Problem 23812

Evaluate the integral ∫x2(x3+13)5dx\int x^{2}\left(x^{3}+13\right)^{5} d x∫x2(x3+13)5dx using a change of variables or a table.

Problem 23813

Find the average rate of change of f(x)=1x+2f(x)=\frac{1}{x+2}f(x)=x+21​ on [9,9+h][9, 9+h][9,9+h]. Express your answer in terms of hhh.

Problem 23814

Find the derivative dydx\frac{d y}{d x}dxdy​ if y=tan⁡−1(2x3)y=\tan^{-1}(2 x^{3})y=tan−1(2x3).

Problem 23815

Evaluate the integral: ∫112x−5dx\int \frac{1}{12 x-5} d x∫12x−51​dx using a change of variables.

Problem 23816

Find the average rate of change of f(x)=x2+3f(x)=x^{2}+3f(x)=x2+3 from f(4)f(4)f(4) to f(10)f(10)f(10).

Problem 23817

Find the acceleration of the particle at t=4t=4t=4 seconds given its position s(t)=t33−2t2+3ts(t)=\frac{t^{3}}{3}-2 t^{2}+3 ts(t)=3t3​−2t2+3t.

Problem 23818

Evaluate the integral using a change of variables: ∫sec⁡17wtan⁡17w dw\int \sec 17 w \tan 17 w \, dw∫sec17wtan17wdw.

Problem 23819

Find the average rate of change of f(x)=4x3+16f(x) = 4x^3 + 16f(x)=4x3+16 from f(6)f(6)f(6) to f(16)f(16)f(16).

Problem 23820

Evaluate the integral: ∫cos⁡xcsc⁡10x dx\int \cos x \csc^{10} x \, dx∫cosxcsc10xdx using a change of variables or a table.

Problem 23821

Find the tangent line equation for f(x)=3xf(x)=\frac{3}{x}f(x)=x3​ at the point (−6,−12)\left(-6,-\frac{1}{2}\right)(−6,−21​). The equation is y=□y=\squarey=□.

Problem 23822

Find the general solution of the differential equation dydx=x2+23y2\frac{d y}{d x}=\frac{x^{2}+2}{3 y^{2}}dxdy​=3y2x2+2​.

Problem 23823

Is there an aaa in [−4,6)[-4, 6)[−4,6) such that the Mean Value Theorem gives a ccc where f′(c)=13f^{\prime}(c)=\frac{1}{3}f′(c)=31​? Justify.

Problem 23824

Find the absolute extreme values of f(x)=5csc⁡xf(x)=5 \csc xf(x)=5cscx on the interval [−5π6,−π6]\left[-\frac{5 \pi}{6},-\frac{\pi}{6}\right][−65π​,−6π​].

Problem 23825

Find the rate of change of f(x)=x3+x+1f(x)=x^{3}+x+1f(x)=x3+x+1 from x=1x=1x=1 to x=3x=3x=3. Options: 14, 8, 20, 28.

Problem 23826

Problem 23827

Evaluate the integral: ∫e2xe2x+3dx\int \frac{e^{2 x}}{e^{2 x}+3} d x∫e2x+3e2x​dx using a change of variables or a table.

Problem 23828

Solve dydt=−34t\frac{d y}{d t}=-\frac{3}{4} \sqrt{t}dtdy​=−43​t​ with the initial condition (0,10)(0,10)(0,10).

Problem 23829

Evaluate the integral using the substitution u=x2+2u=x^{2}+2u=x2+2: ∫2x(x2+2)−3dx=□\int 2 x\left(x^{2}+2\right)^{-3} d x=\square∫2x(x2+2)−3dx=□

Problem 23830

Find the values of t≥0t \geq 0t≥0 where the acceleration of the particle, given v(t)=(t−5)(t−2)2v(t)=(t-5)(t-2)^{2}v(t)=(t−5)(t−2)2, is 000.

Problem 23831

Find the horizontal and vertical asymptotes of the function f(x)=x+3x+2f(x)=\frac{x+3}{x+2}f(x)=x+2x+3​.

Problem 23832

Evaluate the integral ∫0π/15cos⁡5x dx\int_{0}^{\pi / 15} \cos 5 x \, dx∫0π/15​cos5xdx using a change of variables or a table.

Problem 23833

Problem 23834

Determine the concavity of f(x)f(x)f(x) where f′′(x)=x(x−1)(x+1)ex(x−1)f^{\prime \prime}(x)=x(x-1)(x+1)e^{x(x-1)}f′′(x)=x(x−1)(x+1)ex(x−1) in the intervals (−∞,−1)(-\infty,-1)(−∞,−1), (−1,0)(-1,0)(−1,0), (0,1)(0,1)(0,1), (1,+∞)(1,+\infty)(1,+∞).

Problem 23835

Find the sign of f′(x)f'(x)f′(x) for f(x)=sin⁡2(x)+cos⁡(x)f(x)=\sin^2(x)+\cos(x)f(x)=sin2(x)+cos(x) in the intervals around its critical points in (−π2,π2)\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)(−2π​,2π​).

Problem 23836

Evaluate the integral using the substitution u=4xu=4xu=4x: ∫12csc⁡(4x)cot⁡(4x)dx=∫(□)du\int 12 \csc(4x) \cot(4x) dx = \int(\square) du∫12csc(4x)cot(4x)dx=∫(□)du

Problem 23837

Find values of xxx in −4<x<3-4 < x < 3−4<x<3 where fff is continuous but not differentiable, given vertical tangent at x=−2x=-2x=−2.

Problem 23838

Evaluate the integral using u=9−x8u=9-x^{8}u=9−x8: ∫x7(9−x8)2dx=□\int \frac{x^{7}}{\left(9-x^{8}\right)^{2}} d x=\square∫(9−x8)2x7​dx=□

Problem 23839

A train's location is s(t)=110ts(t)=\frac{110}{t}s(t)=t110​ for 3≤t≤93 \leq t \leq 93≤t≤9. Graph s(t)s(t)s(t) and find average velocity from t=3t=3t=3 to t=9t=9t=9.

Problem 23840

Evaluate the integral: ∫12csc⁡(4x)cot⁡(4x) dx=□\int 12 \csc (4 x) \cot (4 x) \, dx = \square∫12csc(4x)cot(4x)dx=□

Problem 23841

Evaluate the integral ∫056e2xdx\int_{0}^{5} 6 e^{2 x} d x∫05​6e2xdx using a change of variables or a table.

Find the surface area from revolving $y=4-x^{2}$ around the $y$ -axis.

Evaluate the integral $\int(14 x+5) \sqrt{7 x^{2}+5 x} dx$ using the substitution $u=7 x^{2}+5 x$ .

Find an antiderivative of $e^{4x+7}$ and verify by differentiating your result.

Calculate the average rate of change of $f(x)=\sqrt{x}$ from $x=2$ to $x=9$ .

Evaluate the integral $\int \sqrt{6 x+2} \, dx$ using the substitution $u = ax + b$ .

Evaluate the integral using a change of variables: $\int 2 x\left(x^{2}-3\right)^{101} d x$ .

Evaluate the integral: $\int \frac{2 x^{3}}{\sqrt{7-8 x^{4}}} d x$ using a change of variables or a table.

Evaluate the integral: $\int \frac{1}{15 x-3} d x$ using a change of variables.

Find the tangent line to $f(x)=\sqrt{x}$ at $(4,2)$ and use it to approximate $f(5)$ . Give your answer as a decimal.

Find $\frac{d y}{d x}$ if $y=\ln (3 x) \csc x$ .

Evaluate the integral $\int x^{2}\left(x^{3}+13\right)^{5} d x$ using a change of variables or a table.

Find the average rate of change of $f(x)=\frac{1}{x+2}$ on $[9, 9+h]$ . Express your answer in terms of $h$ .

Find the derivative $\frac{d y}{d x}$ if $y=\tan^{-1}(2 x^{3})$ .

Evaluate the integral: $\int \frac{1}{12 x-5} d x$ using a change of variables.

Find the average rate of change of $f(x)=x^{2}+3$ from $f(4)$ to $f(10)$ .

Find the acceleration of the particle at $t=4$ seconds given its position $s(t)=\frac{t^{3}}{3}-2 t^{2}+3 t$ .

Evaluate the integral using a change of variables: $\int \sec 17 w \tan 17 w \, dw$ .

Find the average rate of change of $f(x) = 4x^3 + 16$ from $f(6)$ to $f(16)$ .

Evaluate the integral: $\int \cos x \csc^{10} x \, dx$ using a change of variables or a table.

Find the tangent line equation for $f(x)=\frac{3}{x}$ at the point $\left(-6,-\frac{1}{2}\right)$ . The equation is $y=\square$ .

Find the general solution of the differential equation $\frac{d y}{d x}=\frac{x^{2}+2}{3 y^{2}}$ .

Is there an $a$ in $[-4, 6)$ such that the Mean Value Theorem gives a $c$ where $f^{\prime}(c)=\frac{1}{3}$ ? Justify.

Find the absolute extreme values of $f(x)=5 \csc x$ on the interval $\left[-\frac{5 \pi}{6},-\frac{\pi}{6}\right]$ .

Find the rate of change of $f(x)=x^{3}+x+1$ from $x=1$ to $x=3$ . Options: 14, 8, 20, 28.

Evaluate the integral: $\int \frac{e^{2 x}}{e^{2 x}+3} d x$ using a change of variables or a table.

Solve $\frac{d y}{d t}=-\frac{3}{4} \sqrt{t}$ with the initial condition $(0,10)$ .

Evaluate the integral using the substitution $u=x^{2}+2$ : $\int 2 x\left(x^{2}+2\right)^{-3} d x=\square$

Find the values of $t \geq 0$ where the acceleration of the particle, given $v(t)=(t-5)(t-2)^{2}$ , is $0$ .

Find the horizontal and vertical asymptotes of the function $f(x)=\frac{x+3}{x+2}$ .

Evaluate the integral $\int_{0}^{\pi / 15} \cos 5 x \, dx$ using a change of variables or a table.

Determine the concavity of $f(x)$ where $f^{\prime \prime}(x)=x(x-1)(x+1)e^{x(x-1)}$ in the intervals $(-\infty,-1)$ , $(-1,0)$ , $(0,1)$ , $(1,+\infty)$ .

Find the sign of $f'(x)$ for $f(x)=\sin^2(x)+\cos(x)$ in the intervals around its critical points in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ .

Evaluate the integral using the substitution $u=4x$ :
$\int 12 \csc(4x) \cot(4x) dx = \int(\square) du$

Find values of $x$ in $-4 < x < 3$ where $f$ is continuous but not differentiable, given vertical tangent at $x=-2$ .

Evaluate the integral using $u=9-x^{8}$ : $\int \frac{x^{7}}{\left(9-x^{8}\right)^{2}} d x=\square$

A train's location is $s(t)=\frac{110}{t}$ for $3 \leq t \leq 9$ . Graph $s(t)$ and find average velocity from $t=3$ to $t=9$ .

Evaluate the integral: $\int 12 \csc (4 x) \cot (4 x) \, dx = \square$

Evaluate the integral $\int_{0}^{5} 6 e^{2 x} d x$ using a change of variables or a table.

Find and simplify: a. $\frac{f(x)-f(a)}{x-a}$ , b. $\frac{f(x+h)-f(x)}{h}$ for $f(x)=8x^{2}$ .

Evaluate the integral: $\int -\csc(9x-4) \cot(9x-4) \, dx$ . Choose the correct variable change from $x$ to $u$ .

Evaluate the integral using a change of variables: $\int_{0}^{2} \frac{2 x}{\left(x^{2}+2\right)^{3}} d x$ .

Find the limits: (i) $\lim _{x \rightarrow \infty} \frac{4 x^{3}-x+5}{2+3 x^{3}}$ , (ii) $\lim _{x \rightarrow 3} \frac{(x+5)(x-4)}{(x-3)(x+2)}$ (show sign chart).

Rewrite the integral as $u$ substitution: $\int -\csc(9x-4) \cot(9x-4) dx = \int (\square) du$

Evaluate the integral: $\int -\csc(9x-4) \cot(9x-4) \, dx = \square$

Evaluate the integral: $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^{2} x \cos x \, dx$ using a change of variables.

Evaluate the integral $\int r^{2}\left(\frac{r^{3}}{12}+1\right)^{3} d r$ and find the correct substitution for $u$ . Options: A, B, C, D.

Evaluate the integral $\int r^{2}\left(\frac{r^{3}}{12}+1\right)^{3} d r$ and find the correct substitution for $u$ .

Evaluate the integral using geometry: $\int_{1}^{4}|3 x-6| d x$ .

Evaluate the integral: $\int r^{2}\left(\frac{r^{3}}{12}+1\right)^{3} d r=\square$

Invest $\$ 20000$ at $12\%$ continuous interest. Find the average balance over 2 years. Average balance $=$ dollars.

Evaluate the integral $\int \frac{x}{\sqrt{x-7}} d x$ and find the correct change of variables from $x$ to $u$ . Choose A, B, C, or D.

A circle's radius grows at $2 \mathrm{~mm/s}$ . Find the area change rate when the radius is $10 \mathrm{~mm}$ .

Determine the continuity of the function $f$ at $x=1$ based on the given piecewise definition.

Rewrite the integral using the substitution $u = \sqrt{x - 7}$ . Find $\int \frac{x}{u} du$ .

Find the area between $y=14 x$ and $y=x^{3}-2 x$ from $x=1$ to $x=2$ . Area =

Find the area between $y=0$ and $y=x^{2}-49$ from $x=0$ to $x=7$ .

Evaluate the integral: $\int \sqrt{\frac{x^{2}-4}{x^{8}}} d x = \square$

Evaluate the integral using substitution: $\int e^{(-\sin x)}(-\cos x) d x=\square$

Evaluate the integral using geometry: $\int_{1}^{4}|4 x-12| d x$ .

Find the area between $y=x^{2}$ and $y=-5$ from $x=-1$ to $x=1$ . Calculate the area: $\text{Area} =$

Evaluate the integral $\int \frac{4}{(4 x-5) \ln (4 x-5)} d x = \square$

Differentiate $x^{6}+y^{4}=e^{y}$ to find $\frac{d y}{d x}$ . Then find the slope $m$ at point $P(-1,0)$ .

Evaluate the integral $\int_{1/5}^{\sqrt{2}/5} \frac{dx}{x \sqrt{25x^2 - 1}}$ using a change of variables.

Find the area between the curves $y=e^{x}$ and $y=x$ from $x=0$ to $x=2$ . Area =

Evaluate the integral $\int \frac{(\arcsin x)^{2} d x}{\sqrt{1-x^{2}}}$ . What is the result?

A circle's radius grows at $3 \mathrm{~mm/s}$ . Find the area change rate when the radius is $13 \mathrm{~mm}$ . Area change: $\mathfrak{\square}$ (Round to nearest thousandth.)

Evaluate the integral: $\int \frac{(2 r-1) \cos \sqrt{3(2 r-1)^{2}+6}}{\sqrt{3(2 r-1)^{2}+6}} d r=\square$

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

Find the integral of $\cot x = \frac{\cos x}{\sin x}$ ; what is the integral of $\cot x$ ? Answer: $\square$ .