Calculus

Problem 29401

Calculate the integral: $\int x^{3} \ln x \, dx$

See Solution

Problem 29402

Find $\lim _{x \rightarrow-4} f(x)$ if $2x + 4 \leq f(x) \leq x^2 + 10x + 20$ .

See Solution

Problem 29403

Evaluate the limit as $x$ approaches 0: $\lim_{x \rightarrow 0} \left( \frac{1}{t \sqrt{1+t}}-\frac{1}{t} \right)$ . Choose from: $-\frac{2}{3}$ , $\frac{1}{2}$ , 2, $-\frac{1}{2}$ .

See Solution

Problem 29404

Find the limit as $x$ approaches $-\infty$ for $e^{3x} + \frac{\sqrt{1 + 4x^6}}{2 - x^3}$ .

See Solution

Problem 29405

Berechne die Integrale: a) $\int_{0}^{3} (x^{4}-5) \, dx$ , b) $\int_{2}^{3} (3 x^{4}+2 x^{3}) \, dx$ , c) $\int_{1}^{4} (x-2)^{2} \, dx$ .

See Solution

Problem 29406

Find $f^{\prime}(x)$ using the limit definition of a derivative for $f(x)=\sqrt{x}$ . Options: $\frac{1}{\sqrt{x}}$ , $\frac{1}{2 \sqrt{x}}$ , $\frac{\sqrt{x}}{2}$ , $\frac{2}{\sqrt{x}}$ .

See Solution

Problem 29407

Find the derivative of $\frac{e^{x}}{1-e^{x}}$ . Choices: A) $\frac{-e^{x}}{(1-e^{x})^{2}}$ , B) $\frac{e^{x}}{(1-e^{2x})^{2}}$ , C) $\frac{e^{2x}}{(1-e^{x})^{2}}$ , D) $\frac{e^{x}}{(1-e^{x})^{2}}$ .

See Solution

Problem 29408

Find the derivative of $\ln(x \log_{5} x)$ .

See Solution

Problem 29409

Find the derivative: $\frac{d}{d t} \tan \left(e^{\sin (t)}\right)$ .

See Solution

Problem 29410

Find the derivative of $y=(\sqrt{x})^{\frac{1}{x}}$ .

See Solution

Problem 29411

Differentiate these functions: a. $f(x)=\ln(2x^{3})$ , b. $f(x)=e^{x^{7}}$ , c. $f(x)=\ln(11x^{7})$ , d. $f(x)=e^{x^{2}+x^{3}}$ , e. $f(x)=\log_{e}(7x^{-2})$ , f. $f(x)=e^{-x}$ , g. $f(x)=\ln(e^{x}+x^{3})$ , h. $f(x)=\ln(e^{x}x^{3})$ , i. $f(x)=\ln\left(\frac{x^{2}+1}{x^{3}-x}\right)$ .

See Solution

Problem 29412

Find the derivative of $(\sqrt{x})^{\frac{1}{x}}$ with respect to $x$ .

See Solution

Problem 29413

Differentiate these functions: a. $f(x)=\ln(2x^3)$ , b. $f(x)=e^{x^7}$ , c. $f(x)=\ln(11x^7)$ , d. $f(x)=e^{x^2+x^3}$ , e. $f(x)=\log_e(7x^{-2})$ , f. $f(x)=e^{-x}$ , g. $f(x)=\ln(e^x+x^3)$ , h. $f(x)=\ln(e^xx^3)$ , i. $f(x)=\ln\left(\frac{x^2+1}{x^3-x}\right)$ .

See Solution

Problem 29414

Calculate the integral: $\int \frac{5 x^{3-x}}{x^{6}} \, dx$

See Solution

Problem 29415

Find the derivatives of these functions, where $a$ and $b$ are constants in (g), (h), and (p): (a) $y=(x^{2}+1)e^{3x}$ , (b) $y=e^{x^{2}+3x}$ , (c) $y=\frac{e^{2x}+e^{-2x}}{x^{2}}$ , (d) $y=3^{x^{2}+3x}$ , (e) $y=x5^{3x}$ , (f) $y=\frac{3^{x}-3^{-x}}{2}$ , (g) $y=xe^{ax^{2}+1}$ , (h) $y=\sqrt{1+ae^{x}}$ , (i) $y=(2x+e^{x^{2}})^{4}$ , (j) $y=\ln(x^{2}+2x)$ , (k) $y=\log_{2}(3x+4)$ , (l) $y=x\ln(x^{2}+1)$ , (m) $y=\log_{3}(x^{2}+5)$ , (n) $y=\frac{x\ln x}{x^{2}+1}$ , (o) $y=\ln(x+5e^{3x})$ , (p) $y=ax\ln(x^{2}+b^{2})$ , (q) $y=(3x)^{5x}$ , (r) $y=x^{\ln x}$ , (s) $y=(\ln x)^{x}$ , (t) $y=(3x+2)^{2x-1}$ .

See Solution

Problem 29416

Find the derivatives of these functions: (a) $y=(x^{2}+1)e^{3x}$ , (b) $y=e^{x^{2}+3x}$ , (c) $y=\frac{e^{2x}+e^{-2x}}{x^{2}}$ , (d) $y=3^{x^{2}+3x}$ , (e) $y=x5^{3x}$ , (f) $y=\frac{3^{x}-3^{-x}}{2}$ , (g) $y=xe^{ax^{2}+1}$ , (h) $y=\sqrt{1+ae^{x}}$ , (i) $y=(2x+e^{x^{2}})^{4}$ , (j) $y=\ln(x^{2}+2x)$ , (k) $y=\log_{2}(3x+4)$ , (l) $y=x\ln(x^{2}+1)$ , (m) $y=\log_{3}(x^{2}+5)$ , (n) $y=\frac{x\ln x}{x^{2}+1}$ , (o) $y=\ln(x+5e^{3x})$ , (p) $y=ax\ln(x^{2}+b^{2})$ , (q) $y=(3x)^{5x}$ , (r) $y=x^{\ln x}$ , (s) $y=(\ln x)^{x}$ , (t) $y=(3x+2)^{2x-1}$ .

See Solution

Problem 29417

Evaluate the integral: $\int \csc 3 x^{2} \cot 3 x^{2} \, dx$ .

See Solution

Problem 29418

Find the derivative of $\ln(x \log_{5} x)$ . What is $\frac{d}{d x} \ln(x \log_{5} x)$ ?

See Solution

Problem 29419

Calculate the account balance after 5 years for a \$12,000 deposit at a 5.5% continuous interest rate.

See Solution

Problem 29420

Water leaks from a conical tank at $10,000 \mathrm{~cm}^{3}/\mathrm{min}$ . Find the inflow rate when water is $2 \mathrm{~m}$ high.

See Solution

Problem 29421

Find the derivatives of $\sinh x$ and $\cosh x$ in terms of themselves. Then, find the derivatives of $y=x \sinh x$ and $y=\cosh(x^2)$ .

See Solution

Problem 29422

Find the derivative of $\cot^{-1}\left(\frac{1}{x}\right) + x \cos^{-1}(\sqrt{x})$ .

See Solution

Problem 29423

Find the derivatives of $\sinh x$ and $\cosh x$ in terms of themselves. Then find the derivatives of $y=x \sinh x$ and $y=\cosh(x^2)$ .

See Solution

Problem 29424

Find the derivative of $y=(\sqrt{x})^{\frac{1}{x}}$ .

See Solution

Problem 29425

Identify the false statement about the function $f(x)=e^{-0.005 x}$ from the options provided.

See Solution

Problem 29426

Sketch the graph of $F(x)=\frac{x^{2}-1}{|x-1|}$ and find the limits: 1. $\lim _{x \rightarrow 1^{+}} F(x)$ , 2. $\lim _{x \rightarrow 1^{-}} F(x)$ , 3. $\lim _{x \rightarrow 1} F(x)$ .

See Solution

Problem 29427

Find the average rate of change of $f(t)=64-16 t^{2}$ on $0 \leq t \leq 2$ . Choices: A. -4 B. 32 C. -32 D. -64

See Solution

Problem 29428

$f(x)=\frac{\cos x}{x}$ için doğru olanı seçin: A) $x=\frac{\pi}{2}$ sonsuz süreksiz, B) kaldırılabilir süreksiz, C) $x=0$ kaldırılabilir, D) sıçrama, E) $x=0$ sonsuz süreksiz.

See Solution

Problem 29429

How fast is an object falling after $4 \mathrm{sec}$ when dropped near Earth's surface? (Ignore air resistance.)

See Solution

Problem 29430

Find local extrema of $f(x)=2x^3+6x^2-18x+12$ : local max at $x=$ , local min at $x=$ .

See Solution

Problem 29431

Find the derivative $h^{\prime}(x)$ of the function $h(x)=\frac{\cos(5x^{2})}{\sqrt{6x^{3}+1}}$ .

See Solution

Problem 29432

Find the integral of $\cos \frac{x}{2}$ with respect to $x$ .

See Solution

Problem 29433

Evaluate the integral: $\int\left(\frac{4 x^{19}-5 x^{-8}}{x^{2}}\right) d x$

See Solution

Problem 29434

Find the limits of the piecewise function $f(x)$ as $x$ approaches -1 from both sides.

See Solution

Problem 29435

Find the limit of the piecewise function $f(x)$ as $x$ approaches -1.

See Solution

Problem 29436

Given the piecewise function:
1. Find $\lim_{x \to -1^-} f(x)$ .
2. Find $\lim_{x \to -1^+} f(x)$ .
3. Find $\lim_{x \to -1} f(x)$ and identify any discontinuity type.

See Solution

Problem 29437

Find the integral of $(2 x+1)^{2}$ with respect to $x$ .

See Solution

Problem 29438

Find the dimensions of an open box with maximum volume made from a 32-inch square material.

See Solution

Problem 29439

Find the limit of $f(x)$ as $x$ approaches $-3$ from the left for the piecewise function defined above.

See Solution

Problem 29440

Find the limits of the piecewise function $f(x)$ as $x$ approaches 2 from both sides and check for continuity.

See Solution

Problem 29441

Find the limits as $x$ approaches $-3$ from the left and right, and check continuity at $x=-3$ .

See Solution

Problem 29442

Find the limits as $x$ approaches $-3$ from the left and right for the piecewise function $f(x)$ and check continuity at $x=-3$ .

See Solution

Problem 29443

Find the limits: [a] as $x$ approaches $-3^{-}$ for $f(x)$ [b] as $x$ approaches $-3^{+}$ for $f(x)$ .

See Solution

Problem 29444

Find $f(3)$ and $f(4)$ for $f(x)=x^{3}-3 x^{2}-2 x-7$ . Does the IVT guarantee a real zero between 3 and 4?

See Solution

Problem 29445

Evaluate the integral $c \int e^{-c t} dt$ .

See Solution

Problem 29446

Let $f(x)=\frac{x^{2}-x-2}{x-1}$ . Find limits as $x \to 1$ , $\infty$ , verify $f(x)=x-\frac{2}{x-1}$ , and analyze asymptotes.

See Solution

Problem 29447

Find the limits of the piecewise function $f(x)$ at $x = -1$ and $x = 2$ , and check continuity at these points.

See Solution

Problem 29448

Given $f(x)=x^{3}-6 x^{2}+3$ find: (a) $f(-1)$ and $f(5)$ , (b) critical points, (c) max/min in $[-1,5]$ , (d) their values.

See Solution

Problem 29449

Evaluate the integral of $\frac{x d x}{\sqrt{7-2 x^{2}}}$ .

See Solution

Problem 29450

What does $\lim g(x)=6$ as $x \rightarrow 3^{+}$ mean? Choose the correct interpretation: A. $x$ approaches 3, $g(x)$ approaches 6 B. $g(x)$ approaches 6, $x$ approaches 3 from the left C. $x$ approaches 6, $g(x)$ approaches 3 from the right D. $x$ approaches 3 from the right, $g(x)$ approaches 6

See Solution

Problem 29451

Calculate $\int \sin ^{2} 4 x \, dx$ . Choose the correct answer from the options given.

See Solution

Problem 29452

Evaluate the integral of $(5 \sqrt{x}-4)^{2}$ with respect to $x$ .

See Solution

Problem 29453

Evaluate the integral: $\int \sin ^{3} \frac{x}{2} d x$ . Choose the correct answer from the options provided.

See Solution

Problem 29454

Estimate the integral of $f(x)=5 x$ from $x=1$ to $x=7$ using $n=3$ intervals. What are the left Riemann sum $x$ -values?

See Solution

Problem 29455

Estimate the integral of $f(x)=5x$ from $x=1$ to $x=7$ using $n=3$ intervals. What are the right Riemann sum $x$ -values?

See Solution

Problem 29456

Finde die Bedeutung von $f(15) = 8$ , $f'(25) = 0$ , $f''(25) < 0$ und prüfe die Funktion $f(t) = 0,15 \cdot (t - 15) \cdot e^{-0,1 t} + 8$ . Bestimme $h(x) = 5 x^{2} e^{x}$ für den Deich.

See Solution

Problem 29457

Calculate the integral $\int \frac{2 x^{3}-1}{x^{2}+1} d x$ .

See Solution

Problem 29458

Find the limit: $\lim_{{x \to 0}} \frac{x^{2}+2x}{x^{3}-3x}$ .

See Solution

Problem 29459

Calculate the integral: $\int \frac{x \cdot d x}{\sqrt{1-4 x^{2}}}$ .

See Solution

Problem 29460

Calculate the integral: $\int \frac{x d x}{\left(3 x^{2}+4\right)^{3}}$

See Solution

Problem 29461

Find the integral: $\int x \sec^{2} x \, dx$

See Solution

Problem 29462

Calculate the integral $\int x \sin x \, dx$ .

See Solution

Problem 29463

Calculate the integral $\int x^{2} \sin x \, dx$ .

See Solution

Problem 29464

Evaluate the integral: $\int_{1}^{e} \frac{\ln x}{x} d x$

See Solution

Problem 29465

Approximate $f(-1)$ using $f(-2)=-5$ and $f'(-2)=-7$ .

See Solution

Problem 29466

Calculate the integral: $\int x^{2} \cos x \, dx$

See Solution

Problem 29467

Find the integral: $\int \tan^{-1} x \, dx$

See Solution

Problem 29468

Verify if $y=\sqrt{x} \int_{4}^{x} \frac{\cos (t)}{\sqrt{t}} d t$ solves $2 x \frac{d y}{d x}-y=2 x \cos (x)$ .

See Solution

Problem 29469

Find the derivative of the function $f(x)=x^{2} \ln(x^{3})$ .

See Solution

Problem 29470

Find the growth of average computer devices per person over 10 years, given $2\%$ production and $1\%$ population growth.

See Solution

Problem 29471

Find the derivative of $f(x)=\frac{x^{2}+2}{e^{x}}$ .

See Solution

Problem 29472

Find the tangent line equation to $y=6 \sin x$ at $\left(\frac{\pi}{6}, 3\right)$ in the form $y=m x+b$ .

See Solution

Problem 29473

Find the derivative of $f(x)=\frac{x^{2}+2}{e^{x}}$ . Select the correct expression for $y'$ .

See Solution

Problem 29474

Find the tangent line equation for $y=6 \sin x$ at the point $\left(\frac{\pi}{6}, 3\right)$ in the form $y=m x+b$ .

See Solution

Problem 29475

Find the derivative of $f(x)=x^{2} \ln(x^{3})$ . What is $f'(x)$ ?

See Solution

Problem 29476

Find the stationary point of $f(x)=4+(x-8)^{2}$ . Determine if it's a maximum or minimum.

See Solution

Problem 29477

Find the maximum of the function $f(x)=4 \ln(x^{2}) - 2x$ .

See Solution

Problem 29478

Find the maximum of $f(x)=4 \ln \left(x^{2}\right)-2 x$ . Choose from: (4, 2.09), (4, 3.09), (4, 1.09), (4, 0.09).

See Solution

Problem 29479

Find the stationary point of $f(x)=4+(x-8)^{2}$ . Determine if it's a max or min. Options: $16 \text{ min}$ , $28 \text{ min}$ , $37 \text{ max}$ , $45 \text{ max}$ .

See Solution

Problem 29480

Find the stationary point of $f(x)=4+(x-8)^2$ . Determine if it's a max or min. Options: 6 min, 8 min, 7 max, 5 max.

See Solution

Problem 29481

Find the stationary point of $f(x)=4+(x-8)^{2}$ and determine if it's a max or min.

See Solution

Problem 29482

Find the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for $f(x, y)=3 x+2 y-2 x^{2}-y^{2}$ .

See Solution

Problem 29483

Find the stationary point of $f(x, y)=3 x+2 y-2 x^{2}-y^{2}$ . Determine if it's a max, min, or neither using second order conditions.

See Solution

Problem 29484

Find the stationary point of $f(x, y)=3x+2y-2x^{2}-y^{2}$ and classify it as max, min, or neither.

See Solution

Problem 29485

Find the rectangle with maximum area inside the ellipse $\left(\frac{x}{5}\right)^{2}+\left(\frac{y}{7}\right)^{2}=1$ .

See Solution

Problem 29486

Find the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for $f(x, y)=3x+2y-2x^2-y^2$ .

See Solution

Problem 29487

Maximize $z=2 x^{2}+4 y^{2}$ with the constraint $2 x=2 y-4$ . Find $x, y, \lambda$ , and the max value of $z$ .

See Solution

Problem 29488

Find the tangent line equation for $y=x^{3}+3 x^{2}+2$ at its point of inflection.

See Solution

Problem 29489

Find the slope of the tangent line to $y=\ln \left(\frac{x}{2}\right)$ at $x=4$ .

See Solution

Problem 29490

Determine the truth of these statements about the function $f(x)$ : I. $\lim _{x \rightarrow 3} f(x)$ exists. II. $f$ is continuous at $x=3$ . III. $f$ is differentiable at $x=3$ . Options: (A) None (B) I only (C) II only (D) I and II only (E) I, II, and III.

See Solution

Problem 29491

Find the 4th derivative of $f(x)=(2x+1)^{4}$ at $x=0$ .

See Solution

Problem 29492

Determine where the function $f$ with derivative $f^{\prime}(x)=x^{2}-\frac{2}{x}$ is decreasing. Choose from the options.

See Solution

Problem 29493

f(x)에서 $\lim _{x \rightarrow 1} f(x)$ 의 값을 구하시오.

See Solution

Problem 29494

$\lim _{x \rightarrow 2} f(x)$ 의 값을 구하시오, 여기서 $f(x)=\begin{cases} x^{2}-4x+3 & (x \neq 2) \\ 3 & (x=2) \end{cases}$ .

See Solution

Problem 29495

극한값 $\lim _{x \rightarrow 2} f(x)$ 존재 시, 함수 $f(x)$ 에서 상수 $k$ 의 값을 구하라.

See Solution

Problem 29496

$\lim _{x \rightarrow-\infty}\left(\sqrt{1+2 x+x^{2}}+x\right)$ 의 값을 구하시오.

See Solution

Problem 29497

함수 $f(x)=\left\{\begin{array}{cl}2 & (x<0) \\ 2 x(2-x) & (0 \leq x<1) \\ 1-x & (x \geq 1)\end{array}\right.$ 에서 $\lim _{x \rightarrow 0-} f(x)+\lim _{x \rightarrow 1-} f(x)$ 의 값을 구하시오.

See Solution

Problem 29498

두 함수 $f(x)$ , $g(x)$ 에 대해 $\lim _{x \rightarrow 1} f(x)=-\infty$ 이고 $\lim _{x \rightarrow 1}\{f(x)+3 g(x)\}=4$ 일 때, $\lim _{x \rightarrow 1} \frac{f(x)-g(x)}{5 f(x)+2 g(x)}$ 의 값을 구하시오.

See Solution

Problem 29499

함수 $f(x)$ 에 대해 $\lim _{x \rightarrow \infty} \frac{f(x)}{x}=3$ 일 때, $\lim _{x \rightarrow \infty} \frac{f(x)+3 x^{2}}{2 x^{2}-f(x)}$ 의 값을 구하시오.

See Solution

Problem 29500

함수 $f(x)$ 가 $\lim _{x \rightarrow \infty} \frac{f(x)}{x}=3$ 일 때, $\lim _{x \rightarrow \infty} \frac{x^{2}+x f(x)}{x^{2}-f(x)}$ 의 값을 구하라.

See Solution

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Calculus

Problem 29401

Calculate the integral: ∫x3ln⁡x dx\int x^{3} \ln x \, dx∫x3lnxdx

Problem 29402

Find lim⁡x→−4f(x)\lim _{x \rightarrow-4} f(x)limx→−4​f(x) if 2x+4≤f(x)≤x2+10x+202x + 4 \leq f(x) \leq x^2 + 10x + 202x+4≤f(x)≤x2+10x+20.

Problem 29403

Evaluate the limit as x x x approaches 0: lim⁡x→0(1t1+t−1t) \lim_{x \rightarrow 0} \left( \frac{1}{t \sqrt{1+t}}-\frac{1}{t} \right) limx→0​(t1+t​1​−t1​). Choose from: −23-\frac{2}{3}−32​, 12\frac{1}{2}21​, 2, −12-\frac{1}{2}−21​.

Problem 29404

Find the limit as xxx approaches −∞-\infty−∞ for e3x+1+4x62−x3e^{3x} + \frac{\sqrt{1 + 4x^6}}{2 - x^3}e3x+2−x31+4x6​​.

Problem 29405

Berechne die Integrale: a) ∫03(x4−5) dx\int_{0}^{3} (x^{4}-5) \, dx∫03​(x4−5)dx, b) ∫23(3x4+2x3) dx\int_{2}^{3} (3 x^{4}+2 x^{3}) \, dx∫23​(3x4+2x3)dx, c) ∫14(x−2)2 dx\int_{1}^{4} (x-2)^{2} \, dx∫14​(x−2)2dx.

Problem 29406

Find f′(x)f^{\prime}(x)f′(x) using the limit definition of a derivative for f(x)=xf(x)=\sqrt{x}f(x)=x​. Options: 1x\frac{1}{\sqrt{x}}x​1​, 12x\frac{1}{2 \sqrt{x}}2x​1​, x2\frac{\sqrt{x}}{2}2x​​, 2x\frac{2}{\sqrt{x}}x​2​.

Problem 29407

Problem 29408

Find the derivative of ln⁡(xlog⁡5x)\ln(x \log_{5} x)ln(xlog5​x).

Problem 29409

Find the derivative: ddttan⁡(esin⁡(t))\frac{d}{d t} \tan \left(e^{\sin (t)}\right)dtd​tan(esin(t)).

Problem 29410

Find the derivative of y=(x)1xy=(\sqrt{x})^{\frac{1}{x}}y=(x​)x1​.

Problem 29411

Problem 29412

Find the derivative of (x)1x(\sqrt{x})^{\frac{1}{x}}(x​)x1​ with respect to xxx.

Problem 29413

Problem 29414

Calculate the integral: ∫5x3−xx6 dx\int \frac{5 x^{3-x}}{x^{6}} \, dx∫x65x3−x​dx

Problem 29415

Problem 29416

Problem 29417

Evaluate the integral: ∫csc⁡3x2cot⁡3x2 dx\int \csc 3 x^{2} \cot 3 x^{2} \, dx∫csc3x2cot3x2dx.

Problem 29418

Find the derivative of ln⁡(xlog⁡5x)\ln(x \log_{5} x)ln(xlog5​x). What is ddxln⁡(xlog⁡5x)\frac{d}{d x} \ln(x \log_{5} x)dxd​ln(xlog5​x)?

Problem 29419

Calculate the account balance after 5 years for a \$12,000 deposit at a 5.5% continuous interest rate.

Problem 29420

Water leaks from a conical tank at 10,000 cm3/min10,000 \mathrm{~cm}^{3}/\mathrm{min}10,000 cm3/min. Find the inflow rate when water is 2 m2 \mathrm{~m}2 m high.

Problem 29421

Find the derivatives of sinh⁡x\sinh xsinhx and cosh⁡x\cosh xcoshx in terms of themselves. Then, find the derivatives of y=xsinh⁡xy=x \sinh xy=xsinhx and y=cosh⁡(x2)y=\cosh(x^2)y=cosh(x2).

Problem 29422

Find the derivative of cot⁡−1(1x)+xcos⁡−1(x)\cot^{-1}\left(\frac{1}{x}\right) + x \cos^{-1}(\sqrt{x})cot−1(x1​)+xcos−1(x​).

Problem 29423

Find the derivatives of sinh⁡x\sinh xsinhx and cosh⁡x\cosh xcoshx in terms of themselves. Then find the derivatives of y=xsinh⁡xy=x \sinh xy=xsinhx and y=cosh⁡(x2)y=\cosh(x^2)y=cosh(x2).

Problem 29424

Find the derivative of y=(x)1xy=(\sqrt{x})^{\frac{1}{x}}y=(x​)x1​.

Problem 29425

Identify the false statement about the function f(x)=e−0.005xf(x)=e^{-0.005 x}f(x)=e−0.005x from the options provided.

Problem 29426

Problem 29427

Find the average rate of change of f(t)=64−16t2f(t)=64-16 t^{2}f(t)=64−16t2 on 0≤t≤20 \leq t \leq 20≤t≤2. Choices: A. -4 B. 32 C. -32 D. -64

Problem 29428

f(x)=cos⁡xxf(x)=\frac{\cos x}{x}f(x)=xcosx​ için doğru olanı seçin: A) x=π2x=\frac{\pi}{2}x=2π​ sonsuz süreksiz, B) kaldırılabilir süreksiz, C) x=0x=0x=0 kaldırılabilir, D) sıçrama, E) x=0x=0x=0 sonsuz süreksiz.

Problem 29429

How fast is an object falling after 4sec4 \mathrm{sec}4sec when dropped near Earth's surface? (Ignore air resistance.)

Problem 29430

Find local extrema of f(x)=2x3+6x2−18x+12f(x)=2x^3+6x^2-18x+12f(x)=2x3+6x2−18x+12: local max at x=x=x=, local min at x=x=x=.

Problem 29431

Find the derivative h′(x)h^{\prime}(x)h′(x) of the function h(x)=cos⁡(5x2)6x3+1h(x)=\frac{\cos(5x^{2})}{\sqrt{6x^{3}+1}}h(x)=6x3+1​cos(5x2)​.

Problem 29432

Find the integral of cos⁡x2\cos \frac{x}{2}cos2x​ with respect to xxx.

Problem 29433

Evaluate the integral: ∫(4x19−5x−8x2)dx\int\left(\frac{4 x^{19}-5 x^{-8}}{x^{2}}\right) d x∫(x24x19−5x−8​)dx

Problem 29434

Find the limits of the piecewise function f(x)f(x)f(x) as xxx approaches -1 from both sides.

Problem 29435

Find the limit of the piecewise function f(x)f(x)f(x) as xxx approaches -1.

Problem 29436

Given the piecewise function:1. Find lim⁡x→−1−f(x)\lim_{x \to -1^-} f(x)limx→−1−​f(x).2. Find lim⁡x→−1+f(x)\lim_{x \to -1^+} f(x)limx→−1+​f(x).3. Find lim⁡x→−1f(x)\lim_{x \to -1} f(x)limx→−1​f(x) and identify any discontinuity type.

Problem 29437

Find the integral of (2x+1)2(2 x+1)^{2}(2x+1)2 with respect to xxx.

Problem 29438

Find the dimensions of an open box with maximum volume made from a 32-inch square material.

Problem 29439

Find the limit of f(x)f(x)f(x) as xxx approaches −3-3−3 from the left for the piecewise function defined above.

Problem 29440

Find the limits of the piecewise function f(x)f(x)f(x) as xxx approaches 2 from both sides and check for continuity.

Problem 29441

Find the limits as xxx approaches −3-3−3 from the left and right, and check continuity at x=−3x=-3x=−3.

Problem 29442

Find the limits as xxx approaches −3-3−3 from the left and right for the piecewise function f(x)f(x)f(x) and check continuity at x=−3x=-3x=−3.

Problem 29443

Calculate the integral: $\int x^{3} \ln x \, dx$

Find $\lim _{x \rightarrow-4} f(x)$ if $2x + 4 \leq f(x) \leq x^2 + 10x + 20$ .

Evaluate the limit as $x$ approaches 0: $\lim_{x \rightarrow 0} \left( \frac{1}{t \sqrt{1+t}}-\frac{1}{t} \right)$ . Choose from: $-\frac{2}{3}$ , $\frac{1}{2}$ , 2, $-\frac{1}{2}$ .

Find the limit as $x$ approaches $-\infty$ for $e^{3x} + \frac{\sqrt{1 + 4x^6}}{2 - x^3}$ .

Berechne die Integrale: a) $\int_{0}^{3} (x^{4}-5) \, dx$ , b) $\int_{2}^{3} (3 x^{4}+2 x^{3}) \, dx$ , c) $\int_{1}^{4} (x-2)^{2} \, dx$ .

Find $f^{\prime}(x)$ using the limit definition of a derivative for $f(x)=\sqrt{x}$ . Options: $\frac{1}{\sqrt{x}}$ , $\frac{1}{2 \sqrt{x}}$ , $\frac{\sqrt{x}}{2}$ , $\frac{2}{\sqrt{x}}$ .

Find the derivative of $\ln(x \log_{5} x)$ .

Find the derivative: $\frac{d}{d t} \tan \left(e^{\sin (t)}\right)$ .

Find the derivative of $y=(\sqrt{x})^{\frac{1}{x}}$ .

Find the derivative of $(\sqrt{x})^{\frac{1}{x}}$ with respect to $x$ .

Calculate the integral: $\int \frac{5 x^{3-x}}{x^{6}} \, dx$

Evaluate the integral: $\int \csc 3 x^{2} \cot 3 x^{2} \, dx$ .

Find the derivative of $\ln(x \log_{5} x)$ . What is $\frac{d}{d x} \ln(x \log_{5} x)$ ?

Water leaks from a conical tank at $10,000 \mathrm{~cm}^{3}/\mathrm{min}$ . Find the inflow rate when water is $2 \mathrm{~m}$ high.

Find the derivatives of $\sinh x$ and $\cosh x$ in terms of themselves. Then, find the derivatives of $y=x \sinh x$ and $y=\cosh(x^2)$ .

Find the derivative of $\cot^{-1}\left(\frac{1}{x}\right) + x \cos^{-1}(\sqrt{x})$ .

Find the derivatives of $\sinh x$ and $\cosh x$ in terms of themselves. Then find the derivatives of $y=x \sinh x$ and $y=\cosh(x^2)$ .

Find the derivative of $y=(\sqrt{x})^{\frac{1}{x}}$ .

Identify the false statement about the function $f(x)=e^{-0.005 x}$ from the options provided.

Find the average rate of change of $f(t)=64-16 t^{2}$ on $0 \leq t \leq 2$ . Choices: A. -4 B. 32 C. -32 D. -64

$f(x)=\frac{\cos x}{x}$ için doğru olanı seçin: A) $x=\frac{\pi}{2}$ sonsuz süreksiz, B) kaldırılabilir süreksiz, C) $x=0$ kaldırılabilir, D) sıçrama, E) $x=0$ sonsuz süreksiz.

How fast is an object falling after $4 \mathrm{sec}$ when dropped near Earth's surface? (Ignore air resistance.)

Find local extrema of $f(x)=2x^3+6x^2-18x+12$ : local max at $x=$ , local min at $x=$ .

Find the derivative $h^{\prime}(x)$ of the function $h(x)=\frac{\cos(5x^{2})}{\sqrt{6x^{3}+1}}$ .

Find the integral of $\cos \frac{x}{2}$ with respect to $x$ .

Evaluate the integral: $\int\left(\frac{4 x^{19}-5 x^{-8}}{x^{2}}\right) d x$

Find the limits of the piecewise function $f(x)$ as $x$ approaches -1 from both sides.

Find the limit of the piecewise function $f(x)$ as $x$ approaches -1.

Given the piecewise function:
1. Find $\lim_{x \to -1^-} f(x)$ .
2. Find $\lim_{x \to -1^+} f(x)$ .
3. Find $\lim_{x \to -1} f(x)$ and identify any discontinuity type.

Find the integral of $(2 x+1)^{2}$ with respect to $x$ .

Find the limit of $f(x)$ as $x$ approaches $-3$ from the left for the piecewise function defined above.

Find the limits of the piecewise function $f(x)$ as $x$ approaches 2 from both sides and check for continuity.

Find the limits as $x$ approaches $-3$ from the left and right, and check continuity at $x=-3$ .

Find the limits as $x$ approaches $-3$ from the left and right for the piecewise function $f(x)$ and check continuity at $x=-3$ .

Find the limits: [a] as $x$ approaches $-3^{-}$ for $f(x)$ [b] as $x$ approaches $-3^{+}$ for $f(x)$ .

Find $f(3)$ and $f(4)$ for $f(x)=x^{3}-3 x^{2}-2 x-7$ . Does the IVT guarantee a real zero between 3 and 4?

Evaluate the integral $c \int e^{-c t} dt$ .

Let $f(x)=\frac{x^{2}-x-2}{x-1}$ . Find limits as $x \to 1$ , $\infty$ , verify $f(x)=x-\frac{2}{x-1}$ , and analyze asymptotes.

Find the limits of the piecewise function $f(x)$ at $x = -1$ and $x = 2$ , and check continuity at these points.

Given $f(x)=x^{3}-6 x^{2}+3$ find: (a) $f(-1)$ and $f(5)$ , (b) critical points, (c) max/min in $[-1,5]$ , (d) their values.

Evaluate the integral of $\frac{x d x}{\sqrt{7-2 x^{2}}}$ .

Calculate $\int \sin ^{2} 4 x \, dx$ . Choose the correct answer from the options given.

Evaluate the integral of $(5 \sqrt{x}-4)^{2}$ with respect to $x$ .

Evaluate the integral: $\int \sin ^{3} \frac{x}{2} d x$ . Choose the correct answer from the options provided.

Estimate the integral of $f(x)=5 x$ from $x=1$ to $x=7$ using $n=3$ intervals. What are the left Riemann sum $x$ -values?

Estimate the integral of $f(x)=5x$ from $x=1$ to $x=7$ using $n=3$ intervals. What are the right Riemann sum $x$ -values?

Calculate the integral $\int \frac{2 x^{3}-1}{x^{2}+1} d x$ .

Find the limit: $\lim_{{x \to 0}} \frac{x^{2}+2x}{x^{3}-3x}$ .

Calculate the integral: $\int \frac{x \cdot d x}{\sqrt{1-4 x^{2}}}$ .

Calculate the integral: $\int \frac{x d x}{\left(3 x^{2}+4\right)^{3}}$

Find the integral: $\int x \sec^{2} x \, dx$

Calculate the integral $\int x \sin x \, dx$ .

Calculate the integral $\int x^{2} \sin x \, dx$ .

Evaluate the integral: $\int_{1}^{e} \frac{\ln x}{x} d x$

Approximate $f(-1)$ using $f(-2)=-5$ and $f'(-2)=-7$ .

Calculate the integral: $\int x^{2} \cos x \, dx$

Find the integral: $\int \tan^{-1} x \, dx$

Verify if $y=\sqrt{x} \int_{4}^{x} \frac{\cos (t)}{\sqrt{t}} d t$ solves $2 x \frac{d y}{d x}-y=2 x \cos (x)$ .

Find the derivative of the function $f(x)=x^{2} \ln(x^{3})$ .

Find the growth of average computer devices per person over 10 years, given $2\%$ production and $1\%$ population growth.

Find the derivative of $f(x)=\frac{x^{2}+2}{e^{x}}$ .

Find the tangent line equation to $y=6 \sin x$ at $\left(\frac{\pi}{6}, 3\right)$ in the form $y=m x+b$ .

Find the derivative of $f(x)=\frac{x^{2}+2}{e^{x}}$ . Select the correct expression for $y'$ .

Find the tangent line equation for $y=6 \sin x$ at the point $\left(\frac{\pi}{6}, 3\right)$ in the form $y=m x+b$ .