Calculus

Problem 27401

Given $f(x)=\sin x$ and $a=\frac{5 \pi}{3}$ , write the power series in summation notation. Choose A, B, C, or D.

See Solution

Problem 27402

Approximate $\frac{1}{1.19^{2}}$ using the first four nonzero terms of the Taylor series for $f(x)=(1+x)^{-2}$ . First term is $\square$ .

See Solution

Problem 27403

Find the first four nonzero terms of the Taylor series for $(1+2x)^{-2}$ using $(1+x)^{-2}$ .

See Solution

Problem 27404

Approximate $1.17^{-4/5}$ using the first four terms of the Taylor series for $f(x)=(1+x)^{-4/5}$ centered at 0.

See Solution

Problem 27405

Find the first four nonzero terms of the Taylor series for $f(x)=(1+x)^{-4/5}$ and use them to approximate $1.17^{-4/5}$ .

See Solution

Problem 27406

A sandbag is dropped from a balloon at $300 \mathrm{~m}$ and $13 \mathrm{~m/s}$ up. Find its speed on impact and time to ground.
a. Speed: $[-77.8 \mathrm{~m/s}]$ b. Time: $[9.27 \mathrm{~s}]$

See Solution

Problem 27407

Find the first four nonzero terms of the Maclaurin series for $f(x)=\ln(1+4x)$ and its interval of convergence.

See Solution

Problem 27408

Find the first four nonzero terms of the Taylor series for $\frac{1}{(5+8 x)^{2}}$ centered at 0 using $(1+x)^{-2}$ .

See Solution

Problem 27409

Find the limit: $\lim _{x \rightarrow 0} \frac{3 \tan^{-1} x - 3x + x^3}{5x^5}$ .

See Solution

Problem 27410

Calculate the distance change rate from a particle on $y=\sin x$ to the origin, given $\frac{d x}{d t}=2 \mathrm{~cm/s}$ .

See Solution

Problem 27411

Find the volume of the solid formed by rotating the area between $y=(2+x)^{2}$ , $x=0$ , and $x=1$ around the $x$ -axis.

See Solution

Problem 27412

Differentiate the Taylor series for $f(x)=e^{-3 x}$ , identify the function, and find the interval of convergence.

See Solution

Problem 27413

Find the derivative of $f(x) = \sin(x^6)$ , which is $f'(x) = 6x^5 \cos(x^6)$ . Determine the interval of convergence.

See Solution

Problem 27414

Determine if $f(x)=\frac{x^{2}}{x-1}$ has local or absolute extrema at the given points. Select the correct statement.

See Solution

Problem 27415

Determine where the function $f(x)=\frac{x^{2}-2}{x^{2}-1}$ is concave down using $f^{\prime \prime}=\frac{2(3x^{2}+1)}{(x^{2}-1)^{3}}$ .

See Solution

Problem 27416

Find the limit: $\lim _{x \rightarrow 0} \frac{5 x^{3}+8 x^{2}}{3 x^{4}-16 x^{2}}$ .

See Solution

Problem 27417

Find the function for the series $\sum_{k=0}^{\infty} 5 e^{-3 k x}$ and its interval of convergence.

See Solution

Problem 27418

Approximate $\sqrt[3]{1.19}$ using the first four nonzero terms of the Taylor series for $f(x)=\sqrt[3]{1+x}$ at 0.

See Solution

Problem 27419

Find the limit: $L = \lim _{x \rightarrow 0} \frac{5 x^{3}+8 x^{2}}{3 x^{4}-16 x^{2}}$ .

See Solution

Problem 27420

Find the limit: $\lim _{x \rightarrow a} \frac{x^{3}-a^{3}}{x^{4}-a^{4}}$

See Solution

Problem 27421

Find the value of $c$ that satisfies Rolle's Theorem for $f(x)=2 x^{3}-5 x^{2}-4 x+3$ on $x \in\left[\frac{1}{3}, 3\right]$ .

See Solution

Problem 27422

Find the values of $c$ for Rolle's Theorem with $f(x)=\cos(2x)$ on $[0, 2\pi]$ . Options: a, b, c, d.

See Solution

Problem 27423

Find values of $c$ that satisfy Rolle's Theorem for $f(x)=\cos(2x)$ on $[0, 2\pi]$ . Options: a, b, c, d.

See Solution

Problem 27424

Find all $x$ where the curve $y=4x^4-2x^2-4$ has a horizontal tangent. Choices: a, b, c, d.

See Solution

Problem 27425

Find the value of $c$ in Rolle's Theorem for $f(x)=e^{x} \cos x$ on $[0, \pi]$ . Options: a. $\frac{\pi}{2}$ b. $\frac{3 \pi}{4}$ c. $\frac{\pi}{4}$ d. None

See Solution

Problem 27426

Find the limit: $\lim _{x \rightarrow 0} \frac{\operatorname{Sin} 2 x}{2 x^{2}+x}$ .

See Solution

Problem 27427

Find the limit as $y$ approaches infinity for the expression $\frac{3y + 7}{y^2 - 2}$ .

See Solution

Problem 27428

Differentiate the Taylor series for $f(x)=e^{-3x}$ , identify the function, and find the interval of convergence.

See Solution

Problem 27429

Find the function for the power series: $\sum_{k=1}^{\infty}(-1)^{k} \frac{k x^{k+3}}{3^{k}}$ . What is $f(x)=\square$ ?

See Solution

Problem 27430

Find the first four nonzero terms of the Taylor series for sin $\left(\frac{1}{2}\right)$ in order. $\sin \left(\frac{1}{2}\right)=(\square)+(\square)+(\square)+(\square)+\cdots$

See Solution

Problem 27431

Find the first four nonzero terms of the Taylor series for $e^{3}$ . Write them in order:
$e^{3}=\square+\square+\square+\square+\cdots$

See Solution

Problem 27432

Find the function for the power series: $\sum_{k=0}^{\infty}(-1)^{k} \frac{x^{3 k}}{4^{k}}$ . What is $f(x)=\square$ ?

See Solution

Problem 27433

Express $\ln(2)$ as the sum of the first four nonzero terms of its Taylor series: $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots$ .

See Solution

Problem 27434

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin(5x)}{\sin(3x)}$ .

See Solution

Problem 27435

Find the Taylor series for $\mathrm{Si}$ and approximate $\mathrm{Si}(0.7)$ to the nearest thousandth.

See Solution

Problem 27436

Find the area between the curve $y=\sqrt{1-x^{4}}$ and the $x$ -axis. Options: (A) 0.886 (B) 1.253 (C) 1.414 (D) 1.571 (E) 1.748

See Solution

Problem 27437

If the average value of $f$ on $[a, b]$ is 10, find $\int_{a}^{b} f(x) d x$ . Options: (A) $\frac{10}{b-a}$ , (B) $\frac{f(a)+f(b)}{10}$ , (C) $10 b-10 a$ , (D) $\frac{b-a}{10}$ , (E) $\frac{f(a)+f(b)}{20}$ .

See Solution

Problem 27438

Find the average value of $f(x)=\cos x$ over the interval $[1,5]$ . Choose the correct answer from the options.

See Solution

Problem 27439

Find the linearization of $f(x)=\int_{\pi}^{x} \cos ^{3} t \, dt$ at $x=\pi$ . What is it?

See Solution

Problem 27440

Find $f(5)$ if $f^{\prime}(x)=\ln(2+\sin x)$ and $f(3)=4$ . Choose from: (A) 0.040 (B) 0.272 (C) 0.961 (D) 4.555 (E) 6.667.

See Solution

Problem 27441

Find $\lim _{h \rightarrow 0} \frac{1}{h} \int_{x}^{x+h} f(t) d t$ . Choose from: (A) 0, (B) 1, (C) $f^{\prime}(x)$ , (D) $f(x)$ , (E) nonexistent.

See Solution

Problem 27442

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin (\pi-2 x)}{x}$ .

See Solution

Problem 27443

Determine the intervals where the function $f$ is increasing given that $f^{\prime}(x)=\frac{x^{2}-1}{x^{2}+2}$ .

See Solution

Problem 27444

Calculate the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{x}-\sqrt[3]{x}}{\sqrt{x}+\sqrt[3]{x}}$ .

See Solution

Problem 27445

Find the average rate of change of $f(x)=\cot x$ on the interval $\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$ .

See Solution

Problem 27446

Find the slope of the normal line for the function $f(x)=\frac{5}{x^{2}+2}$ at the point where $x=2$ .

See Solution

Problem 27447

Find the absolute maximum of $f(x)=4-x^{2}$ on the interval $[1,3]$ .

See Solution

Problem 27448

Is the antiderivative $F(x)$ of $f(x)=\frac{3}{\sqrt{x}}+\sin 2 x$ with $F\left(\frac{\pi}{4}\right)=3 \sqrt{\pi}$ equal to $F(x)=6 \sqrt{x}-\frac{1}{2} \cos 2 x$ ? True or False.

See Solution

Problem 27449

Find $\int_{-2}^{2} f(x) \, dx$ if the average value $a v(f) = \frac{\pi}{2}$ .

See Solution

Problem 27450

Find $\frac{d^{2} y}{d x^{2}}$ if $x^{3}-y^{3}=1$ .

See Solution

Problem 27451

The acceleration at $t=0$ for the position $s(t)=t^{3}-6 t^{2}+9 t$ is $a(0)=9 \mathrm{~m} / \mathrm{sec}^{2}$ . True or False?

See Solution

Problem 27452

Evaluate the limit: $\lim_{x \to +\infty} (x + \sqrt{x})$ and simplify the expression $\frac{E x e r c i c e}{x+\sqrt{x} \frac{v}{+y}} x$ .

See Solution

Problem 27453

Iodine-123 has a half-life of 13 hours. How long until the patient is $99.9\%$ free of it? Options: 130h, 117h, 180h, 11h.

See Solution

Problem 27454

Differentiate the function $\frac{\sin \left(e^{4 x^{2}-3 x}\right)}{\cos x^{2}-7 \ln x}$ .

See Solution

Problem 27455

Calculate the total number of people entering the escalator line from $t=0$ to $t=300$ given $r(t)$ and exit rate of 0.7 p/s.

See Solution

Problem 27456

Find the Taylor series at $x=0$ for the function $5 \sin (-v)$ using substitution.

See Solution

Problem 27457

Find the Taylor series at $v=0$ for the function $5 \sin(-v)$ using substitution.

See Solution

Problem 27458

Find the Taylor series at $x=0$ for these functions using substitution: 1. $e^{-5 x}$ , 2. $e^{-x / 2}$ , 3. $5 \sin (-x)$ .

See Solution

Problem 27459

Find the Taylor series at $x=0$ for: 1. $e^{-5 x}$ , 2. $e^{-x / 2}$ , 3. $5 \sin (-x)$ .

See Solution

Problem 27460

Find the Taylor series at $x = 0$ for $f(x) = \frac{1}{2-x}$ and $g(x) = e^{-\frac{x}{2}}$ using power series.

See Solution

Problem 27461

Find the Taylor series at $x=0$ for the function $e^{-x / 2}$ using substitution.

See Solution

Problem 27462

Differentiate the function $(6x - 5)^4$ with respect to $x$ .

See Solution

Problem 27463

Analyze the behavior of $f(x) = \log(x-2)$ as $x$ approaches 2.

See Solution

Problem 27464

Verify if the function $y=\sqrt{2 x-x^{2}}$ satisfies the equation $y^{3} y^{\prime \prime}+1=0$ .

See Solution

Problem 27465

Find $(f g)^{\prime}(0)$ and $\left(\frac{f}{g}\right)^{\prime}(0)$ given $f(0)=7, f^{\prime}(0)=2, g(0)=6, g^{\prime}(0)=-10$ .

See Solution

Problem 27466

Evaluate the limit: $\lim _{x \rightarrow 1} \frac{x^{2}-5 x+4}{x^{2}+3 x-4}$ . If it doesn't exist, write DNE.

See Solution

Problem 27467

Find the hydrostatic force on a $40\text{ cm}$ window $2\text{ m}$ below water and the pressure center, with specific gravity $1.025$ .

See Solution

Problem 27468

Find the limit: $\lim _{x \rightarrow \infty} \frac{x+x^{3}+x^{5}}{1-x^{2}+x^{4}}$

See Solution

Problem 27469

Evaluate the limit as $x$ approaches $x$ : $\lim _{x \rightarrow x}\left(x+\sqrt{x^{2}+2 x}\right)$ .

See Solution

Problem 27470

Find the infinite limits, limits at infinity, and asymptotes for $y=\frac{2 x^{2}+1}{\sqrt{x^{4}-4}}$ .

See Solution

Problem 27471

Determine if the derivative $f^{\prime}(1)$ exists for the piecewise function $f(x)=\left\{ \begin{array}{ll} x^{2}+x & x \leq 1 \\ 2 x^{2} & x>1 \end{array} \right.$ .

See Solution

Problem 27472

Find $e^{At}$ given that $\mathcal{L}^{-1}\left[(sI - A)^{-1}\right]$ is the matrix shown.

See Solution

Problem 27473

Let $f(x)=e^{2x}-(x+1)e^{x}$ . Find limits, derivative, and sketch the curve of $f$ .

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Problem 27474

Given the curve (G) of the derivative $f'$ of $f$ , show $f$ is increasing and find the area under (G) from $x=0$ to $x=1$ . Also, find the tangent line at $A(0)$ and analyze $f$ . The function is $f(x) = x(x+1)(a-x)$ .

See Solution

Problem 27475

Given the function $f(x)=x+(x+1)e^{-x}$ , find:
a) 1) $\lim_{x \rightarrow+\infty} f(x)$ and verify the asymptote $y=x$ . 2) Coordinates of intersection $E$ between (C) and $y=x$ . 3) Values of $x$ where (C) is below $y=x$ .
b) 1) $\lim_{x \rightarrow-\infty} f(x)$ 2) Calculate $f(-2)$ .

See Solution

Problem 27476

Given the function $f(x)=x+(x+1)e^{-x}$ , find limits, intersections, and behavior with respect to the line $y=x$ .

See Solution

Problem 27477

Dönüşüm $t=\sqrt{\theta^{2}+11 \theta}$ ile $\int \frac{2 \theta+11}{\sqrt{\theta^{2}+11 \theta}} d \theta$ integralini hesaplayın.

See Solution

Problem 27478

Calculate $\left.\pi r^{2}[x]\right|_{-r} ^{r} - \left.\pi\left[\frac{1}{3} x^{3}\right]\right|_{-r} ^{r}$ .

See Solution

Problem 27479

Given the function $f(x) = x + (x+1)e^{-x}$ , find the following:
1. $\lim_{x \rightarrow +\infty} f(x)$ and check if $y = x$ is an asymptote.
2. Coordinates of intersection point $E$ between $(C)$ and $(d)$ .
3. Values of $x$ where $(C)$ is below $(d)$ .
4. $\lim_{x \rightarrow -\infty} f(x)$ .
5. Calculate $f(-2)$ .
For the derivative function $f'$ :
a) Show $f$ is strictly increasing. b) Find the area under $(G)$ from $x=0$ to $x=1$ .
d) Write the tangent line at point $A$ with abscissa 0. e) Analyze the variations of $f$ and sketch $(C)$ and $(d)$ .

See Solution

Problem 27480

Find the inverse Fourier transform of $e^{-a|\lambda|}$ for $a > 0$ .

See Solution

Problem 27481

Calculate the integral $\int \frac{d x}{x \ln x}$ .

See Solution

Problem 27482

Evaluate the integral $\int \sin x \cdot 5^{\cos x} \, dx$ .

See Solution

Problem 27483

Calculate the integral $\int_{-1}^{2}|x| \, dx$ .

See Solution

Problem 27484

Find the integral for the surface area of the curve $x=\sin y$ from $y=0$ to $y=\pi$ when revolved around the $y$ -axis.

See Solution

Problem 27485

Calculate the curve length for $y=\frac{1}{3}(x^{2}+2)^{3/2}$ from $x=0$ to $x=1$ .

See Solution

Problem 27486

Evaluate the integral $\int_{0}^{t} x e^{x} d x$ .

See Solution

Problem 27487

Calculate the integral from 0 to 1 of $\sin x$ with respect to $x$ : $\int_{0}^{1} \sin x \, dx$ .

See Solution

Problem 27488

Find $R^{\prime}(2)$ for the rainfall function $R(t)$ defined as:
$R(t)=\left\{\begin{array}{ll} -t^{2}+4t+1 & t \leq 3 \\ \frac{t+5}{2} & t>3 \end{array}\right.$
Options: 0 cm/min, 5 cm/min, 0 cm/min², 5 cm/min².

See Solution

Problem 27489

Find the limit: $\lim _{x \rightarrow \infty} \frac{\ln \left(e^{3 x}+x\right)}{x}$ .

See Solution

Problem 27490

A cylinder of cookie dough has height $8 \mathrm{~mm}$ and radius $18 \mathrm{~mm}$ , with radius increasing at $2 \mathrm{~mm/min}$ .
Part A: Find the rate of increase of the circular surface area. Part B: Find the rate of decrease of the height.

See Solution

Problem 27491

Find $\frac{d y}{d x}$ if $x^{2} y - \frac{x}{y} = -2$ . Options: (A) (B) (C) (D)

See Solution

Problem 27492

If $f$ and $g$ are differentiable with $\lim_{x \to 0} f(x) = \lim_{x \to 0} g(x) = 0$ , find $\lim_{x \to 0} \frac{f(x)}{g(x)}$ given $\lim_{x \to 0} \frac{f'(x)}{g'(x)}$ exists.

See Solution

Problem 27493

Find the price $x$ that maximizes revenue given $f(x)=1220-5x$ and revenue $R=x \cdot f(x)$ .

See Solution

Problem 27494

Find the $y$ -coordinates on the curve $e^{x}=\sin y$ where there is a vertical tangent line. Options: (A), (B), (C), (D).

See Solution

Problem 27495

Analyze the curve $x^{2}-\frac{y^{2}}{5}=1$ in Quadrant IV. Is it concave up or down based on $\frac{d^{2} y}{d x^{2}}<0$ ?

See Solution

Problem 27496

计算定积分 $\int_{0}^{1} \sin x \, dx$ 的值。

See Solution

Problem 27497

Find the value of $k$ such that $\int_{0}^{1} \int_{\frac{1}{2}}^{1} k x y \, dx \, dy = 1$ .

See Solution

Problem 27498

Solve the equation $\frac{\partial \mu}{\partial t} = K \frac{\partial^2 \mu}{\partial x^2} + a \mu$ with initial condition $\mu(x, 0) = f(x)$ .

See Solution

Problem 27499

Find local maxima, minima, and inflection points of $f(x)=(x+1)(x+2)^{\frac{1}{3}}$ , for $x \geq -2$ .

See Solution

Problem 27500

If $F \cdot T(f(x))=\frac{\lambda^{2}}{1+\lambda^{5}}$ and $F \cdot T(g(x))=e^{-5 \lambda^{2}}$ , find: a) $F \cdot T(f(x-4))$ b) $F \cdot T(f(6x))$ c) $F \cdot T(f^{(5)}(x))$ d) $F \cdot T(f * g_{(x)})$ e) $F \cdot T\left(\int_{x-ct}^{x+ct} g(s) \, ds\right)$

See Solution

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Calculus

Problem 27401

Given f(x)=sin⁡xf(x)=\sin xf(x)=sinx and a=5π3a=\frac{5 \pi}{3}a=35π​, write the power series in summation notation. Choose A, B, C, or D.

Problem 27402

Approximate 11.192\frac{1}{1.19^{2}}1.1921​ using the first four nonzero terms of the Taylor series for f(x)=(1+x)−2f(x)=(1+x)^{-2}f(x)=(1+x)−2. First term is □\square□.

Problem 27403

Find the first four nonzero terms of the Taylor series for (1+2x)−2(1+2x)^{-2}(1+2x)−2 using (1+x)−2(1+x)^{-2}(1+x)−2.

Problem 27404

Approximate 1.17−4/51.17^{-4/5}1.17−4/5 using the first four terms of the Taylor series for f(x)=(1+x)−4/5f(x)=(1+x)^{-4/5}f(x)=(1+x)−4/5 centered at 0.

Problem 27405

Find the first four nonzero terms of the Taylor series for f(x)=(1+x)−4/5f(x)=(1+x)^{-4/5}f(x)=(1+x)−4/5 and use them to approximate 1.17−4/51.17^{-4/5}1.17−4/5.

Problem 27406

A sandbag is dropped from a balloon at 300 m300 \mathrm{~m}300 m and 13 m/s13 \mathrm{~m/s}13 m/s up. Find its speed on impact and time to ground. a. Speed: [−77.8 m/s][-77.8 \mathrm{~m/s}][−77.8 m/s] b. Time: [9.27 s][9.27 \mathrm{~s}][9.27 s]

Problem 27407

Find the first four nonzero terms of the Maclaurin series for f(x)=ln⁡(1+4x)f(x)=\ln(1+4x)f(x)=ln(1+4x) and its interval of convergence.

Problem 27408

Find the first four nonzero terms of the Taylor series for 1(5+8x)2\frac{1}{(5+8 x)^{2}}(5+8x)21​ centered at 0 using (1+x)−2(1+x)^{-2}(1+x)−2.

Problem 27409

Find the limit: lim⁡x→03tan⁡−1x−3x+x35x5\lim _{x \rightarrow 0} \frac{3 \tan^{-1} x - 3x + x^3}{5x^5}limx→0​5x53tan−1x−3x+x3​.

Problem 27410

Calculate the distance change rate from a particle on y=sin⁡xy=\sin xy=sinx to the origin, given dxdt=2 cm/s\frac{d x}{d t}=2 \mathrm{~cm/s}dtdx​=2 cm/s.

Problem 27411

Find the volume of the solid formed by rotating the area between y=(2+x)2y=(2+x)^{2}y=(2+x)2, x=0x=0x=0, and x=1x=1x=1 around the xxx-axis.

Problem 27412

Differentiate the Taylor series for f(x)=e−3xf(x)=e^{-3 x}f(x)=e−3x, identify the function, and find the interval of convergence.

Problem 27413

Find the derivative of f(x)=sin⁡(x6)f(x) = \sin(x^6)f(x)=sin(x6), which is f′(x)=6x5cos⁡(x6)f'(x) = 6x^5 \cos(x^6)f′(x)=6x5cos(x6). Determine the interval of convergence.

Problem 27414

Determine if f(x)=x2x−1f(x)=\frac{x^{2}}{x-1}f(x)=x−1x2​ has local or absolute extrema at the given points. Select the correct statement.

Problem 27415

Determine where the function f(x)=x2−2x2−1f(x)=\frac{x^{2}-2}{x^{2}-1}f(x)=x2−1x2−2​ is concave down using f′′=2(3x2+1)(x2−1)3f^{\prime \prime}=\frac{2(3x^{2}+1)}{(x^{2}-1)^{3}}f′′=(x2−1)32(3x2+1)​.

Problem 27416

Find the limit: lim⁡x→05x3+8x23x4−16x2\lim _{x \rightarrow 0} \frac{5 x^{3}+8 x^{2}}{3 x^{4}-16 x^{2}}limx→0​3x4−16x25x3+8x2​.

Problem 27417

Find the function for the series ∑k=0∞5e−3kx\sum_{k=0}^{\infty} 5 e^{-3 k x}∑k=0∞​5e−3kx and its interval of convergence.

Problem 27418

Approximate 1.193\sqrt[3]{1.19}31.19​ using the first four nonzero terms of the Taylor series for f(x)=1+x3f(x)=\sqrt[3]{1+x}f(x)=31+x​ at 0.

Problem 27419

Find the limit: L=lim⁡x→05x3+8x23x4−16x2L = \lim _{x \rightarrow 0} \frac{5 x^{3}+8 x^{2}}{3 x^{4}-16 x^{2}}L=limx→0​3x4−16x25x3+8x2​.

Problem 27420

Find the limit: lim⁡x→ax3−a3x4−a4\lim _{x \rightarrow a} \frac{x^{3}-a^{3}}{x^{4}-a^{4}}limx→a​x4−a4x3−a3​

Problem 27421

Find the value of ccc that satisfies Rolle's Theorem for f(x)=2x3−5x2−4x+3f(x)=2 x^{3}-5 x^{2}-4 x+3f(x)=2x3−5x2−4x+3 on x∈[13,3]x \in\left[\frac{1}{3}, 3\right]x∈[31​,3].

Problem 27422

Find the values of ccc for Rolle's Theorem with f(x)=cos⁡(2x)f(x)=\cos(2x)f(x)=cos(2x) on [0,2π][0, 2\pi][0,2π]. Options: a, b, c, d.

Problem 27423

Find values of ccc that satisfy Rolle's Theorem for f(x)=cos⁡(2x)f(x)=\cos(2x)f(x)=cos(2x) on [0,2π][0, 2\pi][0,2π]. Options: a, b, c, d.

Problem 27424

Find all xxx where the curve y=4x4−2x2−4y=4x^4-2x^2-4y=4x4−2x2−4 has a horizontal tangent. Choices: a, b, c, d.

Problem 27425

Find the value of ccc in Rolle's Theorem for f(x)=excos⁡xf(x)=e^{x} \cos xf(x)=excosx on [0,π][0, \pi][0,π]. Options: a. π2\frac{\pi}{2}2π​ b. 3π4\frac{3 \pi}{4}43π​ c. π4\frac{\pi}{4}4π​ d. None

Problem 27426

Find the limit: lim⁡x→0Sin⁡2x2x2+x\lim _{x \rightarrow 0} \frac{\operatorname{Sin} 2 x}{2 x^{2}+x}limx→0​2x2+xSin2x​.

Problem 27427

Find the limit as yyy approaches infinity for the expression 3y+7y2−2\frac{3y + 7}{y^2 - 2}y2−23y+7​.

Problem 27428

Differentiate the Taylor series for f(x)=e−3xf(x)=e^{-3x}f(x)=e−3x, identify the function, and find the interval of convergence.

Problem 27429

Find the function for the power series: ∑k=1∞(−1)kkxk+33k\sum_{k=1}^{\infty}(-1)^{k} \frac{k x^{k+3}}{3^{k}}∑k=1∞​(−1)k3kkxk+3​. What is f(x)=□f(x)=\squaref(x)=□?

Problem 27430

Find the first four nonzero terms of the Taylor series for sin (12)\left(\frac{1}{2}\right)(21​) in order. sin⁡(12)=(□)+(□)+(□)+(□)+⋯ \sin \left(\frac{1}{2}\right)=(\square)+(\square)+(\square)+(\square)+\cdots sin(21​)=(□)+(□)+(□)+(□)+⋯

Problem 27431

Find the first four nonzero terms of the Taylor series for e3e^{3}e3. Write them in order: e3=□+□+□+□+⋯ e^{3}=\square+\square+\square+\square+\cdots e3=□+□+□+□+⋯

Problem 27432

Find the function for the power series: ∑k=0∞(−1)kx3k4k\sum_{k=0}^{\infty}(-1)^{k} \frac{x^{3 k}}{4^{k}}∑k=0∞​(−1)k4kx3k​. What is f(x)=□f(x)=\squaref(x)=□?

Problem 27433

Express ln⁡(2)\ln(2)ln(2) as the sum of the first four nonzero terms of its Taylor series: 1−12+13−14+⋯1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots1−21​+31​−41​+⋯.

Problem 27434

Find the limit: lim⁡x→0sin⁡(5x)sin⁡(3x)\lim _{x \rightarrow 0} \frac{\sin(5x)}{\sin(3x)}limx→0​sin(3x)sin(5x)​.

Problem 27435

Find the Taylor series for Si\mathrm{Si}Si and approximate Si(0.7)\mathrm{Si}(0.7)Si(0.7) to the nearest thousandth.

Problem 27436

Find the area between the curve y=1−x4y=\sqrt{1-x^{4}}y=1−x4​ and the xxx-axis. Options: (A) 0.886 (B) 1.253 (C) 1.414 (D) 1.571 (E) 1.748

Problem 27437

Problem 27438

Find the average value of f(x)=cos⁡xf(x)=\cos xf(x)=cosx over the interval [1,5][1,5][1,5]. Choose the correct answer from the options.

Problem 27439

Find the linearization of f(x)=∫πxcos⁡3t dtf(x)=\int_{\pi}^{x} \cos ^{3} t \, dtf(x)=∫πx​cos3tdt at x=πx=\pix=π. What is it?

Problem 27440

Find f(5)f(5)f(5) if f′(x)=ln⁡(2+sin⁡x)f^{\prime}(x)=\ln(2+\sin x)f′(x)=ln(2+sinx) and f(3)=4f(3)=4f(3)=4. Choose from: (A) 0.040 (B) 0.272 (C) 0.961 (D) 4.555 (E) 6.667.

Given $f(x)=\sin x$ and $a=\frac{5 \pi}{3}$ , write the power series in summation notation. Choose A, B, C, or D.

Approximate $\frac{1}{1.19^{2}}$ using the first four nonzero terms of the Taylor series for $f(x)=(1+x)^{-2}$ . First term is $\square$ .

Find the first four nonzero terms of the Taylor series for $(1+2x)^{-2}$ using $(1+x)^{-2}$ .

Approximate $1.17^{-4/5}$ using the first four terms of the Taylor series for $f(x)=(1+x)^{-4/5}$ centered at 0.

Find the first four nonzero terms of the Taylor series for $f(x)=(1+x)^{-4/5}$ and use them to approximate $1.17^{-4/5}$ .

A sandbag is dropped from a balloon at $300 \mathrm{~m}$ and $13 \mathrm{~m/s}$ up. Find its speed on impact and time to ground.
a. Speed: $[-77.8 \mathrm{~m/s}]$ b. Time: $[9.27 \mathrm{~s}]$

Find the first four nonzero terms of the Maclaurin series for $f(x)=\ln(1+4x)$ and its interval of convergence.

Find the first four nonzero terms of the Taylor series for $\frac{1}{(5+8 x)^{2}}$ centered at 0 using $(1+x)^{-2}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{3 \tan^{-1} x - 3x + x^3}{5x^5}$ .

Calculate the distance change rate from a particle on $y=\sin x$ to the origin, given $\frac{d x}{d t}=2 \mathrm{~cm/s}$ .

Find the volume of the solid formed by rotating the area between $y=(2+x)^{2}$ , $x=0$ , and $x=1$ around the $x$ -axis.

Differentiate the Taylor series for $f(x)=e^{-3 x}$ , identify the function, and find the interval of convergence.

Find the derivative of $f(x) = \sin(x^6)$ , which is $f'(x) = 6x^5 \cos(x^6)$ . Determine the interval of convergence.

Determine if $f(x)=\frac{x^{2}}{x-1}$ has local or absolute extrema at the given points. Select the correct statement.

Determine where the function $f(x)=\frac{x^{2}-2}{x^{2}-1}$ is concave down using $f^{\prime \prime}=\frac{2(3x^{2}+1)}{(x^{2}-1)^{3}}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{5 x^{3}+8 x^{2}}{3 x^{4}-16 x^{2}}$ .

Find the function for the series $\sum_{k=0}^{\infty} 5 e^{-3 k x}$ and its interval of convergence.

Approximate $\sqrt[3]{1.19}$ using the first four nonzero terms of the Taylor series for $f(x)=\sqrt[3]{1+x}$ at 0.

Find the limit: $L = \lim _{x \rightarrow 0} \frac{5 x^{3}+8 x^{2}}{3 x^{4}-16 x^{2}}$ .

Find the limit: $\lim _{x \rightarrow a} \frac{x^{3}-a^{3}}{x^{4}-a^{4}}$

Find the value of $c$ that satisfies Rolle's Theorem for $f(x)=2 x^{3}-5 x^{2}-4 x+3$ on $x \in\left[\frac{1}{3}, 3\right]$ .

Find the values of $c$ for Rolle's Theorem with $f(x)=\cos(2x)$ on $[0, 2\pi]$ . Options: a, b, c, d.

Find values of $c$ that satisfy Rolle's Theorem for $f(x)=\cos(2x)$ on $[0, 2\pi]$ . Options: a, b, c, d.

Find all $x$ where the curve $y=4x^4-2x^2-4$ has a horizontal tangent. Choices: a, b, c, d.

Find the value of $c$ in Rolle's Theorem for $f(x)=e^{x} \cos x$ on $[0, \pi]$ . Options: a. $\frac{\pi}{2}$ b. $\frac{3 \pi}{4}$ c. $\frac{\pi}{4}$ d. None

Find the limit: $\lim _{x \rightarrow 0} \frac{\operatorname{Sin} 2 x}{2 x^{2}+x}$ .

Find the limit as $y$ approaches infinity for the expression $\frac{3y + 7}{y^2 - 2}$ .

Differentiate the Taylor series for $f(x)=e^{-3x}$ , identify the function, and find the interval of convergence.

Find the function for the power series: $\sum_{k=1}^{\infty}(-1)^{k} \frac{k x^{k+3}}{3^{k}}$ . What is $f(x)=\square$ ?

Find the first four nonzero terms of the Taylor series for sin $\left(\frac{1}{2}\right)$ in order. $\sin \left(\frac{1}{2}\right)=(\square)+(\square)+(\square)+(\square)+\cdots$

Find the first four nonzero terms of the Taylor series for $e^{3}$ . Write them in order:
$e^{3}=\square+\square+\square+\square+\cdots$

Find the function for the power series: $\sum_{k=0}^{\infty}(-1)^{k} \frac{x^{3 k}}{4^{k}}$ . What is $f(x)=\square$ ?

Express $\ln(2)$ as the sum of the first four nonzero terms of its Taylor series: $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin(5x)}{\sin(3x)}$ .

Find the Taylor series for $\mathrm{Si}$ and approximate $\mathrm{Si}(0.7)$ to the nearest thousandth.

Find the area between the curve $y=\sqrt{1-x^{4}}$ and the $x$ -axis. Options: (A) 0.886 (B) 1.253 (C) 1.414 (D) 1.571 (E) 1.748

Find the average value of $f(x)=\cos x$ over the interval $[1,5]$ . Choose the correct answer from the options.

Find the linearization of $f(x)=\int_{\pi}^{x} \cos ^{3} t \, dt$ at $x=\pi$ . What is it?

Find $f(5)$ if $f^{\prime}(x)=\ln(2+\sin x)$ and $f(3)=4$ . Choose from: (A) 0.040 (B) 0.272 (C) 0.961 (D) 4.555 (E) 6.667.

Find $\lim _{h \rightarrow 0} \frac{1}{h} \int_{x}^{x+h} f(t) d t$ . Choose from: (A) 0, (B) 1, (C) $f^{\prime}(x)$ , (D) $f(x)$ , (E) nonexistent.

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin (\pi-2 x)}{x}$ .

Determine the intervals where the function $f$ is increasing given that $f^{\prime}(x)=\frac{x^{2}-1}{x^{2}+2}$ .

Calculate the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{x}-\sqrt[3]{x}}{\sqrt{x}+\sqrt[3]{x}}$ .

Find the average rate of change of $f(x)=\cot x$ on the interval $\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$ .

Find the slope of the normal line for the function $f(x)=\frac{5}{x^{2}+2}$ at the point where $x=2$ .

Find the absolute maximum of $f(x)=4-x^{2}$ on the interval $[1,3]$ .

Is the antiderivative $F(x)$ of $f(x)=\frac{3}{\sqrt{x}}+\sin 2 x$ with $F\left(\frac{\pi}{4}\right)=3 \sqrt{\pi}$ equal to $F(x)=6 \sqrt{x}-\frac{1}{2} \cos 2 x$ ? True or False.

Find $\int_{-2}^{2} f(x) \, dx$ if the average value $a v(f) = \frac{\pi}{2}$ .

Find $\frac{d^{2} y}{d x^{2}}$ if $x^{3}-y^{3}=1$ .

The acceleration at $t=0$ for the position $s(t)=t^{3}-6 t^{2}+9 t$ is $a(0)=9 \mathrm{~m} / \mathrm{sec}^{2}$ . True or False?

Evaluate the limit: $\lim_{x \to +\infty} (x + \sqrt{x})$ and simplify the expression $\frac{E x e r c i c e}{x+\sqrt{x} \frac{v}{+y}} x$ .

Iodine-123 has a half-life of 13 hours. How long until the patient is $99.9\%$ free of it? Options: 130h, 117h, 180h, 11h.

Differentiate the function $\frac{\sin \left(e^{4 x^{2}-3 x}\right)}{\cos x^{2}-7 \ln x}$ .

Calculate the total number of people entering the escalator line from $t=0$ to $t=300$ given $r(t)$ and exit rate of 0.7 p/s.

Find the Taylor series at $x=0$ for the function $5 \sin (-v)$ using substitution.

Find the Taylor series at $v=0$ for the function $5 \sin(-v)$ using substitution.

Find the Taylor series at $x=0$ for these functions using substitution: 1. $e^{-5 x}$ , 2. $e^{-x / 2}$ , 3. $5 \sin (-x)$ .

Find the Taylor series at $x=0$ for: 1. $e^{-5 x}$ , 2. $e^{-x / 2}$ , 3. $5 \sin (-x)$ .

Find the Taylor series at $x = 0$ for $f(x) = \frac{1}{2-x}$ and $g(x) = e^{-\frac{x}{2}}$ using power series.

Find the Taylor series at $x=0$ for the function $e^{-x / 2}$ using substitution.

Differentiate the function $(6x - 5)^4$ with respect to $x$ .

Analyze the behavior of $f(x) = \log(x-2)$ as $x$ approaches 2.

Verify if the function $y=\sqrt{2 x-x^{2}}$ satisfies the equation $y^{3} y^{\prime \prime}+1=0$ .

Find $(f g)^{\prime}(0)$ and $\left(\frac{f}{g}\right)^{\prime}(0)$ given $f(0)=7, f^{\prime}(0)=2, g(0)=6, g^{\prime}(0)=-10$ .

Evaluate the limit: $\lim _{x \rightarrow 1} \frac{x^{2}-5 x+4}{x^{2}+3 x-4}$ . If it doesn't exist, write DNE.

Find the hydrostatic force on a $40\text{ cm}$ window $2\text{ m}$ below water and the pressure center, with specific gravity $1.025$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{x+x^{3}+x^{5}}{1-x^{2}+x^{4}}$

Evaluate the limit as $x$ approaches $x$ : $\lim _{x \rightarrow x}\left(x+\sqrt{x^{2}+2 x}\right)$ .

Find the infinite limits, limits at infinity, and asymptotes for $y=\frac{2 x^{2}+1}{\sqrt{x^{4}-4}}$ .

Determine if the derivative $f^{\prime}(1)$ exists for the piecewise function $f(x)=\left\{ \begin{array}{ll} x^{2}+x & x \leq 1 \\ 2 x^{2} & x>1 \end{array} \right.$ .

Find $e^{At}$ given that $\mathcal{L}^{-1}\left[(sI - A)^{-1}\right]$ is the matrix shown.

Let $f(x)=e^{2x}-(x+1)e^{x}$ . Find limits, derivative, and sketch the curve of $f$ .

Given the curve (G) of the derivative $f'$ of $f$ , show $f$ is increasing and find the area under (G) from $x=0$ to $x=1$ . Also, find the tangent line at $A(0)$ and analyze $f$ . The function is $f(x) = x(x+1)(a-x)$ .