Calculus

Problem 27701

Find the limit: $\lim _{x \rightarrow-\infty} \frac{x^{3}+3}{x^{2}+2 x-9}$ . Options: a. $\infty$ , b. $-\frac{1}{3}$ , c. $-\infty$ , d. 0.

See Solution

Problem 27702

Find the value of $\int_{0}^{\frac{\pi}{6}}(\sec x+\tan x)^{2} d x$ . Choose the correct option.

See Solution

Problem 27703

Find the value of $\int_{0}^{\frac{\pi}{4}}(\cos x+\sec x)^{2} d x$ . Choose the correct answer from the options.

See Solution

Problem 27704

Find the volume of a solid with a base under $y=9-x^{2}$ and square cross-sections: $V=\int_{0}^{3}\left(9-x^{2}\right)^{2} d x$ . Choose the correct integral.

See Solution

Problem 27705

Calculate $\int_{0}^{\frac{\pi}{6}}(\cos x+\sec x)^{2} d x$ and choose the correct answer from the options.

See Solution

Problem 27706

Given the function $f(x)=\frac{x^{4}+1}{x^{2}}$ , determine the true statement about its critical points and concavity.

See Solution

Problem 27707

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=\sqrt{2}$ , $x=-\frac{\pi}{4}$ , and $x=\frac{\pi}{4}$ about the x-axis. Select the correct integral expression.

See Solution

Problem 27708

Find the value of the integral $\int_{0}^{\frac{\pi}{4}}(\cos x+\sec x)^{2} d x$ .

See Solution

Problem 27709

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=0$ , $x=-\frac{\pi}{4}$ , $x=\frac{\pi}{4}$ around $y=2$ .

See Solution

Problem 27710

Find the volume of a solid with a square cross-section under the parabola $y=4-x^{2}$ in the first quadrant.

See Solution

Problem 27711

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=0$ , $x=-\frac{\pi}{4}$ , $x=\frac{\pi}{4}$ around $y=2$ .

See Solution

Problem 27712

Find the area between the curves $y=2 \sqrt{x}$ and $y=2 x^{2}$ . What is the area?

See Solution

Problem 27713

Find the volume of a solid with a base under $y=9-x^{2}$ and square cross-sections perpendicular to the $y$ -axis.

See Solution

Problem 27714

Find the area between the curves $y=7 \sqrt{x}$ and $y=7 x$ . What is the area? Choices: $\frac{7}{6}, \text{none}, \frac{14}{3}, \frac{3}{6}, \frac{5}{6}$ .

See Solution

Problem 27715

Given $f'(3)=0$ , $f'(5)=0$ , $f''(3)=-4$ , and $f''(5)=5$ , determine the true statement about $f$ .

See Solution

Problem 27716

Find the slope of the tangent to $f(x)=-\frac{4}{x+1}$ at $x=1$ . Provide just the numerical answer.

See Solution

Problem 27717

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=\sqrt{2}$ , $x=-\frac{\pi}{4}$ , $x=\frac{\pi}{4}$ around the x-axis. Select one: none, $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \pi(2-\sec^2 x) dx$ , $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} 2\pi x(\sec^2 x - \sqrt{2}) dx$ , $\pi(2-(\sqrt{2}-\sec x)^2) dx$ , $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \pi(\sqrt{2}-\sec x)^2 dx$ .

See Solution

Problem 27718

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=\sqrt{2}$ , $x=0$ , and $x=\frac{\pi}{4}$ around $x=\frac{\pi}{4}$ .

See Solution

Problem 27719

Find the value(s) of $x$ where the function $f(x)=\frac{1}{3} x^{3}-x^{2}+3$ has a horizontal tangent.

See Solution

Problem 27720

Find the volume of a solid with a disk base $x^{2}+y^{2} \leq 1$ and equilateral triangle cross-sections.

See Solution

Problem 27721

Find the volume of a solid with a base under $y=4-x^{2}$ and square cross-sections perpendicular to the $y$ -axis. Select one: $V=\int_{0}^{4}(4-y^{2}) dy$ $V=\int_{0}^{4}(4-x^{2}) dx$ $V=\int_{-2}^{2} 4 x^{2} dx$ $V=\int_{0}^{4} 4(4-y) dy$

See Solution

Problem 27722

Given $f'(3)=0$ , $f'(5)=0$ , $f''(3)=4$ , and $f''(5)=-5$ , which statement about $f$ is true?

See Solution

Problem 27723

Find the volume of a solid with semicircular cross-sections over the region bounded by $y=x^{2}$ and $y=2-x^{2}$ .

See Solution

Problem 27724

Find the value(s) of $x$ where the function $f(x)=\frac{1}{3} x^{3}-x^{2}+3$ has a horizontal tangent.

See Solution

Problem 27725

Find the value of $\int_{0}^{\frac{\pi}{3}}(\sec x+\tan x)^{2} d x$ . Choose one option: 2 $\sqrt{3}$ - $\frac{\pi}{3}$ +2, 2 $\sqrt{3}$ + $\frac{\pi}{3}$ +2, 2 $\sqrt{3}$ - $\frac{\pi}{3}$ -2, none, or 2 $\sqrt{3}$ + $\frac{\pi}{3}$ -2.

See Solution

Problem 27726

Given $f(x)$ with $f^{\prime}(2)=0$ , which must be true? Select one:
1. $f$ changes from increasing to decreasing or vice versa.
2. $f$ has a local max or min at $(2, f(2))$ .
3. The tangent line at $(2, f(2))$ is vertical.
4. The tangent line at $(2, f(2))$ is horizontal.
5. $(2, f(2))$ is an inflection point.

See Solution

Problem 27727

Find the volume of a solid with a disk base $x^{2}+y^{2} \leq 1$ and triangular cross-sections.

See Solution

Problem 27728

Find the rate of change of marginal revenue for $R(x)=70 x+0.5 x^{2}-.001 x^{3}$ at $x=100$ .

See Solution

Problem 27729

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=\sqrt{2}$ , $x=0$ , $x=\frac{\pi}{4}$ around $x=\frac{\pi}{2}$ .

See Solution

Problem 27730

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

See Solution

Problem 27731

Find the marginal cost to produce the $5^{\text{th}}$ unit for $C(x)=4 x^{\frac{5}{2}}+100 x$ . Options: a. 180 b. 32.82 c. 140 d. 25.98

See Solution

Problem 27732

Evaluate the integral $\int_{0}^{\frac{\pi}{4}}(\cos x+\sec x)^{2} \, dx$ .

See Solution

Problem 27733

Calculate the value of $\int_{0}^{\frac{\pi}{4}}(\cos x+\sec x)^{2} dx$ . Choose the correct answer from the options provided.

See Solution

Problem 27734

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=\sqrt{2}$ , $x=0$ , $x=\frac{\pi}{4}$ around $x=-1$ . Choose the correct integral.

See Solution

Problem 27735

Find the volume of a solid with a base under $y=9-x^{2}$ and square cross-sections: $V=\int_{-3}^{3} 4(9-y)^{2} dy$ .

See Solution

Problem 27736

Calculate the value of $\int_{0}^{\frac{\pi}{4}}(\cos x+\sec x)^{2} d x$ and select the correct answer.

See Solution

Problem 27737

Find the instantaneous velocity at $t=3 \mathrm{~s}$ and $t=4 \mathrm{~s}$ , and calculate average and instantaneous acceleration.

See Solution

Problem 27738

Find $f(x)$ if $f^{\prime}(x)=\lim _{\Delta x \rightarrow 0} \frac{-2(x+\Delta x)^{2}+2 x^{2}}{\Delta x}$ . Options: A) $-4 x$ , B) $4 x$ , C) $2 x^{2}$ , D) $-2 x^{2}$ .

See Solution

Problem 27739

Given the function $f(x)=\frac{x^{4}+1}{x^{2}}$ , which statement is true?
1. No vertical asymptotes
2. Local maximum at $x=0$
3. Only one critical point at $x=0$
4. Concave up on $(-\infty, 0) \cup(0, \infty)$ Correct answer: 4.

See Solution

Problem 27740

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{f(2+\Delta x)-f(2)}{\Delta x}$ for $f(x)=\frac{-4}{5-x}$ . Options: A) $\frac{4}{3}$ B) $\frac{-4}{3}$ C) $\frac{4}{9}$ D) $\frac{-4}{9}$

See Solution

Problem 27741

Use the Power Law to solve: $\lim _{y \rightarrow \frac{1}{2}}\left(16 y^{2}-5\right)^{4}=\left(\lim _{y \rightarrow \frac{1}{2}}\left(16 y^{2}-5\right)\right)^{4}$ .

See Solution

Problem 27742

Evaluate the limit: $$\lim _{x \rightarrow-2} \sqrt{x^{2}-5 x+3$$. Identify the limit laws applied.

See Solution

Problem 27743

If $f$ is even and $f^{\prime}(c)=-\frac{2}{5}$ , find $f^{\prime}(-c)$ . A) $-\frac{5}{2}$ B) $-\frac{2}{5}$ C) $\frac{5}{2}$ D) $\frac{2}{5}$

See Solution

Problem 27744

Find $h^{\prime}(-1)$ for $h(x)=f(x)+3g(x)$ given $f^{\prime}(-1)=2g^{\prime}(-1)$ and $\lim_{x \rightarrow 1} \frac{f(x)-f(-1)}{x+1}=-6$ . Choices: A) -15 B) -2 C) -18 D) 0

See Solution

Problem 27745

A turtle's position is given by $x(t)=50 \mathrm{~cm}+(2 \mathrm{~cm/s}) t-(0.0625 \mathrm{~cm/s^2}) t^2$ . Find: (a) avg velocity $t=0$ to $t=10 \mathrm{~s}$ , (b) instant velocity at $t=5 \mathrm{~s}$ , (c) instant acceleration at $t=5 \mathrm{~s}$ . Convert to meters.

See Solution

Problem 27746

Evaluate the limits as $x$ approaches 9 for:
1. $\lim _{x \rightarrow 9}(10 x^{2}-81 x-81)=\square$
2. $\lim _{x \rightarrow 9}(2 x^{2}-162)=\square$
What type of limit is $f(x)$ at $x=9$ ?

See Solution

Problem 27747

Evaluate the limit:
$\lim _{y \rightarrow \frac{1}{2}}\left(16 y^{2}-5\right)^{4}$
using limit laws.

See Solution

Problem 27748

Calculate the limit: $\lim _{x \rightarrow \infty}(x+1)[\ln (x+1)-\ln x]$ .

See Solution

Problem 27749

Prove that $a_{n}=\frac{\sqrt{n+5}}{n}$ approaches 0 as $n$ goes to infinity.

See Solution

Problem 27750

Evaluate the limit: $\lim _{x \rightarrow-2} \sqrt{x^{2}-5 x+3}$ . Identify the limit laws used.

See Solution

Problem 27751

Find the derivative of the function $\frac{1}{x^{3}} - \frac{6}{x}$ .

See Solution

Problem 27752

Determine the convexity and concavity of the function $y=\frac{1}{x^{3}}-\frac{6}{x}$ .

See Solution

Problem 27753

Find the limits of the following sequences:
1. $a_{n}=n\left(\sqrt[3]{8 n^{3}+2 n-1}-2 n-1\right)$
2. $a_{n}=\frac{2}{\sqrt[n]{2^{n}+\pi^{n}+n^{\pi n}}}$
3. $a_{n}=\frac{1^{4}+2^{4}+\cdots+n^{4}}{1^{4}+2^{4}+\cdots+n^{4}+(n+1)^{4}}$

See Solution

Problem 27754

Find the limit: $\lim_{n \to \infty} n\left(\sqrt[3]{8 n^{3}+2 n-1}-2 n-1\right)$ .

See Solution

Problem 27755

Find the limit as $n$ approaches infinity: $\lim_{n \to \infty} a_{n} = \lim_{n \to \infty} \frac{2}{\sqrt[4]{2^{n}+\pi^{n}+n^{10}}}$ .

See Solution

Problem 27756

Investigate the convergence of these series: 1. $\sum_{n=1}^{\infty} \frac{\sqrt{n-2}}{n+5}$ , 2. $\sum_{n=1}^{\infty} \frac{n-1}{(3 n-1)(2 n+5)}$ .

See Solution

Problem 27757

Evaluate the integral of $\cos (\ln x)$ with respect to $x$ : $\int \cos (\ln x) d x$ .

See Solution

Problem 27758

Evaluate the integral $\int_{0}^{1} \frac{r^{3}}{\sqrt{4+r^{2}}} d r$ .

See Solution

Problem 27759

Two planes fly at the same height: A at 250 mi/h east and B at 300 mi/h north. Find the rate of distance change to the airport.

See Solution

Problem 27760

Evaluate the limit: $$\lim _{x \rightarrow 0} \frac{5+e^{x}}{1-e^{x^{2}}$$. If it doesn't exist, write DNE.

See Solution

Problem 27761

A field hockey ball is launched at an angle. Find horizontal and vertical accelerations at: (a) max height, (b) halfway up, (c) halfway down.

See Solution

Problem 27762

Minimize the cost of a cylinder cup with volume 100 cm $^3$ and lateral surface area $S = 2\pi r h$ . Which equation helps?

See Solution

Problem 27763

Find when the object with velocity $v(t)=-t^{3}+4 t^{2}+2 t$ reaches maximum acceleration for $0 \leq t \leq 8$ .

See Solution

Problem 27764

Find the tangent line to $y=x^{3}-2x$ for $x \geq 0$ that minimizes the absolute slope.

See Solution

Problem 27765

Find $f(2)$ , $\lim _{x \rightarrow 2^{-}} f(x)$ , $\lim _{x \rightarrow 2^{+}} f(x)$ , $\lim _{x \rightarrow 2} f(x)$ ; explain any non-existences. Also, find asymptotes of $f(x)=\frac{x+1}{x-1}$ .

See Solution

Problem 27766

The function $F(x)=x^{4}-4 x^{3}+4 x^{2}+6$ is increasing in the intervals: $0<x<1$ and $x>2$ .

See Solution

Problem 27767

Find the limits: (a) $\lim _{x \rightarrow 2}\left(\frac{x^{3}-8}{x-2}\right)$ , (b) $\lim _{x \rightarrow 1}\left(\frac{\sqrt{x^{2}+48}-7}{x-1}\right)$ , (c) $\lim _{x \rightarrow-2}\left(\frac{\frac{1}{2}+\frac{1}{x}}{2+x}\right)$ . Justify without graphs or tables.

See Solution

Problem 27768

Find the tangent line equation to the curve $y=x^{2}-4 x$ at the $y$ -axis intersection. Choose from: (a) $y=8 x-4$ , (b) $y=-4 x$ , (c) $y=-4$ , (d) $y=4 x$ , (e) $y=4 x-8$ .

See Solution

Problem 27769

Find the slope of the tangent line to the curve $y^{3}-x y^{2}=4$ at the point where $y=2$ .

See Solution

Problem 27770

Determine the relative extrema of the function $G(x)=x^{4}-4 x^{2}$ from the options: (a) 1 min, 2 max (b) 1 min, 1 max (c) 0 min, 2 max (d) 2 min, 0 max (e) 2 min, 1 max.

See Solution

Problem 27771

Find the total relative extrema of $F(x)$ if $F^{\prime}(x)=x(x-3)^{2}(x+1)^{4}$ . Options: (a) 0 (b) 1 (c) 2 (d) 3 (e) none.

See Solution

Problem 27772

Find the value of $c$ defined by Rolle's Theorem for $F(x)=2 x^{3}-6 x$ on $0 \leq x \leq \sqrt{3}$ .

See Solution

Problem 27773

Find the derivative of the function $U(x) = x + 2 \sqrt{20 - 2x}$ .

See Solution

Problem 27774

Find the volume of the solid of rotation for $y=4-x^2$ using these methods: (a) disk, (b) washer, (c) shell, (d) shell about $y=2$ .

See Solution

Problem 27775

Find the interval where the average rate of change of $f$ is greatest: A. $[-3,-2]$ , B. $[-3,1]$ , C. $[-2,1]$ , D. $[3,5]$ .

See Solution

Problem 27776

Find $\frac{d^{2} y}{d x^{2}}$ at (1,2) given $\frac{1}{3 x-y} \cdot \frac{d y}{d x}=x+y$ . A. 0 B. 3 C. 4 D. 8

See Solution

Problem 27777

Find the third derivative of $y=2 \sin x-e^{3 x}$ . What is $\frac{d^{3} y}{d x^{3}}$ ? A. $2 \cos x-3 e^{3 x}$ B. $-2 \sin x-9 e^{3 x}$ C. $-2 \cos x-e^{3 x}$ D. $-2 \cos x-27 e^{3 x}$

See Solution

Problem 27778

Find $\frac{d y}{d x}$ for the equation $y^{3}-3 x^{3} y=9$ . Choices: A, B, C, D.

See Solution

Problem 27779

Find $\frac{d y}{d x}$ for $y^{3}-3 x^{3} y=9$ . Choose from: A, B, C, or D.

See Solution

Problem 27780

Find the slope of the tangent line to $y=x \sin x$ at $x=\pi$ . A. $-\pi$ B. -1 C. 0 D. $\pi$

See Solution

Problem 27781

Given $f(-2)=-3$ , $g(-2)=-2$ , $f'(-2)=4$ , $g'(-2)=1$ , find $h'(-2)$ for $h(x)=\frac{f(x)}{g(x)}$ . A. $-\frac{11}{4}$ B. $-\frac{5}{4}$ C. $-\frac{2}{5}$ D. $\frac{5}{4}$

See Solution

Problem 27782

Given $f(x)=g(h(x))$ , find $f^{\prime}(-1)$ using the table values. A. -26 B. -16 C. -8 D. 3

See Solution

Problem 27783

Find the limit: $\lim _{x \rightarrow-2^{-}} \frac{x+2}{|x+2|}$ . What is the value? A. -2 B. -1 C. 1 D. DNE

See Solution

Problem 27784

Evaluate the integral: $\int \sec \theta \tan \theta e^{\sec \theta} d \theta$

See Solution

Problem 27785

Evaluate the integral: $\int_{0}^{1} x \sqrt{1-x^{2}} d x$

See Solution

Problem 27786

Evaluate the integral: $\int \frac{4 x^{6}+5 x^{2}}{x^{3}} d x$

See Solution

Problem 27787

Calculate the integral: $\int_{-1}^{1} \frac{1}{x+2} d x$

See Solution

Problem 27788

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

See Solution

Problem 27789

Write the definite integral as a Riemann Sum limit: $\int_{1}^{3}(x+1) dx$ . Do not evaluate.

See Solution

Problem 27790

Translate the Riemann sum to a definite integral without evaluating: $\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(\cos \left(\frac{\pi}{n} k\right)\right) \frac{\pi}{2 n}$

See Solution

Problem 27791

Evaluate the integral $\int_{\sqrt{2}}^{5}\left[x\left(5 x^{2}-2\right)^{14}\right] d x$ using $u=5 x^{2}-2$ . Find new limits.

See Solution

Problem 27792

Find the time(s), $t$ , when the acceleration of the object with position $s(t)=\frac{1}{4} t^{4}-\frac{7}{6} t^{3}+t^{2}+1$ is 0.

See Solution

Problem 27793

Find the initial population size and size after 9 years for $P(t)=\frac{520}{1+9 e^{-0.4 t}}$ . Round to whole numbers.

See Solution

Problem 27794

Find the soda's temperature after 10 and 20 minutes using $T(x)=-7+25 e^{-0.034 x}$ . Round to the nearest degree.

See Solution

Problem 27795

Find local minima and maxima of $y = \frac{\ln x}{x^{2}}$ by finding critical points and analyzing $y'$ and $y''$ .

See Solution

Problem 27796

Differentiate the function $f(x) = \frac{1}{3x + 5}$ .

See Solution

Problem 27797

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=0$ , $x=-\frac{\pi}{4}$ , $x=\frac{\pi}{4}$ around $y=-2$ . Select one:
$\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \pi\left((\sec x+2)^{2}-4\right) dx$
none
$\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \pi\left(\sec^{2} x-2\right) dx$
$\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \pi(\sec x+4)^{2} dx$
$\int_{\frac{\pi}{4}} \pi(\sec x-2)^{2} dx$

See Solution

Problem 27798

Differentiate $f(x) = \frac{1}{3x + 5}$ using first principles: $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$ .

See Solution

Problem 27799

Find the derivative $f'(x)$ of the function $f(x) = 5 \sqrt{x}$ using the limit definition.

See Solution

Problem 27800

Find where the function $y = \frac{2x}{\sqrt{x^2 + 1}}$ is concave up and concave down.

See Solution

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Calculus

Problem 27701

Find the limit: lim⁡x→−∞x3+3x2+2x−9\lim _{x \rightarrow-\infty} \frac{x^{3}+3}{x^{2}+2 x-9}limx→−∞​x2+2x−9x3+3​. Options: a. ∞\infty∞, b. −13-\frac{1}{3}−31​, c. −∞-\infty−∞, d. 0.

Problem 27702

Find the value of ∫0π6(sec⁡x+tan⁡x)2dx\int_{0}^{\frac{\pi}{6}}(\sec x+\tan x)^{2} d x∫06π​​(secx+tanx)2dx. Choose the correct option.

Problem 27703

Find the value of ∫0π4(cos⁡x+sec⁡x)2dx\int_{0}^{\frac{\pi}{4}}(\cos x+\sec x)^{2} d x∫04π​​(cosx+secx)2dx. Choose the correct answer from the options.

Problem 27704

Find the volume of a solid with a base under y=9−x2y=9-x^{2}y=9−x2 and square cross-sections: V=∫03(9−x2)2dxV=\int_{0}^{3}\left(9-x^{2}\right)^{2} d xV=∫03​(9−x2)2dx. Choose the correct integral.

Problem 27705

Calculate ∫0π6(cos⁡x+sec⁡x)2dx\int_{0}^{\frac{\pi}{6}}(\cos x+\sec x)^{2} d x∫06π​​(cosx+secx)2dx and choose the correct answer from the options.

Problem 27706

Given the function f(x)=x4+1x2f(x)=\frac{x^{4}+1}{x^{2}}f(x)=x2x4+1​, determine the true statement about its critical points and concavity.

Problem 27707

Find the volume of the solid formed by revolving the area between y=sec⁡xy=\sec xy=secx, y=2y=\sqrt{2}y=2​, x=−π4x=-\frac{\pi}{4}x=−4π​, and x=π4x=\frac{\pi}{4}x=4π​ about the x-axis. Select the correct integral expression.

Problem 27708

Find the value of the integral ∫0π4(cos⁡x+sec⁡x)2dx\int_{0}^{\frac{\pi}{4}}(\cos x+\sec x)^{2} d x∫04π​​(cosx+secx)2dx.

Problem 27709

Find the volume of the solid formed by revolving the area between y=sec⁡xy=\sec xy=secx, y=0y=0y=0, x=−π4x=-\frac{\pi}{4}x=−4π​, x=π4x=\frac{\pi}{4}x=4π​ around y=2y=2y=2.

Problem 27710

Find the volume of a solid with a square cross-section under the parabola y=4−x2y=4-x^{2}y=4−x2 in the first quadrant.

Problem 27711

Find the volume of the solid formed by revolving the area between y=sec⁡xy=\sec xy=secx, y=0y=0y=0, x=−π4x=-\frac{\pi}{4}x=−4π​, x=π4x=\frac{\pi}{4}x=4π​ around y=2y=2y=2.

Problem 27712

Find the area between the curves y=2xy=2 \sqrt{x}y=2x​ and y=2x2y=2 x^{2}y=2x2. What is the area?

Problem 27713

Find the volume of a solid with a base under y=9−x2y=9-x^{2}y=9−x2 and square cross-sections perpendicular to the yyy-axis.

Problem 27714

Find the area between the curves y=7xy=7 \sqrt{x}y=7x​ and y=7xy=7 xy=7x. What is the area? Choices: 76,none,143,36,56\frac{7}{6}, \text{none}, \frac{14}{3}, \frac{3}{6}, \frac{5}{6}67​,none,314​,63​,65​.

Problem 27715

Given f′(3)=0f'(3)=0f′(3)=0, f′(5)=0f'(5)=0f′(5)=0, f′′(3)=−4f''(3)=-4f′′(3)=−4, and f′′(5)=5f''(5)=5f′′(5)=5, determine the true statement about fff.

Problem 27716

Find the slope of the tangent to f(x)=−4x+1f(x)=-\frac{4}{x+1}f(x)=−x+14​ at x=1x=1x=1. Provide just the numerical answer.

Problem 27717

Problem 27718

Find the volume of the solid formed by revolving the area between y=sec⁡xy=\sec xy=secx, y=2y=\sqrt{2}y=2​, x=0x=0x=0, and x=π4x=\frac{\pi}{4}x=4π​ around x=π4x=\frac{\pi}{4}x=4π​.

Problem 27719

Find the value(s) of xxx where the function f(x)=13x3−x2+3f(x)=\frac{1}{3} x^{3}-x^{2}+3f(x)=31​x3−x2+3 has a horizontal tangent.

Problem 27720

Find the volume of a solid with a disk base x2+y2≤1x^{2}+y^{2} \leq 1x2+y2≤1 and equilateral triangle cross-sections.

Problem 27721

Problem 27722

Given f′(3)=0f'(3)=0f′(3)=0, f′(5)=0f'(5)=0f′(5)=0, f′′(3)=4f''(3)=4f′′(3)=4, and f′′(5)=−5f''(5)=-5f′′(5)=−5, which statement about fff is true?

Problem 27723

Find the volume of a solid with semicircular cross-sections over the region bounded by y=x2y=x^{2}y=x2 and y=2−x2y=2-x^{2}y=2−x2.

Problem 27724

Find the value(s) of xxx where the function f(x)=13x3−x2+3f(x)=\frac{1}{3} x^{3}-x^{2}+3f(x)=31​x3−x2+3 has a horizontal tangent.

Problem 27725

Problem 27726

Problem 27727

Find the volume of a solid with a disk base x2+y2≤1x^{2}+y^{2} \leq 1x2+y2≤1 and triangular cross-sections.

Problem 27728

Find the rate of change of marginal revenue for R(x)=70x+0.5x2−.001x3R(x)=70 x+0.5 x^{2}-.001 x^{3}R(x)=70x+0.5x2−.001x3 at x=100x=100x=100.

Problem 27729

Find the volume of the solid formed by revolving the area between y=sec⁡xy=\sec xy=secx, y=2y=\sqrt{2}y=2​, x=0x=0x=0, x=π4x=\frac{\pi}{4}x=4π​ around x=π2x=\frac{\pi}{2}x=2π​.

Problem 27730

Find h′(4)h^{\prime}(4)h′(4) for h(x)=3(f(x))2−g(x)+2xh(x)=3(f(x))^{2}-g(x)+2 xh(x)=3(f(x))2−g(x)+2x given f(4)=2,f′(4)=−1,g′(4)=−3f(4)=2, f^{\prime}(4)=-1, g^{\prime}(4)=-3f(4)=2,f′(4)=−1,g′(4)=−3.

Problem 27731

Find the marginal cost to produce the 5th5^{\text{th}}5th unit for C(x)=4x52+100xC(x)=4 x^{\frac{5}{2}}+100 xC(x)=4x25​+100x. Options: a. 180 b. 32.82 c. 140 d. 25.98

Problem 27732

Evaluate the integral ∫0π4(cos⁡x+sec⁡x)2 dx\int_{0}^{\frac{\pi}{4}}(\cos x+\sec x)^{2} \, dx∫04π​​(cosx+secx)2dx.

Problem 27733

Calculate the value of ∫0π4(cos⁡x+sec⁡x)2dx\int_{0}^{\frac{\pi}{4}}(\cos x+\sec x)^{2} dx∫04π​​(cosx+secx)2dx. Choose the correct answer from the options provided.

Problem 27734

Find the volume of the solid formed by revolving the area between y=sec⁡xy=\sec xy=secx, y=2y=\sqrt{2}y=2​, x=0x=0x=0, x=π4x=\frac{\pi}{4}x=4π​ around x=−1x=-1x=−1. Choose the correct integral.

Problem 27735

Find the volume of a solid with a base under y=9−x2y=9-x^{2}y=9−x2 and square cross-sections: V=∫−334(9−y)2dyV=\int_{-3}^{3} 4(9-y)^{2} dyV=∫−33​4(9−y)2dy.

Problem 27736

Calculate the value of ∫0π4(cos⁡x+sec⁡x)2dx\int_{0}^{\frac{\pi}{4}}(\cos x+\sec x)^{2} d x∫04π​​(cosx+secx)2dx and select the correct answer.

Problem 27737

Find the instantaneous velocity at t=3 st=3 \mathrm{~s}t=3 s and t=4 st=4 \mathrm{~s}t=4 s, and calculate average and instantaneous acceleration.

Problem 27738

Find f(x)f(x)f(x) if f′(x)=lim⁡Δx→0−2(x+Δx)2+2x2Δxf^{\prime}(x)=\lim _{\Delta x \rightarrow 0} \frac{-2(x+\Delta x)^{2}+2 x^{2}}{\Delta x}f′(x)=limΔx→0​Δx−2(x+Δx)2+2x2​. Options: A) −4x-4 x−4x, B) 4x4 x4x, C) 2x22 x^{2}2x2, D) −2x2-2 x^{2}−2x2.

Problem 27739

Given the function f(x)=x4+1x2f(x)=\frac{x^{4}+1}{x^{2}}f(x)=x2x4+1​, which statement is true? 1. No vertical asymptotes 2. Local maximum at x=0x=0x=0 3. Only one critical point at x=0x=0x=0 4. Concave up on (−∞,0)∪(0,∞)(-\infty, 0) \cup(0, \infty)(−∞,0)∪(0,∞) Correct answer: 4.

Problem 27740

Problem 27741

Use the Power Law to solve: lim⁡y→12(16y2−5)4=(lim⁡y→12(16y2−5))4\lim _{y \rightarrow \frac{1}{2}}\left(16 y^{2}-5\right)^{4}=\left(\lim _{y \rightarrow \frac{1}{2}}\left(16 y^{2}-5\right)\right)^{4}y→21​lim​(16y2−5)4=(y→21​lim​(16y2−5))4.

Problem 27742

Evaluate the limit: $$\lim _{x \rightarrow-2} \sqrt{x^{2}-5 x+3$$. Identify the limit laws applied.

Find the limit: $\lim _{x \rightarrow-\infty} \frac{x^{3}+3}{x^{2}+2 x-9}$ . Options: a. $\infty$ , b. $-\frac{1}{3}$ , c. $-\infty$ , d. 0.

Find the value of $\int_{0}^{\frac{\pi}{6}}(\sec x+\tan x)^{2} d x$ . Choose the correct option.

Find the value of $\int_{0}^{\frac{\pi}{4}}(\cos x+\sec x)^{2} d x$ . Choose the correct answer from the options.

Find the volume of a solid with a base under $y=9-x^{2}$ and square cross-sections: $V=\int_{0}^{3}\left(9-x^{2}\right)^{2} d x$ . Choose the correct integral.

Calculate $\int_{0}^{\frac{\pi}{6}}(\cos x+\sec x)^{2} d x$ and choose the correct answer from the options.

Given the function $f(x)=\frac{x^{4}+1}{x^{2}}$ , determine the true statement about its critical points and concavity.

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=\sqrt{2}$ , $x=-\frac{\pi}{4}$ , and $x=\frac{\pi}{4}$ about the x-axis. Select the correct integral expression.

Find the value of the integral $\int_{0}^{\frac{\pi}{4}}(\cos x+\sec x)^{2} d x$ .

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=0$ , $x=-\frac{\pi}{4}$ , $x=\frac{\pi}{4}$ around $y=2$ .

Find the volume of a solid with a square cross-section under the parabola $y=4-x^{2}$ in the first quadrant.

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=0$ , $x=-\frac{\pi}{4}$ , $x=\frac{\pi}{4}$ around $y=2$ .

Find the area between the curves $y=2 \sqrt{x}$ and $y=2 x^{2}$ . What is the area?

Find the volume of a solid with a base under $y=9-x^{2}$ and square cross-sections perpendicular to the $y$ -axis.

Find the area between the curves $y=7 \sqrt{x}$ and $y=7 x$ . What is the area? Choices: $\frac{7}{6}, \text{none}, \frac{14}{3}, \frac{3}{6}, \frac{5}{6}$ .

Given $f'(3)=0$ , $f'(5)=0$ , $f''(3)=-4$ , and $f''(5)=5$ , determine the true statement about $f$ .

Find the slope of the tangent to $f(x)=-\frac{4}{x+1}$ at $x=1$ . Provide just the numerical answer.

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=\sqrt{2}$ , $x=0$ , and $x=\frac{\pi}{4}$ around $x=\frac{\pi}{4}$ .

Find the value(s) of $x$ where the function $f(x)=\frac{1}{3} x^{3}-x^{2}+3$ has a horizontal tangent.

Find the volume of a solid with a disk base $x^{2}+y^{2} \leq 1$ and equilateral triangle cross-sections.

Given $f'(3)=0$ , $f'(5)=0$ , $f''(3)=4$ , and $f''(5)=-5$ , which statement about $f$ is true?

Find the volume of a solid with semicircular cross-sections over the region bounded by $y=x^{2}$ and $y=2-x^{2}$ .

Find the value(s) of $x$ where the function $f(x)=\frac{1}{3} x^{3}-x^{2}+3$ has a horizontal tangent.

Find the volume of a solid with a disk base $x^{2}+y^{2} \leq 1$ and triangular cross-sections.

Find the rate of change of marginal revenue for $R(x)=70 x+0.5 x^{2}-.001 x^{3}$ at $x=100$ .

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=\sqrt{2}$ , $x=0$ , $x=\frac{\pi}{4}$ around $x=\frac{\pi}{2}$ .

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

Find the marginal cost to produce the $5^{\text{th}}$ unit for $C(x)=4 x^{\frac{5}{2}}+100 x$ . Options: a. 180 b. 32.82 c. 140 d. 25.98

Evaluate the integral $\int_{0}^{\frac{\pi}{4}}(\cos x+\sec x)^{2} \, dx$ .

Calculate the value of $\int_{0}^{\frac{\pi}{4}}(\cos x+\sec x)^{2} dx$ . Choose the correct answer from the options provided.

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=\sqrt{2}$ , $x=0$ , $x=\frac{\pi}{4}$ around $x=-1$ . Choose the correct integral.

Find the volume of a solid with a base under $y=9-x^{2}$ and square cross-sections: $V=\int_{-3}^{3} 4(9-y)^{2} dy$ .

Calculate the value of $\int_{0}^{\frac{\pi}{4}}(\cos x+\sec x)^{2} d x$ and select the correct answer.

Find the instantaneous velocity at $t=3 \mathrm{~s}$ and $t=4 \mathrm{~s}$ , and calculate average and instantaneous acceleration.

Find $f(x)$ if $f^{\prime}(x)=\lim _{\Delta x \rightarrow 0} \frac{-2(x+\Delta x)^{2}+2 x^{2}}{\Delta x}$ . Options: A) $-4 x$ , B) $4 x$ , C) $2 x^{2}$ , D) $-2 x^{2}$ .

Given the function $f(x)=\frac{x^{4}+1}{x^{2}}$ , which statement is true?
1. No vertical asymptotes
2. Local maximum at $x=0$
3. Only one critical point at $x=0$
4. Concave up on $(-\infty, 0) \cup(0, \infty)$ Correct answer: 4.

Use the Power Law to solve: $\lim _{y \rightarrow \frac{1}{2}}\left(16 y^{2}-5\right)^{4}=\left(\lim _{y \rightarrow \frac{1}{2}}\left(16 y^{2}-5\right)\right)^{4}$ .

If $f$ is even and $f^{\prime}(c)=-\frac{2}{5}$ , find $f^{\prime}(-c)$ . A) $-\frac{5}{2}$ B) $-\frac{2}{5}$ C) $\frac{5}{2}$ D) $\frac{2}{5}$

Evaluate the limit:
$\lim _{y \rightarrow \frac{1}{2}}\left(16 y^{2}-5\right)^{4}$
using limit laws.

Calculate the limit: $\lim _{x \rightarrow \infty}(x+1)[\ln (x+1)-\ln x]$ .

Prove that $a_{n}=\frac{\sqrt{n+5}}{n}$ approaches 0 as $n$ goes to infinity.

Evaluate the limit: $\lim _{x \rightarrow-2} \sqrt{x^{2}-5 x+3}$ . Identify the limit laws used.

Find the derivative of the function $\frac{1}{x^{3}} - \frac{6}{x}$ .

Determine the convexity and concavity of the function $y=\frac{1}{x^{3}}-\frac{6}{x}$ .

Find the limit: $\lim_{n \to \infty} n\left(\sqrt[3]{8 n^{3}+2 n-1}-2 n-1\right)$ .

Find the limit as $n$ approaches infinity: $\lim_{n \to \infty} a_{n} = \lim_{n \to \infty} \frac{2}{\sqrt[4]{2^{n}+\pi^{n}+n^{10}}}$ .

Investigate the convergence of these series: 1. $\sum_{n=1}^{\infty} \frac{\sqrt{n-2}}{n+5}$ , 2. $\sum_{n=1}^{\infty} \frac{n-1}{(3 n-1)(2 n+5)}$ .

Evaluate the integral of $\cos (\ln x)$ with respect to $x$ : $\int \cos (\ln x) d x$ .

Evaluate the integral $\int_{0}^{1} \frac{r^{3}}{\sqrt{4+r^{2}}} d r$ .

Minimize the cost of a cylinder cup with volume 100 cm $^3$ and lateral surface area $S = 2\pi r h$ . Which equation helps?

Find when the object with velocity $v(t)=-t^{3}+4 t^{2}+2 t$ reaches maximum acceleration for $0 \leq t \leq 8$ .

Find the tangent line to $y=x^{3}-2x$ for $x \geq 0$ that minimizes the absolute slope.

The function $F(x)=x^{4}-4 x^{3}+4 x^{2}+6$ is increasing in the intervals: $0<x<1$ and $x>2$ .

Find the tangent line equation to the curve $y=x^{2}-4 x$ at the $y$ -axis intersection. Choose from: (a) $y=8 x-4$ , (b) $y=-4 x$ , (c) $y=-4$ , (d) $y=4 x$ , (e) $y=4 x-8$ .

Find the slope of the tangent line to the curve $y^{3}-x y^{2}=4$ at the point where $y=2$ .

Determine the relative extrema of the function $G(x)=x^{4}-4 x^{2}$ from the options: (a) 1 min, 2 max (b) 1 min, 1 max (c) 0 min, 2 max (d) 2 min, 0 max (e) 2 min, 1 max.

Find the total relative extrema of $F(x)$ if $F^{\prime}(x)=x(x-3)^{2}(x+1)^{4}$ . Options: (a) 0 (b) 1 (c) 2 (d) 3 (e) none.

Find the value of $c$ defined by Rolle's Theorem for $F(x)=2 x^{3}-6 x$ on $0 \leq x \leq \sqrt{3}$ .

Find the derivative of the function $U(x) = x + 2 \sqrt{20 - 2x}$ .

Find the volume of the solid of rotation for $y=4-x^2$ using these methods: (a) disk, (b) washer, (c) shell, (d) shell about $y=2$ .