Calculus

Problem 28501

Differentiate $y=x^{2} \sqrt{3-4 x}$ with respect to $x$ .

See Solution

Problem 28502

Find the limit: $\lim _{x \rightarrow 0} \frac{e^{x}-1}{x}$ .

See Solution

Problem 28503

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin (4 x)}{x}$ .

See Solution

Problem 28504

Find the limit: $\lim _{x \rightarrow \frac{\pi}{2}} \frac{3 \sec (x)+5}{\tan (x)}$ .

See Solution

Problem 28505

Which statement is guaranteed by the Intermediate Value Theorem for the function $f$ continuous on $[2,4]$ with $f(2)=10$ and $f(4)=20$ ?

See Solution

Problem 28506

Dane są zbieżny szereg $\sum_{n=1}^{+\infty} a_{n}$ i bezwzględnie zbieżny $\sum_{n=1}^{+\infty} b_{n}$ . Udowodnij zbieżność $\sum_{n=1}^{+\infty} c_{n}$ , gdzie $c_{n}:=b_{n} \cdot\left(3+4 a_{n}+5 \sum_{k=1}^{n} a_{k}\right)$ . Czy jest bezwzględnie zbieżny?

See Solution

Problem 28507

Find the minimum value of $f(x)=x \ln x$ . Options: (A) $-e$ , (B) -1, (C) $-\frac{1}{e}$ , (D) 0, (E) no minimum.

See Solution

Problem 28508

Differentiate the function $\frac{(3 x^{2}+2)^{4}}{\sqrt{2 x-1}}$ .

See Solution

Problem 28509

Find the limit as $x$ approaches 15: $\lim _{x \rightarrow 15} 4=\square$

See Solution

Problem 28510

Find the limit: $\lim _{x \rightarrow 0} \frac{\tan ^{-1}(x)-x}{8 x^{3}}$ .

See Solution

Problem 28511

Evaluate the limit as $z$ approaches -1 for the expression $7 z^{2} - 5 z$ .

See Solution

Problem 28512

Calculate the limit: $\lim _{x \rightarrow \pi} \frac{\sin (x)}{1-\cos (x)}$ .

See Solution

Problem 28513

Evaluate the limit as x approaches 5 for the expression 5x - 3. What is the result?

See Solution

Problem 28514

Find the limit as $x$ approaches infinity of $\frac{7 x}{\ln(150 x + e^{x})}$ .

See Solution

Problem 28515

Find the limit as $a$ approaches 6 for the expression $\frac{a^{3}-216}{a-6}$ .

See Solution

Problem 28516

Find the limit as $t$ approaches 6 for the expression $\frac{\frac{1}{t}-\frac{1}{6}}{t-6}$ .

See Solution

Problem 28517

Find the limit: $\lim _{x \rightarrow 2} \frac{x^{2}-6 x+8}{x-2}$

See Solution

Problem 28518

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

See Solution

Problem 28519

Calculate the limit: $\lim _{x \rightarrow 16} \frac{\sqrt{x}-4}{x-16}$ .

See Solution

Problem 28520

Find the limit as $x$ approaches $\frac{\pi}{2}$ for the expression $(\tan (x)-\sec (x))$ .

See Solution

Problem 28521

Find the remaining mass of a 100-mg Cesium-137 sample after 180 years, given its half-life is 30 years.

See Solution

Problem 28522

Evaluate the limit as $a$ approaches 5: $\lim _{a \rightarrow 5} \frac{\frac{1}{a+8}-\frac{1}{13}}{a-5}$ .

See Solution

Problem 28523

Find the limit as $s$ approaches 25 for the expression $\frac{25-s}{5-\sqrt{s}}$ .

See Solution

Problem 28524

Find the limit as $x$ approaches 49 for the expression $\frac{\sqrt{x}-7}{x-49}$ .

See Solution

Problem 28525

Calculate the limit as $x$ approaches 4 for the expression $\frac{x^{3}-5 x^{2}+2 x+8}{x-4}$ .

See Solution

Problem 28526

Find the limits for the piecewise function $f(x)=\left\{\begin{array}{ll}4-x-x^{2} & \text { if } x \leq 2 \\ 2 x-7 & \text { if } x>2\end{array}\right.$ at $x=2$ .

See Solution

Problem 28527

Sketch the graph of $f(x)=\left\{\begin{array}{ll}6-x, & x<-3 \\ x, & -3 \leq x<3 \\ (x-3)^{2}, & x \geq 3\end{array}\right.$ and find $\lim _{x \rightarrow-3^{-}} f(x)$ .

See Solution

Problem 28528

Is there a limit to the instantaneous rate of change for $f(x)=x^{2}$ ? Explain your reasoning.

See Solution

Problem 28529

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{3x + 16} - 4}{x}$

See Solution

Problem 28530

Find the limit as $x$ approaches 0 for the expression $\frac{|x|}{x}$ . What is the result?

See Solution

Problem 28531

Find the limit: $\lim _{h \rightarrow 0} \frac{(3+h)^{2}-9}{h}$

See Solution

Problem 28532

Find the lower Riemann sum for $f(x)=2-x^{2}$ on $[0,1]$ using the partition $P=[0, \frac{1}{2}, \frac{3}{4}, 1]$ .

See Solution

Problem 28533

Find critical numbers for: (a) $y=x^{\frac{4}{5}}(x-4)^{2}$ , (b) $g(\theta)=4\theta-\tan\theta$ .

See Solution

Problem 28534

Find the local maximum of $f(x) = x^{6} e^{-x}$ by finding critical points with $f'(x)=0$ and using the derivative test.

See Solution

Problem 28535

Solve the differential equation $y(x+1) \frac{d y}{d x}=x(1+y^{2})$ .

See Solution

Problem 28536

Find the upper Riemann sum for $f(x)=x^{2}$ on $[-1,1]$ using the partition $P=\left[-1,-\frac{1}{2}, \frac{1}{2}, \frac{3}{4}, 1\right]$ .

See Solution

Problem 28537

Estimate the integral $\int_{0}^{12} 2 x^{2} d x$ using Riemann sums with the right endpoint method and $n=6$ .

See Solution

Problem 28538

Find the lower Riemann sum for $f(x)=\sin(x)$ on $[0, \pi]$ using the partition $P=\left[0, \frac{\pi}{3}, \frac{3 \pi}{4}, \pi\right]$ .

See Solution

Problem 28539

Given the curve on the xy-plane, find the intervals where $f$ is increasing based on the points provided. (Use interval notation.)

See Solution

Problem 28540

A balloon rises at $8 \mathrm{ft/sec}$ from 150 feet away. How fast is the distance to the observer increasing at 50 feet high?

See Solution

Problem 28541

Find the diver's speed upon hitting the water if they jump with $2.00 \, \mathrm{m/s}$ from a $10 \, \mathrm{m}$ board.

See Solution

Problem 28542

A person 2m tall moves away from an 8m light. If their shadow grows at $4/9$ m/s, find their walking speed in m/s.

See Solution

Problem 28543

An acute angle in a right triangle increases at 3 rad/min. If the hypotenuse is $5 \mathrm{~cm}$ , find the rate of the opposite side's increase when it is $3 \mathrm{~cm}$ .

See Solution

Problem 28544

A square's sides grow at $3 \mathrm{~cm/s}$ . What is the area increase rate when sides are $5 \mathrm{~cm}$ ?
An observer 50 m south of an intersection sees a car moving east at 45 m/s. How fast is it moving away after 5 seconds?

See Solution

Problem 28545

Evaluate the surface integral:
$\iint_{S} f(x, y, z) \, dS = 2\pi \int_{-1}^{1} g(z_{0} + t) \, dt$ .

See Solution

Problem 28546

Find the radius increase rate of a marshmallow (cylinder) with radius $1 \mathrm{~cm}$ , height $3 \mathrm{~cm}$ , expanding at $2 \mathrm{~cm}^{3}/\mathrm{sec}$ when height is $6 \mathrm{~cm}$ .

See Solution

Problem 28547

A pretzel maker rolls dough into a cylinder. If the length increases at $0.5 \mathrm{~cm/sec}$ , find the radius change rate at radius $1 \mathrm{~cm}$ and length $5 \mathrm{~cm}$ .

See Solution

Problem 28548

Check if the function $f$ is continuous at $x=a$ where $f(x)=\begin{cases}2x-3 & , x \leq 2 \\ x^2 & , x>2\end{cases}$ and $a=2$ .

See Solution

Problem 28549

Find the linear approximation of $f(x)=3 x e^{2 x-10}$ at $x=5$ .

See Solution

Problem 28550

Find the linear approximation of $h(t)=t^{4}-6 t^{3}+3 t-7$ at $t=-3$ .

See Solution

Problem 28551

Determine if $f(x)=0$ for some $x$ in $[-5,-1]$ given $f(-5)=-6$ and $f(-1)=6$ . Choices: Sometimes true, Always false, Always true.

See Solution

Problem 28552

True or False: If $f(x)$ is continuous at $x=-3$ and $f(-3)=4$ , then $\lim_{x \to -3} f(x) = 4$ .

See Solution

Problem 28553

Find the linear approximation for the following functions at the specified points:
1. $f(x)=\cos(2x)$ at $x=\pi$
2. $h(z)=\ln(z^{2}+5)$ at $z=2$

See Solution

Problem 28554

Calculate (a) the zero-point energy of a hydrogen atom in a 1D box of length $1.00 \mathrm{~cm}$ and (b) the ratio to thermal energy $k_{B} T$ at $300 \mathrm{~K}$ .

See Solution

Problem 28555

Find the linear approximation of $f(x)=\cos(2x)$ at the point $x=\pi$ .

See Solution

Problem 28556

Given the piecewise function $f(x)=\left\{\begin{array}{ll}2 x+19 & \text { if } x<-5 \\ \sqrt{x+86} & \text { if } x>-5 \\ 2 & \text { if } \quad x=-5\end{array}\right.$ , evaluate statements about continuity at $x=-5$ .

See Solution

Problem 28557

Find the constant $k$ that makes the function $f$ continuous at $x=0$ where:
$f(x)=\begin{cases} B+\frac{x-3}{2-\sqrt{x+1}}, & x>3 \\ Ax, & x=3 \\ \frac{\sin(3x-9)}{x-3}+\frac{|x-3|}{|x|-3}, & x<3 \end{cases}$

See Solution

Problem 28558

Find the value of the infinite series: $\sum_{n=4}^{\infty}\left(\frac{1}{2}\right)^{n}$ .

See Solution

Problem 28559

Find the limit: $\lim _{x \rightarrow \pi} \frac{\sin (x)}{1-\cos (x)}=N u m b$

See Solution

Problem 28560

Find the limit: $\lim _{x \rightarrow \pi} \frac{\sin (x)}{1-\cos (x)}$

See Solution

Problem 28561

Find $x$ values for which the series $\sum_{n=0}^{\infty} \frac{(x+9)^{n}}{2^{n}}$ converges and its sum.

See Solution

Problem 28562

Find the limit as $n$ approaches infinity for $\sum_{n=1}^{\infty} \frac{1}{2^{n}}$ and show it equals 1.

See Solution

Problem 28563

Find the limit of $c_{n}=\ln \left(\frac{9 n-7}{13 n+4}\right)$ as $n$ approaches infinity.

See Solution

Problem 28564

For problems 1-4, find linear approximations at the specified points. 1. $f(x)=\cos(2x)$ at $x=\pi$ 2. $h(z)=\ln(z^2+5)$ at $z=2$ 3. $g(x)=2-9x-3x^2-x^3$ at $x=-1$ 4. $g(t)=e^{\sin(t)}$ at $t=-4$ . For problem 5, approximate $\sin(2)$ and $\sin(15)$ using $h(y)=\sin(y+1)$ at $y=0$ . For problem 6, approximate $\sqrt[5]{31}$ and $\sqrt[5]{3}$ using $R(t)=\sqrt[5]{t}$ at $t=32$ . For problem 7, approximate $e^{-4}e^{-4}$ using $h(x)=e^{1-x}$ at $x=1$ .

See Solution

Problem 28565

Calculate the sum of the series: $\sum_{k=1}^{\infty} \frac{4^{k+3}}{5^{k-1}}$ .

See Solution

Problem 28566

Find the volume $V = 2\pi \int_{0}^{2} (6 - x - x^2) (x + 1) \, dx$ .

See Solution

Problem 28567

Find the limit: $\lim _{n \rightarrow \infty} c_{n}$ where $c_{n}=\ln \left(\frac{9 n-7}{13 n+4}\right)$ .

See Solution

Problem 28568

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

See Solution

Problem 28569

Solve $\frac{\partial^{2} u}{\partial t^{2}}=\frac{1}{16} \frac{\partial^{2} u}{\partial x^{2}}$ with B.C. $u(0, t)=0$ , $\frac{\partial u}{\partial x}(1, t)=0$ and I.C. $u(x, 0)=x^{2}-x$ , $\frac{\partial u}{\partial t}(x, 0)=x+1$ for $0<x<1$ . Find $u\left(\frac{1}{4}, 8\right)$ .

See Solution

Problem 28570

Find the length of the curve $y=\sqrt{x+2}$ for $-1 \leq x \leq 0$ . Which integral represents this length?

See Solution

Problem 28571

Find the length of the curve $y=\sqrt{x+4}$ for $-3 \leq x \leq -2$ . Select the correct integral:
$\int_{-3}^{-2} \sqrt{\frac{4 x+17}{4 x+16}} dx$

See Solution

Problem 28572

Find the surface area of the curve $x=2\sqrt{y}$ from $y=\frac{1}{4}$ to $y=\frac{1}{2}$ around the y-axis. Select one:
1. $\int_{\frac{1}{4}}^{\frac{1}{2}} 4 \pi \sqrt{y+1} dy$
2. None
3. $\int_{\frac{1}{4}}^{\frac{1}{2}} 2 \pi \sqrt{2y+1} dy$
4. $\int_{\frac{1}{4}}^{\frac{1}{2}} 2 \pi \sqrt{y+1} dy$
5. $\int_{\frac{1}{4}}^{\frac{1}{2}} \pi \sqrt{y+1} dy$

See Solution

Problem 28573

Find the length of the curve $y=\int_{1}^{x} \sqrt{t^{2}-1} d t$ for $1 \leq x \leq \sqrt{3}$ .

See Solution

Problem 28574

Find the curve length for $x=(y+2)^{2}$ from $y=1$ to $y=4$ : $\int_{1}^{4}(2y+4) dy$ .

See Solution

Problem 28575

Find $f^{\prime \prime}\left(\frac{\pi}{2}\right)$ for $f(x)=\sin(3x)$ .

See Solution

Problem 28576

Find the length of the curve defined by $x=(y-1)^{2}$ for $1 \leq y \leq 4$ .

See Solution

Problem 28577

Find the surface area of the curve $x=3 \sqrt{y}$ from $y=\frac{1}{9}$ to $y=\frac{2}{9}$ rotated about the $y$ -axis.

See Solution

Problem 28578

Find the length of the curve $y=\frac{2}{3}(x+1)^{3/2}$ for $3 \leq x \leq 4$ . Choose the correct answer.

See Solution

Problem 28579

Find the surface area from revolving $y=\frac{1}{\pi} \sin (5 x)$ , $0 \leq x \leq \frac{\pi}{10}$ about the $x$ -axis.

See Solution

Problem 28580

Find the surface area of the curve $y=\frac{1}{\pi} \sin (x)$ from $0$ to $\frac{\pi}{2}$ revolved around the $x$ -axis.

See Solution

Problem 28581

Find the length of the curve $x=12 \sqrt{y}+43, 1 \leq y \leq 4$ . Select the correct integral expression for this length.

See Solution

Problem 28582

Displacement of a slide valve is given by $x=A \cos (\omega t)+B \sin (\omega t)$ . Find velocity, acceleration, and times when velocity is zero.

See Solution

Problem 28583

Differentiate the function $y=\frac{x \sqrt{5-x^{2}}}{6-x}$ .

See Solution

Problem 28584

Find the length of the curve $x=(y+1)^{2}$ for $1 \leq y \leq 4$ . Choose the correct integral expression.

See Solution

Problem 28585

Find the surface area of the curve $x=2 \sqrt{\cos (5 y)}$ for $-\frac{\pi}{20} \leq y \leq \frac{\pi}{20}$ revolved around the $y$ -axis.

See Solution

Problem 28586

Find the surface area of the curve $y=\frac{2}{\pi x^{2}}$ from $x=1$ to $x=4$ rotated around the $x$ -axis.

See Solution

Problem 28587

Find the surface area of the curve $x=2 \sqrt{\cos (3 y)}$ from $y=-\frac{\pi}{12}$ to $y=\frac{\pi}{12}$ about the $y$ -axis.

See Solution

Problem 28588

Find the curve length of $y=(1 / 3)(x^{2}+2)^{3 / 2}$ from $x=0$ to $x=3$ .

See Solution

Problem 28589

Find $\frac{d y}{d x}$ given $x^{3}+y^{3}=3 x y$ . Options: $\frac{y-x^{2}}{y^{2}-x}$ , $\frac{x^{2}}{1-y^{2}}$ , $\frac{x+y-x^{2}}{y^{2}}$ .

See Solution

Problem 28590

Find the length of the curve $y=\frac{2}{3}(x+1)^{3/2}$ for $4 \leq x \leq 5$ . Choose the correct answer.

See Solution

Problem 28591

Find the surface area of the curve $x=2 \sqrt{\cos (5 y)}$ from $y=-\frac{\pi}{20}$ to $y=\frac{\pi}{20}$ revolved around the $y$ -axis.

See Solution

Problem 28592

Find the value of $k$ if $y=x e^{k x}$ and $\frac{d^{2} y}{d x^{2}}=6$ at $x=0$ .

See Solution

Problem 28593

Differentiate $\sin(5y) + e^{4x}$ and find the result: $5 \cos(5y) \frac{dy}{dx} + 4 e^{4x}$ .

See Solution

Problem 28594

Find the length of the curve $y=\frac{2}{5}(x+3)^{5/2}$ for $-2 \leq x \leq -1$ . Which integral represents it?

See Solution

Problem 28595

Find the surface area of the curve $y=\frac{1}{\pi} \sin (x)$ from $0$ to $\frac{\pi}{2}$ revolved about the $x$ -axis.

See Solution

Problem 28596

Analysiere die Funktion $O(t)=-0,64 \cdot t^{3}+2,93 \cdot t^{2}-3,3 \cdot t+12,2$ für $0 \leq t \leq 3$ . a) Zeichne $O$ , $O'$ , $O''$ . b) Deute die Nullstellen von $O'$ und $O''$ . c) Berechne $O(1)$ , $O'(1)$ , $O''(1)$ und interpretiere $O(1)$ und $O'(1)$ .

See Solution

Problem 28597

Find the surface area of the curve $y=\frac{1}{\pi} \sin (x)$ , $0 \leq x \leq \frac{\pi}{2}$ , revolved around the $x$ -axis.

See Solution

Problem 28598

Find the surface area of the curve $y=\frac{1}{\pi} \sin (x)$ , $0 \leq x \leq \frac{\pi}{2}$ , revolved around the $x$ -axis.

See Solution

Problem 28599

Find the length of the curve $y=\frac{2}{3}(x+1)^{3 / 2}$ for $0 \leq x \leq 1$ . Is it $6 \sqrt{3}-4 \sqrt{2}$ ?

See Solution

Problem 28600

Find the lengths of the following curves from specified intervals: 1. $y=(1/3)(x^2+2)^{3/2}$ , $x=0$ to $3$ ; 2. $y=x^{3/2}$ , $x=0$ to $4$ ; 3. $x=(y^3/3)+1/(4y)$ , $y=1$ to $3$ ; 4. $x=(y^{3/2}/3)-y^{1/2}$ , $y=1$ to $9$ ; 5. $x=(y^4/4)+1/(8y^2)$ , $y=1$ to $2$ ; 6. $x=(y^3/6)+1/(2y)$ , $y=2$ to $3$ ; 7. $y=(3/4)x^{4/3}-(3/8)x^{2/3}+5$ , $1 \leq x \leq 8$ ; 8. $y=(x^3/3)+x^2+x+1/(4x+4)$ , $0 \leq x \leq 2$ .

See Solution

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Calculus

Problem 28501

Differentiate y=x23−4xy=x^{2} \sqrt{3-4 x}y=x23−4x​ with respect to xxx.

Problem 28502

Find the limit: lim⁡x→0ex−1x\lim _{x \rightarrow 0} \frac{e^{x}-1}{x}limx→0​xex−1​.

Problem 28503

Find the limit: lim⁡x→0sin⁡(4x)x\lim _{x \rightarrow 0} \frac{\sin (4 x)}{x}limx→0​xsin(4x)​.

Problem 28504

Find the limit: lim⁡x→π23sec⁡(x)+5tan⁡(x)\lim _{x \rightarrow \frac{\pi}{2}} \frac{3 \sec (x)+5}{\tan (x)}limx→2π​​tan(x)3sec(x)+5​.

Problem 28505

Which statement is guaranteed by the Intermediate Value Theorem for the function fff continuous on [2,4][2,4][2,4] with f(2)=10f(2)=10f(2)=10 and f(4)=20f(4)=20f(4)=20?

Problem 28506

Problem 28507

Find the minimum value of f(x)=xln⁡xf(x)=x \ln xf(x)=xlnx. Options: (A) −e-e−e, (B) -1, (C) −1e-\frac{1}{e}−e1​, (D) 0, (E) no minimum.

Problem 28508

Differentiate the function (3x2+2)42x−1\frac{(3 x^{2}+2)^{4}}{\sqrt{2 x-1}}2x−1​(3x2+2)4​.

Problem 28509

Find the limit as xxx approaches 15: lim⁡x→154=□\lim _{x \rightarrow 15} 4=\squarelimx→15​4=□

Problem 28510

Find the limit: lim⁡x→0tan⁡−1(x)−x8x3\lim _{x \rightarrow 0} \frac{\tan ^{-1}(x)-x}{8 x^{3}}limx→0​8x3tan−1(x)−x​.

Problem 28511

Evaluate the limit as zzz approaches -1 for the expression 7z2−5z7 z^{2} - 5 z7z2−5z.

Problem 28512

Calculate the limit: lim⁡x→πsin⁡(x)1−cos⁡(x)\lim _{x \rightarrow \pi} \frac{\sin (x)}{1-\cos (x)}limx→π​1−cos(x)sin(x)​.

Problem 28513

Evaluate the limit as x approaches 5 for the expression 5x - 3. What is the result?

Problem 28514

Find the limit as xxx approaches infinity of 7xln⁡(150x+ex)\frac{7 x}{\ln(150 x + e^{x})}ln(150x+ex)7x​.

Problem 28515

Find the limit as aaa approaches 6 for the expression a3−216a−6\frac{a^{3}-216}{a-6}a−6a3−216​.

Problem 28516

Find the limit as ttt approaches 6 for the expression 1t−16t−6\frac{\frac{1}{t}-\frac{1}{6}}{t-6}t−6t1​−61​​.

Problem 28517

Find the limit: lim⁡x→2x2−6x+8x−2\lim _{x \rightarrow 2} \frac{x^{2}-6 x+8}{x-2}limx→2​x−2x2−6x+8​

Problem 28518

Find the limit: lim⁡x→0−ex+e−x−2sin⁡(x)+tan⁡(x)\lim _{x \rightarrow 0^{-}} \frac{e^{x}+e^{-x}-2}{\sin (x)+\tan (x)}limx→0−​sin(x)+tan(x)ex+e−x−2​.

Problem 28519

Calculate the limit: lim⁡x→16x−4x−16\lim _{x \rightarrow 16} \frac{\sqrt{x}-4}{x-16}limx→16​x−16x​−4​.

Problem 28520

Find the limit as xxx approaches π2\frac{\pi}{2}2π​ for the expression (tan⁡(x)−sec⁡(x))(\tan (x)-\sec (x))(tan(x)−sec(x)).

Problem 28521

Find the remaining mass of a 100-mg Cesium-137 sample after 180 years, given its half-life is 30 years.

Problem 28522

Evaluate the limit as aaa approaches 5: lim⁡a→51a+8−113a−5\lim _{a \rightarrow 5} \frac{\frac{1}{a+8}-\frac{1}{13}}{a-5}lima→5​a−5a+81​−131​​.

Problem 28523

Find the limit as sss approaches 25 for the expression 25−s5−s\frac{25-s}{5-\sqrt{s}}5−s​25−s​.

Problem 28524

Find the limit as xxx approaches 49 for the expression x−7x−49\frac{\sqrt{x}-7}{x-49}x−49x​−7​.

Problem 28525

Calculate the limit as xxx approaches 4 for the expression x3−5x2+2x+8x−4\frac{x^{3}-5 x^{2}+2 x+8}{x-4}x−4x3−5x2+2x+8​.

Problem 28526

Find the limits for the piecewise function f(x)={4−x−x2 if x≤22x−7 if x>2f(x)=\left\{\begin{array}{ll}4-x-x^{2} & \text { if } x \leq 2 \\ 2 x-7 & \text { if } x>2\end{array}\right.f(x)={4−x−x22x−7​ if x≤2 if x>2​ at x=2x=2x=2.

Problem 28527

Problem 28528

Is there a limit to the instantaneous rate of change for f(x)=x2f(x)=x^{2}f(x)=x2? Explain your reasoning.

Problem 28529

Evaluate the limit: lim⁡x→03x+16−4x\lim _{x \rightarrow 0} \frac{\sqrt{3x + 16} - 4}{x}limx→0​x3x+16​−4​

Problem 28530

Find the limit as xxx approaches 0 for the expression ∣x∣x\frac{|x|}{x}x∣x∣​. What is the result?

Problem 28531

Find the limit: lim⁡h→0(3+h)2−9h\lim _{h \rightarrow 0} \frac{(3+h)^{2}-9}{h}limh→0​h(3+h)2−9​

Problem 28532

Find the lower Riemann sum for f(x)=2−x2f(x)=2-x^{2}f(x)=2−x2 on [0,1][0,1][0,1] using the partition P=[0,12,34,1]P=[0, \frac{1}{2}, \frac{3}{4}, 1]P=[0,21​,43​,1].

Problem 28533

Find critical numbers for: (a) y=x45(x−4)2y=x^{\frac{4}{5}}(x-4)^{2}y=x54​(x−4)2, (b) g(θ)=4θ−tan⁡θg(\theta)=4\theta-\tan\thetag(θ)=4θ−tanθ.

Problem 28534

Find the local maximum of f(x)=x6e−x f(x) = x^{6} e^{-x} f(x)=x6e−x by finding critical points with f′(x)=0 f'(x)=0 f′(x)=0 and using the derivative test.

Problem 28535

Solve the differential equation y(x+1)dydx=x(1+y2)y(x+1) \frac{d y}{d x}=x(1+y^{2})y(x+1)dxdy​=x(1+y2).

Problem 28536

Find the upper Riemann sum for f(x)=x2f(x)=x^{2}f(x)=x2 on [−1,1][-1,1][−1,1] using the partition P=[−1,−12,12,34,1]P=\left[-1,-\frac{1}{2}, \frac{1}{2}, \frac{3}{4}, 1\right]P=[−1,−21​,21​,43​,1].

Problem 28537

Estimate the integral ∫0122x2dx\int_{0}^{12} 2 x^{2} d x∫012​2x2dx using Riemann sums with the right endpoint method and n=6n=6n=6.

Problem 28538

Find the lower Riemann sum for f(x)=sin⁡(x)f(x)=\sin(x)f(x)=sin(x) on [0,π][0, \pi][0,π] using the partition P=[0,π3,3π4,π]P=\left[0, \frac{\pi}{3}, \frac{3 \pi}{4}, \pi\right]P=[0,3π​,43π​,π].

Problem 28539

Given the curve on the xy-plane, find the intervals where fff is increasing based on the points provided. (Use interval notation.)

Problem 28540

A balloon rises at 8ft/sec8 \mathrm{ft/sec}8ft/sec from 150 feet away. How fast is the distance to the observer increasing at 50 feet high?

Problem 28541

Differentiate $y=x^{2} \sqrt{3-4 x}$ with respect to $x$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{e^{x}-1}{x}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin (4 x)}{x}$ .

Find the limit: $\lim _{x \rightarrow \frac{\pi}{2}} \frac{3 \sec (x)+5}{\tan (x)}$ .

Which statement is guaranteed by the Intermediate Value Theorem for the function $f$ continuous on $[2,4]$ with $f(2)=10$ and $f(4)=20$ ?

Find the minimum value of $f(x)=x \ln x$ . Options: (A) $-e$ , (B) -1, (C) $-\frac{1}{e}$ , (D) 0, (E) no minimum.

Differentiate the function $\frac{(3 x^{2}+2)^{4}}{\sqrt{2 x-1}}$ .

Find the limit as $x$ approaches 15: $\lim _{x \rightarrow 15} 4=\square$

Find the limit: $\lim _{x \rightarrow 0} \frac{\tan ^{-1}(x)-x}{8 x^{3}}$ .

Evaluate the limit as $z$ approaches -1 for the expression $7 z^{2} - 5 z$ .

Calculate the limit: $\lim _{x \rightarrow \pi} \frac{\sin (x)}{1-\cos (x)}$ .

Find the limit as $x$ approaches infinity of $\frac{7 x}{\ln(150 x + e^{x})}$ .

Find the limit as $a$ approaches 6 for the expression $\frac{a^{3}-216}{a-6}$ .

Find the limit as $t$ approaches 6 for the expression $\frac{\frac{1}{t}-\frac{1}{6}}{t-6}$ .

Find the limit: $\lim _{x \rightarrow 2} \frac{x^{2}-6 x+8}{x-2}$

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

Calculate the limit: $\lim _{x \rightarrow 16} \frac{\sqrt{x}-4}{x-16}$ .

Find the limit as $x$ approaches $\frac{\pi}{2}$ for the expression $(\tan (x)-\sec (x))$ .

Evaluate the limit as $a$ approaches 5: $\lim _{a \rightarrow 5} \frac{\frac{1}{a+8}-\frac{1}{13}}{a-5}$ .

Find the limit as $s$ approaches 25 for the expression $\frac{25-s}{5-\sqrt{s}}$ .

Find the limit as $x$ approaches 49 for the expression $\frac{\sqrt{x}-7}{x-49}$ .

Calculate the limit as $x$ approaches 4 for the expression $\frac{x^{3}-5 x^{2}+2 x+8}{x-4}$ .

Find the limits for the piecewise function $f(x)=\left\{\begin{array}{ll}4-x-x^{2} & \text { if } x \leq 2 \\ 2 x-7 & \text { if } x>2\end{array}\right.$ at $x=2$ .

Is there a limit to the instantaneous rate of change for $f(x)=x^{2}$ ? Explain your reasoning.

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{3x + 16} - 4}{x}$

Find the limit as $x$ approaches 0 for the expression $\frac{|x|}{x}$ . What is the result?

Find the limit: $\lim _{h \rightarrow 0} \frac{(3+h)^{2}-9}{h}$

Find the lower Riemann sum for $f(x)=2-x^{2}$ on $[0,1]$ using the partition $P=[0, \frac{1}{2}, \frac{3}{4}, 1]$ .

Find critical numbers for: (a) $y=x^{\frac{4}{5}}(x-4)^{2}$ , (b) $g(\theta)=4\theta-\tan\theta$ .

Find the local maximum of $f(x) = x^{6} e^{-x}$ by finding critical points with $f'(x)=0$ and using the derivative test.

Solve the differential equation $y(x+1) \frac{d y}{d x}=x(1+y^{2})$ .

Find the upper Riemann sum for $f(x)=x^{2}$ on $[-1,1]$ using the partition $P=\left[-1,-\frac{1}{2}, \frac{1}{2}, \frac{3}{4}, 1\right]$ .

Estimate the integral $\int_{0}^{12} 2 x^{2} d x$ using Riemann sums with the right endpoint method and $n=6$ .

Find the lower Riemann sum for $f(x)=\sin(x)$ on $[0, \pi]$ using the partition $P=\left[0, \frac{\pi}{3}, \frac{3 \pi}{4}, \pi\right]$ .

Given the curve on the xy-plane, find the intervals where $f$ is increasing based on the points provided. (Use interval notation.)

A balloon rises at $8 \mathrm{ft/sec}$ from 150 feet away. How fast is the distance to the observer increasing at 50 feet high?

Find the diver's speed upon hitting the water if they jump with $2.00 \, \mathrm{m/s}$ from a $10 \, \mathrm{m}$ board.

A person 2m tall moves away from an 8m light. If their shadow grows at $4/9$ m/s, find their walking speed in m/s.

An acute angle in a right triangle increases at 3 rad/min. If the hypotenuse is $5 \mathrm{~cm}$ , find the rate of the opposite side's increase when it is $3 \mathrm{~cm}$ .

A square's sides grow at $3 \mathrm{~cm/s}$ . What is the area increase rate when sides are $5 \mathrm{~cm}$ ?
An observer 50 m south of an intersection sees a car moving east at 45 m/s. How fast is it moving away after 5 seconds?

Evaluate the surface integral:
$\iint_{S} f(x, y, z) \, dS = 2\pi \int_{-1}^{1} g(z_{0} + t) \, dt$ .

Find the radius increase rate of a marshmallow (cylinder) with radius $1 \mathrm{~cm}$ , height $3 \mathrm{~cm}$ , expanding at $2 \mathrm{~cm}^{3}/\mathrm{sec}$ when height is $6 \mathrm{~cm}$ .

A pretzel maker rolls dough into a cylinder. If the length increases at $0.5 \mathrm{~cm/sec}$ , find the radius change rate at radius $1 \mathrm{~cm}$ and length $5 \mathrm{~cm}$ .

Check if the function $f$ is continuous at $x=a$ where $f(x)=\begin{cases}2x-3 & , x \leq 2 \\ x^2 & , x>2\end{cases}$ and $a=2$ .

Find the linear approximation of $f(x)=3 x e^{2 x-10}$ at $x=5$ .

Find the linear approximation of $h(t)=t^{4}-6 t^{3}+3 t-7$ at $t=-3$ .

Determine if $f(x)=0$ for some $x$ in $[-5,-1]$ given $f(-5)=-6$ and $f(-1)=6$ . Choices: Sometimes true, Always false, Always true.

True or False: If $f(x)$ is continuous at $x=-3$ and $f(-3)=4$ , then $\lim_{x \to -3} f(x) = 4$ .

Find the linear approximation for the following functions at the specified points:
1. $f(x)=\cos(2x)$ at $x=\pi$
2. $h(z)=\ln(z^{2}+5)$ at $z=2$

Calculate (a) the zero-point energy of a hydrogen atom in a 1D box of length $1.00 \mathrm{~cm}$ and (b) the ratio to thermal energy $k_{B} T$ at $300 \mathrm{~K}$ .

Find the linear approximation of $f(x)=\cos(2x)$ at the point $x=\pi$ .

Find the value of the infinite series: $\sum_{n=4}^{\infty}\left(\frac{1}{2}\right)^{n}$ .

Find the limit: $\lim _{x \rightarrow \pi} \frac{\sin (x)}{1-\cos (x)}=N u m b$

Find the limit: $\lim _{x \rightarrow \pi} \frac{\sin (x)}{1-\cos (x)}$

Find $x$ values for which the series $\sum_{n=0}^{\infty} \frac{(x+9)^{n}}{2^{n}}$ converges and its sum.

Find the limit as $n$ approaches infinity for $\sum_{n=1}^{\infty} \frac{1}{2^{n}}$ and show it equals 1.

Find the limit of $c_{n}=\ln \left(\frac{9 n-7}{13 n+4}\right)$ as $n$ approaches infinity.

Calculate the sum of the series: $\sum_{k=1}^{\infty} \frac{4^{k+3}}{5^{k-1}}$ .

Find the volume $V = 2\pi \int_{0}^{2} (6 - x - x^2) (x + 1) \, dx$ .

Find the limit: $\lim _{n \rightarrow \infty} c_{n}$ where $c_{n}=\ln \left(\frac{9 n-7}{13 n+4}\right)$ .

Find the length of the curve $y=\sqrt{x+2}$ for $-1 \leq x \leq 0$ . Which integral represents this length?

Find the length of the curve $y=\sqrt{x+4}$ for $-3 \leq x \leq -2$ . Select the correct integral:
$\int_{-3}^{-2} \sqrt{\frac{4 x+17}{4 x+16}} dx$

Find the length of the curve $y=\int_{1}^{x} \sqrt{t^{2}-1} d t$ for $1 \leq x \leq \sqrt{3}$ .

Find the curve length for $x=(y+2)^{2}$ from $y=1$ to $y=4$ : $\int_{1}^{4}(2y+4) dy$ .