Calculus

Problem 29501

함수 $f(x)$ 가 $\lim _{x \rightarrow 1}\{(x^{2}+1) f(x)\}=6$ 일 때, $\lim _{x \rightarrow 1}\{(x+4) f(x)\}$ 값을 구하시오.

See Solution

Problem 29502

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 29503

함수 $f(x)$ 에 대해 $\lim _{x \rightarrow 1} \frac{f(x)-2}{x-1}=3$ 일 때, $\lim _{x \rightarrow 1} \frac{\{f(x)\}^{2}-2 f(x)}{x^{2} f(x)-f(x)}$ 의 값을 구하세요. ( $f(x) \neq 0$ )

See Solution

Problem 29504

$\lim _{x \rightarrow 0} \frac{3 x^{2}+4 f(x)}{2 x^{2}-f(x)}$ 의 값을 구하라, 단 $\lim _{x \rightarrow 0} \frac{f(x)}{x}=2$ .

See Solution

Problem 29505

두 함수 $f(x), g(x)$ 의 극한을 이용해 가장 작은 값을 찾으세요: (1) $\lim _{x \rightarrow 5}\{2 f(x)+3\}$ , (2) $\lim _{x \rightarrow 5}\{3 f(x)-4 g(x)\}$ , (3) $\lim _{x \rightarrow 5}\{5 f(x) g(x)\}$ , (4) $\lim _{x \rightarrow 5} \frac{6 f(x)}{g(x)}$ , (5) $\lim _{x \rightarrow 5} \frac{3 f(x)-2 g(x)}{4 g(x)}$ .

See Solution

Problem 29506

두 함수 $f(x), g(x)$ 가 주어진 조건을 만족할 때, $\lim _{x \rightarrow-1} \frac{5 f(x)+2 x}{\{g(x)\}^{2}}$ 의 값을 구하시오.

See Solution

Problem 29507

Find the average rate of change of $f(x)=-3x^{2}+4$ between $x=4$ and $x=7$ . Simplify your answer.

See Solution

Problem 29508

Verify the Mean Value Theorem for $f(x)=\left\{\begin{array}{ll}3-x^{2} & \text { if } x \leq 1 \\ \frac{2}{x} & \text { if } x>1\end{array}\right.$ on $[0,2]$ and find values of $c$ .

See Solution

Problem 29509

Calculate the indefinite integral: $\int \sqrt{x}(3 x-4)^{2} \, dx$

See Solution

Problem 29510

Find $\int_{-2}^{7}[3 f(x)-9] d x$ given that $\int_{-2}^{7} f(x) d x=-12$ .

See Solution

Problem 29511

Solve the equation $y' = 5 \sin x - 5 x^4 + 12 e^{-4x}$ with the initial condition $y(0) = 12$ .

See Solution

Problem 29512

Calculate the area between the curves $f(x)=x^{2}+2x$ and $g(x)=-x+4$ from $x=-4$ to $x=2$ .

See Solution

Problem 29513

Calculate the average value of $f(x)=4 \cos ^{3} x \sin x$ on $[0, \frac{2 \pi}{3}]$ . Options: a.) $\frac{5 \pi}{8}$ b.) $\frac{45 \pi}{32}$ c.) $\frac{45}{32 \pi}$ d.) $\frac{15}{16}$

See Solution

Problem 29514

Find the average value and $c$ from the mean value theorem for $f(x)=\cos x$ on $[0, \frac{\pi}{6}]$ .

See Solution

Problem 29515

Find the definite integral: $\int_{1}^{3} x(2 x-5)^{4} d x$ . Choose the correct answer: a.) 24, b.) $\frac{92}{3}$ , c.) $-\frac{274}{3}$ , d.) $-\frac{92}{3}$ .

See Solution

Problem 29516

Estimate $\int_{3}^{5} \frac{2}{x^{2}+4} dx$ using $n=4$ with Simpson's rule. Round to five decimal places.

See Solution

Problem 29517

Create a second-order ODE with $y(x)$ , find $y'(x)$ and $y''(x)$ , then verify a solution $y_C(x)=y(x)+C$ . For what $C$ does $y_C(x_0)=k$ ? Provide an IVP example and another ODE to verify.

See Solution

Problem 29518

Find all $\delta$ values that satisfy $\lim _{x \rightarrow 2} 5 x+4=14$ with $\varepsilon=0.5$ . Options: $\delta=0.0333$ , $\delta=0.01$ , $\delta=0.1$ , $\delta=0.2$ .

See Solution

Problem 29519

Find $\frac{d y}{d x}$ for the equation $3 x^{2} y^{4}-5 \cos (3 x)=y^{5}+2$ .

See Solution

Problem 29520

Find all values of $x$ where the function $f(x)$ is not differentiable, given points at $(1,2)$ , $(1,4)$ , and a sharp point at $(-4,3)$ .

See Solution

Problem 29521

Find the derivative of $y=\frac{x^{9} \sqrt[5]{x^{2}+13}}{x^{3}+27}$ using logarithmic differentiation.

See Solution

Problem 29522

Is there an $x$ in $[0,4]$ such that $f(x)=0$ given $f(0)=-10$ and $f(4)=10$ ? Choose: Always true, Always false, Sometimes true.

See Solution

Problem 29523

Find the derivative of $y=\left(\log _{7}\left(4 x^{2}+25\right)\right)^{3}$ .

See Solution

Problem 29524

Solve the equation $x^{n} \frac{d y}{d x}=x^{m} y$ : (a) Find a trivial solution. (b) Find all solutions $y=y(x)$ . (c) Is there a unique solution for $y(x_0)=y_0$ ?

See Solution

Problem 29525

Find the average velocity of the object with position $s(t)=5 \ln t$ over the interval $[1.5,1.7]$ . Round to 4 decimal places.

See Solution

Problem 29526

Find the derivative of $y=\tan(3^{4x}+1)$ . What is $\frac{dy}{dx}$ ?

See Solution

Problem 29527

Find the rate of change of bacteria $A(t)=16 e^{0.32 t}$ after 12 hours. Choices: 24,619.97, 744.41, 84.61, 238.21.

See Solution

Problem 29528

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

See Solution

Problem 29529

Solve the equation $\frac{1}{x^{n}} \frac{d y}{d x}=y^{m}$ for $y=y(x)$ . Discuss trivial/extraneous solutions and FTOC usage. Is the solution unique for $y(x_0)=y_0$ ?

See Solution

Problem 29530

Find $\lim _{x \rightarrow-1^{+}} f(x)$ for the piecewise function $f(x)$ defined as: $f(x) = -2 + 2x^{2}$ for $x \leq -1$ and $f(x) = -1 + x$ for $x > -1$ .

See Solution

Problem 29531

Find $\lim _{x \rightarrow-3^{+}} f(x)$ for the piecewise function $f(x)$ defined as: $f(x)=x^{2}-8$ for $x \geq-3$ and $f(x)=-5x-10$ for $x<-3$ .

See Solution

Problem 29532

Find the derivative of $f(x)=\ln \left(\frac{x^{6}}{2 x^{5}-7}\right)$ .

See Solution

Problem 29533

Estimate the instantaneous velocity of $s(t)=5 \ln t$ at $t=1.5$ . Consider the average velocity on $[1.5,1.7]$ .

See Solution

Problem 29534

(a) Create an IVP with a unique solution per the Existence and Uniqueness Theorem and verify it. What do you conclude? (b) Create an IVP without a guaranteed unique solution and demonstrate this. What can you conclude? (c) Can an IVP have a solution despite the Existence and Uniqueness Theorem not guaranteeing it? Provide an example and verify.

See Solution

Problem 29535

Find the derivative of $f(x)=-15 \sin x+\frac{11}{x^{4}}+3 \sqrt[7]{x}-5$ .

See Solution

Problem 29536

Find $\lim _{x \rightarrow-2} f(x)$ for the piecewise function $f(x) = \{-2 x^{2}+3 \text{ if } x<-2, 2 x+8 \text{ if } x \geq-2\}$ .

See Solution

Problem 29537

Find $\lim _{x \rightarrow 4} f(x)$ for the piecewise function $f(x) = \{ x^{2}-10 \text{ if } x<4, -9+2x \text{ if } x>4 \}$ .

See Solution

Problem 29538

Estimate the max error in the volume of a sphere with radius error of $0.5 \mathrm{in}$ using $V(r)=\frac{4}{3} \pi r^{3}$ .

See Solution

Problem 29539

Estimate the derivative of $f(x)=\frac{1}{2} x^{3}$ at $x=-2$ . Choose from: $-6$ , $\frac{1}{6}$ , $-\frac{1}{6}$ , 6.

See Solution

Problem 29540

Is the function $f(x)=\sqrt[3]{x}$ differentiable at $x=-27$ ? If not, explain why.

See Solution

Problem 29541

Find the derivative of $y=\arctan(5x^2-1)$ . What is $\frac{dy}{dx}$ ?

See Solution

Problem 29542

Find the derivative of $y=\sin(3x^2-4x+1)$ . What is $\frac{dy}{dx}$ ?

See Solution

Problem 29543

Find the linear approximation of $f(x)=\sqrt[5]{x}$ at $x=-1$ . Which formula represents $L(x)$ ?

See Solution

Problem 29544

Find the tangent line equation for $f(x)=\frac{1}{x^{5}}$ at $x=1$ . Options: $y=\frac{1}{5} x+\frac{4}{5}$ , $y=-5 x-4$ , $y=-5 x+6$ , $y=5 x-4$ .

See Solution

Problem 29545

Find the error in the linear approximation of $f(x)=\sqrt[3]{x}$ near $x=-1$ for estimating $\sqrt[3]{-0.75}$ .

See Solution

Problem 29546

Determine if the function $f(x)$ is continuous at $x=-3$ :
$f(x)=\left\{\begin{array}{ll} 14-3 x^{2}, & x>-3 \\ -13-x, & x<-3 \end{array}\right.$

See Solution

Problem 29547

A cube's edges grow at 6.2 cm/s. Find the volume change rate when edges are 10 cm. Options: $5580 \frac{\mathrm{cm}^{3}}{\mathrm{sec}}$ , $2700 \frac{\mathrm{cm}^{3}}{\mathrm{sec}}$ , $300 \frac{\mathrm{cm}^{3}}{\mathrm{sec}}$ , $1860 \frac{g m^{3}}{s e c}$ .

See Solution

Problem 29548

Find the derivative of $y=e^{5 x} \cos (3 x)$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 29549

Check if the function $f(x)$ is continuous at $x=-2$ , where $f(x)=\begin{cases} 9-x^{2}, & x<-2 \\ 3-x, & x \geq-2 \end{cases}$ .

See Solution

Problem 29550

Find the derivative of $f(x)=\sqrt[3]{x^{6}-64}$ . What is $f'(x)$ ?

See Solution

Problem 29551

Find the derivative of $f(x)=-12 \sin x+x^{7} \cos x$ .

See Solution

Problem 29552

Use Newton's method to solve $-x^{3}-3x+65=0$ and find the third iteration value. Verify your answer.

See Solution

Problem 29553

Check if the function $f(x)$ is continuous at $x=3$ , where $f(x)=5+x^{2}$ for $x<3$ and $f(x)=5+3x$ for $x>3$ .

See Solution

Problem 29554

Find the area between $f(x)=\cos x$ and $g(x)=2-\cos x$ from $x=0$ to $x=2\pi$ .

See Solution

Problem 29555

Find the limit as $y$ approaches $\frac{\pi}{2}$ for the expression $\frac{\sin y}{\cos y - 1}$ .

See Solution

Problem 29556

Evaluate the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{x^{6}+4 x}}{5 x+x^{3}}$ and determine if it equals $\infty$ , $-\infty$ , or a specific value.

See Solution

Problem 29557

Calculate the area between the $x$ -axis and $f(x)=-x^{2}+4$ from $x=-1$ to $x=3$ . Options: $\frac{34}{3}$ , $\frac{28}{3}$ , $\frac{29}{3}$ , $\frac{20}{3}$ .

See Solution

Problem 29558

Calculate the indefinite integral: $\int(6 \csc x \cot x - \frac{25}{x^{4}}) \, dx$ .

See Solution

Problem 29559

Find the total area between $f(x)=3x^{3}-x^{2}-10x$ and $g(x)=-x^{2}+2x$ . Options: 24, 32, 12, or 8 units².

See Solution

Problem 29560

Evaluate the integral $\int_{0}^{\frac{3 \pi}{4}} 15 \tan ^{9} x \sec ^{2} x \, dx$ . What is its value?

See Solution

Problem 29561

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

See Solution

Problem 29562

Evaluate the integral: $\int_{0}^{\frac{3 \pi}{4}} 15 \tan ^{9} x \sec ^{2} x d x$ . What is its value?

See Solution

Problem 29563

Determine if the limit is $\infty$ , $-\infty$ , or a specific value: $\lim _{x \rightarrow-\infty}\left(1+x-6 x^{5}\right)$

See Solution

Problem 29564

Determine if the limit $\lim _{x \rightarrow-\infty} \frac{\sqrt{9 x^{6}+9 x^{5}}}{4-3 x^{3}}$ equals $\infty$ , $-\infty$ , or a specific value.

See Solution

Problem 29565

Find the limit as $x$ approaches $-\infty$ : $\lim _{x \rightarrow-\infty} \frac{\sqrt{4 x^{2}-4 x}}{6 x^{2}+5 x^{3}}$ .

See Solution

Problem 29566

Find the limit: $\lim _{x \rightarrow \infty} \frac{2 x-e^{x}}{4-2 x}$ ; is it $\infty$ , $-\infty$ , or a specific value?

See Solution

Problem 29567

Find the average value and values of $c$ from the mean value theorem for $f(x)=3x^{2}+2x$ on $[0,2]$ .

See Solution

Problem 29568

Approximate the area between $f(x)=x^{2}+1$ and the $x$ -axis on $[1,4]$ using a Riemann sum with 6 rectangles.

See Solution

Problem 29569

Find the limit: $\lim _{x \rightarrow \infty} \frac{2 x-e^{x}}{4-2 x}$ and determine if it's $\infty$ , $-\infty$ , or a specific value.

See Solution

Problem 29570

Determine the limit as $x$ approaches $-\infty$ for $\frac{\sqrt{9 x^{6}+9 x^{3}}}{x^{2}-2 x^{3}}$ . Is it $\infty$ , $-\infty$ , or a specific value?

See Solution

Problem 29571

Approximate the area between $y=-\frac{1}{4} x^{2}+9$ , the $x$ -axis, $x=-4$ , and $x=0$ using rectangles.

See Solution

Problem 29572

Calculate the average value of $f(x)=e^{2 x}+3$ on $[0, \ln 3]$ .

See Solution

Problem 29573

Calculate the indefinite integral: $\int \frac{12 x^{9}-15 x^{5}+27 x^{4}}{3 x^{5}} d x$ .

See Solution

Problem 29574

Find the derivative of the functions $f(x) = 1 - x^{2}$ and $f(x) = 2x^{2} + 1$ using $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ .

See Solution

Problem 29575

Find $\int_{-5}^{6}[-f(x)+8] d x$ given $\int_{-5}^{6} f(x) d x=-15$ .

See Solution

Problem 29576

Evaluate the integral: $\int \cos x \sin ^{7} x \, dx$ . Choose the correct answer from the options provided.

See Solution

Problem 29577

Find the limit: $\lim _{x \rightarrow-\infty}(2 x-3)^{3}(x+1)^{3}$ .

See Solution

Problem 29578

Find the derivative $F'(x)$ of the function $F(x)=\int_{-1}^{x} \sqrt{t^{2}+16} dt$ .

See Solution

Problem 29579

Evaluate the integral: $\int\left(\sqrt{x}-2 \theta^{x}+9 x^{6}\right) d x$ . What is the result?

See Solution

Problem 29580

Determine if the limit $\lim _{x \rightarrow-\infty} \frac{3 x^{3}-4}{3 x^{3}-e^{x}}$ equals $\infty$ , $-\infty$ , or a specific value.

See Solution

Problem 29581

Find the limit: $\lim _{x \rightarrow-\infty} \frac{\sqrt{9 x^{6}+9 x^{3}}}{x^{2}-2 x^{3}}$ . Is it $\infty$ , $-\infty$ , or a specific value?

See Solution

Problem 29582

Find the derivative $f^{\prime}(x)$ using the definition for $f(x)=3 x^{2}-4 x+5$ .

See Solution

Problem 29583

Evaluate the integral $\int_{0}^{\frac{\pi}{28}} \tan ^{2}(7 x) d x$ and choose the correct answer from the options.

See Solution

Problem 29584

Find the limit: $\lim _{n \rightarrow \infty} \frac{\log _{2}\left(n^{2}\right)}{\log _{10}\left(n^{10}\right)}$ .

See Solution

Problem 29585

Find the limit: $\lim _{n \rightarrow \infty} \frac{n^{5} \cdot 2^{3 n}}{3^{2 n}}$ .

See Solution

Problem 29586

Find $f(a)$ , $f(a+h)$ , and $\frac{f(a+h)-f(a)}{h}$ for $f(x)=-2+3x^{2}$ .

See Solution

Problem 29587

Solve these integrals and show your work: 1.) $\int \frac{(2 x+7) d x}{x^{2}+7 x+6}$ 2.) $\int \frac{x d x}{\sqrt{x^{2}-9}}$ 3.) $\int \tan ^{5} 2 x \sec ^{2} 2 x d x$ 4.) $\int \frac{\ln 2 x}{x} d x$

See Solution

Problem 29588

Determine if the series $-60 + (-54.0) + (-48.6) + (-43.7) + \ldots$ is convergent/divergent and if it has a sum.

See Solution

Problem 29589

Evaluate the integral $\int e^{-x} \cos 2 x d x$ using Integration by Parts. Which statements are true?

See Solution

Problem 29590

Solve these integrals and highlight your final answers:
1. $\int 3^{y} 2^{y} d y$
2. $\int \frac{d x}{1-\sin x}$
3. $\int(\sin x+\cos x)^{2} d x$
4. $\int\left(e^{x}-e^{-x}\right) d x$

See Solution

Problem 29591

MULTIPLE CHOICE. Answer with the LETTER before the number.
1. Is $\mathbf{7}^{\mathbf{1 - 3 x}}$ an algebraic function? A. TRUE B. FALSE
2. Which method solves $\int \sin x \cos x d x$ ? A. $F_{4}$ B. $F_{8}$ C. $F_{4}$ and $F_{8}$ D. $F_{4}, F_{8}$ , and $F_{9}$
3. For $\int \frac{d y}{y^{2} \sqrt{y^{2}+16}}$ , which substitution? A. $y=16 \tan \theta$ B. $y=4 \tan \theta$ C. $y=4 \sin \theta$ D. $y=4 \cos \theta$
4. What is $d y$ ? A. $d y=16 \sec ^{2} \theta d \theta$ B. $d y=16 \sec \theta d \theta$ C. $d y=4 \cos \theta d \theta$ D. $d y=4 \sec ^{2} \theta d \theta$
5. Evaluate $\int \frac{d y}{y^{2} \sqrt{y^{2}+16}}$ . A. $\frac{\sqrt{y^{2}+16}}{y}+C$ B. $\frac{\sqrt{y^{2}+16}}{y^{2}}+C$ C. $-\frac{\sqrt{y^{2}+16}}{16 y}+C$ D. $\frac{\sqrt{y^{2}+16}}{16 y}+C$

See Solution

Problem 29592

Evaluate the following integrals and choose the correct answer:
8. $\int_{2}^{4} \frac{d z}{z}$ : A.) $\ln \frac{1}{2}$ B.) $\ln 2$ C.) $-\ln 2$ D.) 0
9. $\int_{0}^{\frac{\pi}{2}} \sin ^{5} x \cos x d x$ : A.) 9.55 B.) $\frac{1}{6}$ C.) 6 D.) cannot be determined
10. $\int_{0}^{\ln 4} \frac{d x}{e^{x}+2}$ : A.) $\frac{1}{2}$ B.) $\ln \sqrt{2}$ C.) $\ln 2$ D.) $\ln \frac{3}{2}$
11. $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{\sec ^{2} \theta}{\tan \theta} d \theta$ : A.) $\ln 3$ B.) $\ln 2$ C.) $\ln \sqrt{3}$ D.) $\ln \frac{1}{\sqrt{3}}$
12. $\int_{1}^{2} \frac{x^{3}+8}{x+2} d x$ : A.) $\frac{10}{3}$ B.) $\frac{3}{10}$ C.) $\frac{20}{3}$ D.) 3
13. $\int_{2}^{1}\left(x^{2}-4\right)^{5} x d x$ : A.) $\frac{4}{243}$ B.) $-\frac{4}{243}$ C.) $\frac{243}{4}$ D.) $-\frac{243}{4}$
14. $\int_{0}^{1} \frac{2(x-1) d x}{x^{2}-2 x+5}$ : A.) $\ln 4$ B.) $\ln 5$ C.) $\ln \frac{5}{4}$ D.) $\ln \frac{4}{5}$
15. Area bounded by $x+y=4, x=0, y=0$ is sq. units: A.) 4 B.) 6 C.) 8 D.) 16

See Solution

Problem 29593

Evaluate these integrals and choose the correct answer:
8. $\int_{0}^{1} e^{x} d x$ A.) $e$ B.) $e-1$ C.) $e^{2}$ D.) $1-e$
9. $\int_{0}^{1}(\sec ^{2} 3 x-\tan ^{2} 3 x) d x$ A. 1 B. 3 C.) $3 x$ D.) -1
10. $\int_{3}^{4} \frac{d x}{25-x^{2}}$ A.) $\frac{1}{2} \ln \frac{3}{5}$ B.) $\frac{1}{3} \ln \frac{2}{5}$ C.) $\frac{1}{5} \ln \frac{3}{2}$ D.) $\frac{1}{5} \ln \frac{2}{3}$
11. $\int_{0}^{\frac{\pi}{4}} \tan x d x$ A.) $\ln \sqrt{2}$ B.) $\ln \frac{\sqrt{2}}{2}$ C.) $\ln 2$ D.) $\ln \frac{2}{\sqrt{2}}$
12. $\int_{0}^{\frac{\pi}{2}} \sin 2 x d x$ A.) 0 B.) 1 C.) -1 D.) 2
13. $\int_{0}^{\pi} \cos ^{2} x d x$ A.) $\pi$ B.) $2 \pi$ C.) $\frac{\pi}{2}$ D.) $-\frac{\pi}{2}$
14. $\int_{0}^{1} x e^{x} d x$ A.) 1 B.) $2 e$ C.) $e$ D.) $e-1$
15. $\int_{0}^{4} \frac{2 d x}{x^{2}+16}$ A.) $\frac{\pi}{4}$ B.) $\frac{\pi}{12}$ C.) $\frac{\pi}{8}$ D.) $\frac{\pi}{3}$

See Solution

Problem 29594

1. What is the indefinite integral? A.) specific value B.) particular integral only C.) particular integral + constant D.) integrand + differential. Evaluate $\int_{\ln 4}^{\ln 11} \frac{e^{x}}{\sqrt{e^{x}+5}} d x$ .
2. Which formula finds the antiderivative? A.) $F_{4}$ B.) $F_{5}$ C.) $F_{6}$ D.) $F_{7}$ .
3. What is the particular integral? A.) $\sqrt{e^{x}+5}$ B.) $\frac{1}{2} \sqrt{e^{x}+5}$ C.) $2 \sqrt{e^{x}+5}$ D.) $\frac{2}{3} \sqrt{e^{x}+5}$ .
4. What is the value of the integral? A.) 2 B.) 4 C.) $\ln \frac{11}{4}$ D.) $e^{2}+5$ .
5. Is $\int \frac{8 x-2}{2 x^{2}-x-3} d x$ a proper rational fraction? A.) TRUE B.) FALSE.
6. Which formula can be used for the antiderivative? A.) $F_{4}$ B.) $F_{5}$ C.) $F_{6}$ D.) $F_{7}$ .
7. Find $\int \frac{8 x-2}{2 x^{2}-x-3} d x$ . A.) $\ln \frac{4 x-1}{2 x^{2}-x-3}+C$ B.) $\ln \left|2 x^{2}-x-3\right|+C$ C.) $\ln \left(2 x^{2}-x-3\right)^{2}+C$ D.) $\ln 2(4 x-1)+C$ .

See Solution

Problem 29595

Find $h^{\prime}(4)$ for $h(x)=f(x) \cdot g(x)$ , $h(x)=\frac{f(x)}{g(x)}$ , and $h(x)=f(g(x))$ .

See Solution

Problem 29596

Find $F^{\prime}(a)$ and $G^{\prime}(a)$ for $F(x)=f\left(x^{6}\right)$ and $G(x)=(f(x))^{6}$ , given $a^{5}=13$ , $f(a)=2$ , $f^{\prime}(a)=12$ , $f^{\prime}\left(a^{6}\right)=11$ .

See Solution

Problem 29597

Find $t$ . Given $y=e^{5 x^{4}}$ , determine $\frac{d y}{d x}=a x^{n} e^{b x^{4}}$ .

See Solution

Problem 29598

Find the derivative of the function $f_{a}(x) = (-x-a) \cdot \frac{10}{x^{2}}$ .

See Solution

Problem 29599

Find the derivative of $e^{\sin (x)}$ and show it equals $e^{\sin (x)} \cos ^{n}(x)$ .

See Solution

Problem 29600

Find the derivative of $e^{\frac{e}{1-x}}$ with respect to $x$ .

See Solution

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Calculus

Problem 29501

함수 f(x)f(x)f(x) 가 lim⁡x→1{(x2+1)f(x)}=6\lim _{x \rightarrow 1}\{(x^{2}+1) f(x)\}=6limx→1​{(x2+1)f(x)}=6 일 때, lim⁡x→1{(x+4)f(x)}\lim _{x \rightarrow 1}\{(x+4) f(x)\}limx→1​{(x+4)f(x)} 값을 구하시오.

Problem 29502

Problem 29503

Problem 29504

lim⁡x→03x2+4f(x)2x2−f(x)\lim _{x \rightarrow 0} \frac{3 x^{2}+4 f(x)}{2 x^{2}-f(x)}limx→0​2x2−f(x)3x2+4f(x)​의 값을 구하라, 단 lim⁡x→0f(x)x=2\lim _{x \rightarrow 0} \frac{f(x)}{x}=2limx→0​xf(x)​=2.

Problem 29505

Problem 29506

두 함수 f(x),g(x)f(x), g(x)f(x),g(x)가 주어진 조건을 만족할 때, lim⁡x→−15f(x)+2x{g(x)}2\lim _{x \rightarrow-1} \frac{5 f(x)+2 x}{\{g(x)\}^{2}}limx→−1​{g(x)}25f(x)+2x​의 값을 구하시오.

Problem 29507

Find the average rate of change of f(x)=−3x2+4f(x)=-3x^{2}+4f(x)=−3x2+4 between x=4x=4x=4 and x=7x=7x=7. Simplify your answer.

Problem 29508

Verify the Mean Value Theorem for f(x)={3−x2 if x≤12x if x>1f(x)=\left\{\begin{array}{ll}3-x^{2} & \text { if } x \leq 1 \\ \frac{2}{x} & \text { if } x>1\end{array}\right.f(x)={3−x2x2​​ if x≤1 if x>1​ on [0,2][0,2][0,2] and find values of ccc.

Problem 29509

Calculate the indefinite integral: ∫x(3x−4)2 dx\int \sqrt{x}(3 x-4)^{2} \, dx∫x​(3x−4)2dx

Problem 29510

Find ∫−27[3f(x)−9]dx\int_{-2}^{7}[3 f(x)-9] d x∫−27​[3f(x)−9]dx given that ∫−27f(x)dx=−12\int_{-2}^{7} f(x) d x=-12∫−27​f(x)dx=−12.

Problem 29511

Solve the equation y′=5sin⁡x−5x4+12e−4xy' = 5 \sin x - 5 x^4 + 12 e^{-4x}y′=5sinx−5x4+12e−4x with the initial condition y(0)=12y(0) = 12y(0)=12.

Problem 29512

Calculate the area between the curves f(x)=x2+2xf(x)=x^{2}+2xf(x)=x2+2x and g(x)=−x+4g(x)=-x+4g(x)=−x+4 from x=−4x=-4x=−4 to x=2x=2x=2.

Problem 29513

Calculate the average value of f(x)=4cos⁡3xsin⁡xf(x)=4 \cos ^{3} x \sin xf(x)=4cos3xsinx on [0,2π3][0, \frac{2 \pi}{3}][0,32π​]. Options: a.) 5π8\frac{5 \pi}{8}85π​ b.) 45π32\frac{45 \pi}{32}3245π​ c.) 4532π\frac{45}{32 \pi}32π45​ d.) 1516\frac{15}{16}1615​

Problem 29514

Find the average value and ccc from the mean value theorem for f(x)=cos⁡xf(x)=\cos xf(x)=cosx on [0,π6][0, \frac{\pi}{6}][0,6π​].

Problem 29515

Find the definite integral: ∫13x(2x−5)4dx\int_{1}^{3} x(2 x-5)^{4} d x∫13​x(2x−5)4dx. Choose the correct answer: a.) 24, b.) 923\frac{92}{3}392​, c.) −2743-\frac{274}{3}−3274​, d.) −923-\frac{92}{3}−392​.

Problem 29516

Estimate ∫352x2+4dx\int_{3}^{5} \frac{2}{x^{2}+4} dx∫35​x2+42​dx using n=4n=4n=4 with Simpson's rule. Round to five decimal places.

Problem 29517

Create a second-order ODE with y(x)y(x)y(x), find y′(x)y'(x)y′(x) and y′′(x)y''(x)y′′(x), then verify a solution yC(x)=y(x)+Cy_C(x)=y(x)+CyC​(x)=y(x)+C. For what CCC does yC(x0)=ky_C(x_0)=kyC​(x0​)=k? Provide an IVP example and another ODE to verify.

Problem 29518

Find all δ\deltaδ values that satisfy lim⁡x→25x+4=14\lim _{x \rightarrow 2} 5 x+4=14limx→2​5x+4=14 with ε=0.5\varepsilon=0.5ε=0.5. Options: δ=0.0333\delta=0.0333δ=0.0333, δ=0.01\delta=0.01δ=0.01, δ=0.1\delta=0.1δ=0.1, δ=0.2\delta=0.2δ=0.2.

Problem 29519

Find dydx\frac{d y}{d x}dxdy​ for the equation 3x2y4−5cos⁡(3x)=y5+23 x^{2} y^{4}-5 \cos (3 x)=y^{5}+23x2y4−5cos(3x)=y5+2.

Problem 29520

Find all values of xxx where the function f(x)f(x)f(x) is not differentiable, given points at (1,2)(1,2)(1,2), (1,4)(1,4)(1,4), and a sharp point at (−4,3)(-4,3)(−4,3).

Problem 29521

Find the derivative of y=x9x2+135x3+27y=\frac{x^{9} \sqrt[5]{x^{2}+13}}{x^{3}+27}y=x3+27x95x2+13​​ using logarithmic differentiation.

Problem 29522

Is there an xxx in [0,4][0,4][0,4] such that f(x)=0f(x)=0f(x)=0 given f(0)=−10f(0)=-10f(0)=−10 and f(4)=10f(4)=10f(4)=10? Choose: Always true, Always false, Sometimes true.

Problem 29523

Find the derivative of y=(log⁡7(4x2+25))3y=\left(\log _{7}\left(4 x^{2}+25\right)\right)^{3}y=(log7​(4x2+25))3.

Problem 29524

Solve the equation xndydx=xmyx^{n} \frac{d y}{d x}=x^{m} yxndxdy​=xmy: (a) Find a trivial solution. (b) Find all solutions y=y(x)y=y(x)y=y(x). (c) Is there a unique solution for y(x0)=y0y(x_0)=y_0y(x0​)=y0​?

Problem 29525

Find the average velocity of the object with position s(t)=5ln⁡ts(t)=5 \ln ts(t)=5lnt over the interval [1.5,1.7][1.5,1.7][1.5,1.7]. Round to 4 decimal places.

Problem 29526

Find the derivative of y=tan⁡(34x+1)y=\tan(3^{4x}+1)y=tan(34x+1). What is dydx\frac{dy}{dx}dxdy​?

Problem 29527

Find the rate of change of bacteria A(t)=16e0.32tA(t)=16 e^{0.32 t}A(t)=16e0.32t after 12 hours. Choices: 24,619.97, 744.41, 84.61, 238.21.

Problem 29528

Find the limit: lim⁡x→−∞x5−x2x3−2x\lim _{x \rightarrow-\infty} \frac{x^{5}-x^{2}}{x^{3}-2 x}limx→−∞​x3−2xx5−x2​.

Problem 29529

Solve the equation 1xndydx=ym\frac{1}{x^{n}} \frac{d y}{d x}=y^{m}xn1​dxdy​=ym for y=y(x)y=y(x)y=y(x). Discuss trivial/extraneous solutions and FTOC usage. Is the solution unique for y(x0)=y0y(x_0)=y_0y(x0​)=y0​?

Problem 29530

Find lim⁡x→−1+f(x)\lim _{x \rightarrow-1^{+}} f(x)limx→−1+​f(x) for the piecewise function f(x)f(x)f(x) defined as: f(x)=−2+2x2f(x) = -2 + 2x^{2}f(x)=−2+2x2 for x≤−1x \leq -1x≤−1 and f(x)=−1+xf(x) = -1 + xf(x)=−1+x for x>−1x > -1x>−1.

Problem 29531

Find lim⁡x→−3+f(x)\lim _{x \rightarrow-3^{+}} f(x)limx→−3+​f(x) for the piecewise function f(x)f(x)f(x) defined as: f(x)=x2−8f(x)=x^{2}-8f(x)=x2−8 for x≥−3x \geq-3x≥−3 and f(x)=−5x−10f(x)=-5x-10f(x)=−5x−10 for x<−3x<-3x<−3.

Problem 29532

Find the derivative of f(x)=ln⁡(x62x5−7)f(x)=\ln \left(\frac{x^{6}}{2 x^{5}-7}\right)f(x)=ln(2x5−7x6​).

Problem 29533

Estimate the instantaneous velocity of s(t)=5ln⁡ts(t)=5 \ln ts(t)=5lnt at t=1.5t=1.5t=1.5. Consider the average velocity on [1.5,1.7][1.5,1.7][1.5,1.7].

Problem 29534

Problem 29535

Find the derivative of f(x)=−15sin⁡x+11x4+3x7−5f(x)=-15 \sin x+\frac{11}{x^{4}}+3 \sqrt[7]{x}-5f(x)=−15sinx+x411​+37x​−5.

Problem 29536

Find lim⁡x→−2f(x)\lim _{x \rightarrow-2} f(x)limx→−2​f(x) for the piecewise function f(x)={−2x2+3 if x<−2,2x+8 if x≥−2}f(x) = \{-2 x^{2}+3 \text{ if } x<-2, 2 x+8 \text{ if } x \geq-2\}f(x)={−2x2+3 if x<−2,2x+8 if x≥−2}.

Problem 29537

Find lim⁡x→4f(x)\lim _{x \rightarrow 4} f(x)limx→4​f(x) for the piecewise function f(x)={x2−10 if x<4,−9+2x if x>4}f(x) = \{ x^{2}-10 \text{ if } x<4, -9+2x \text{ if } x>4 \}f(x)={x2−10 if x<4,−9+2x if x>4}.

Problem 29538

Estimate the max error in the volume of a sphere with radius error of 0.5in0.5 \mathrm{in}0.5in using V(r)=43πr3V(r)=\frac{4}{3} \pi r^{3}V(r)=34​πr3.

Problem 29539

Estimate the derivative of f(x)=12x3f(x)=\frac{1}{2} x^{3}f(x)=21​x3 at x=−2x=-2x=−2. Choose from: −6-6−6, 16\frac{1}{6}61​, −16-\frac{1}{6}−61​, 6.

Problem 29540

Is the function f(x)=x3f(x)=\sqrt[3]{x}f(x)=3x​ differentiable at x=−27x=-27x=−27? If not, explain why.

Problem 29541

Find the derivative of y=arctan⁡(5x2−1)y=\arctan(5x^2-1)y=arctan(5x2−1). What is dydx\frac{dy}{dx}dxdy​?

Problem 29542

함수 $f(x)$ 가 $\lim _{x \rightarrow 1}\{(x^{2}+1) f(x)\}=6$ 일 때, $\lim _{x \rightarrow 1}\{(x+4) f(x)\}$ 값을 구하시오.

$\lim _{x \rightarrow 0} \frac{3 x^{2}+4 f(x)}{2 x^{2}-f(x)}$ 의 값을 구하라, 단 $\lim _{x \rightarrow 0} \frac{f(x)}{x}=2$ .

두 함수 $f(x), g(x)$ 가 주어진 조건을 만족할 때, $\lim _{x \rightarrow-1} \frac{5 f(x)+2 x}{\{g(x)\}^{2}}$ 의 값을 구하시오.

Find the average rate of change of $f(x)=-3x^{2}+4$ between $x=4$ and $x=7$ . Simplify your answer.

Verify the Mean Value Theorem for $f(x)=\left\{\begin{array}{ll}3-x^{2} & \text { if } x \leq 1 \\ \frac{2}{x} & \text { if } x>1\end{array}\right.$ on $[0,2]$ and find values of $c$ .

Calculate the indefinite integral: $\int \sqrt{x}(3 x-4)^{2} \, dx$

Find $\int_{-2}^{7}[3 f(x)-9] d x$ given that $\int_{-2}^{7} f(x) d x=-12$ .

Solve the equation $y' = 5 \sin x - 5 x^4 + 12 e^{-4x}$ with the initial condition $y(0) = 12$ .

Calculate the area between the curves $f(x)=x^{2}+2x$ and $g(x)=-x+4$ from $x=-4$ to $x=2$ .

Calculate the average value of $f(x)=4 \cos ^{3} x \sin x$ on $[0, \frac{2 \pi}{3}]$ . Options: a.) $\frac{5 \pi}{8}$ b.) $\frac{45 \pi}{32}$ c.) $\frac{45}{32 \pi}$ d.) $\frac{15}{16}$

Find the average value and $c$ from the mean value theorem for $f(x)=\cos x$ on $[0, \frac{\pi}{6}]$ .

Find the definite integral: $\int_{1}^{3} x(2 x-5)^{4} d x$ . Choose the correct answer: a.) 24, b.) $\frac{92}{3}$ , c.) $-\frac{274}{3}$ , d.) $-\frac{92}{3}$ .

Estimate $\int_{3}^{5} \frac{2}{x^{2}+4} dx$ using $n=4$ with Simpson's rule. Round to five decimal places.

Create a second-order ODE with $y(x)$ , find $y'(x)$ and $y''(x)$ , then verify a solution $y_C(x)=y(x)+C$ . For what $C$ does $y_C(x_0)=k$ ? Provide an IVP example and another ODE to verify.

Find all $\delta$ values that satisfy $\lim _{x \rightarrow 2} 5 x+4=14$ with $\varepsilon=0.5$ . Options: $\delta=0.0333$ , $\delta=0.01$ , $\delta=0.1$ , $\delta=0.2$ .

Find $\frac{d y}{d x}$ for the equation $3 x^{2} y^{4}-5 \cos (3 x)=y^{5}+2$ .

Find all values of $x$ where the function $f(x)$ is not differentiable, given points at $(1,2)$ , $(1,4)$ , and a sharp point at $(-4,3)$ .

Find the derivative of $y=\frac{x^{9} \sqrt[5]{x^{2}+13}}{x^{3}+27}$ using logarithmic differentiation.

Is there an $x$ in $[0,4]$ such that $f(x)=0$ given $f(0)=-10$ and $f(4)=10$ ? Choose: Always true, Always false, Sometimes true.

Find the derivative of $y=\left(\log _{7}\left(4 x^{2}+25\right)\right)^{3}$ .

Solve the equation $x^{n} \frac{d y}{d x}=x^{m} y$ : (a) Find a trivial solution. (b) Find all solutions $y=y(x)$ . (c) Is there a unique solution for $y(x_0)=y_0$ ?

Find the average velocity of the object with position $s(t)=5 \ln t$ over the interval $[1.5,1.7]$ . Round to 4 decimal places.

Find the derivative of $y=\tan(3^{4x}+1)$ . What is $\frac{dy}{dx}$ ?

Find the rate of change of bacteria $A(t)=16 e^{0.32 t}$ after 12 hours. Choices: 24,619.97, 744.41, 84.61, 238.21.

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

Solve the equation $\frac{1}{x^{n}} \frac{d y}{d x}=y^{m}$ for $y=y(x)$ . Discuss trivial/extraneous solutions and FTOC usage. Is the solution unique for $y(x_0)=y_0$ ?

Find $\lim _{x \rightarrow-1^{+}} f(x)$ for the piecewise function $f(x)$ defined as: $f(x) = -2 + 2x^{2}$ for $x \leq -1$ and $f(x) = -1 + x$ for $x > -1$ .

Find $\lim _{x \rightarrow-3^{+}} f(x)$ for the piecewise function $f(x)$ defined as: $f(x)=x^{2}-8$ for $x \geq-3$ and $f(x)=-5x-10$ for $x<-3$ .

Find the derivative of $f(x)=\ln \left(\frac{x^{6}}{2 x^{5}-7}\right)$ .

Estimate the instantaneous velocity of $s(t)=5 \ln t$ at $t=1.5$ . Consider the average velocity on $[1.5,1.7]$ .

Find the derivative of $f(x)=-15 \sin x+\frac{11}{x^{4}}+3 \sqrt[7]{x}-5$ .

Find $\lim _{x \rightarrow-2} f(x)$ for the piecewise function $f(x) = \{-2 x^{2}+3 \text{ if } x<-2, 2 x+8 \text{ if } x \geq-2\}$ .

Find $\lim _{x \rightarrow 4} f(x)$ for the piecewise function $f(x) = \{ x^{2}-10 \text{ if } x<4, -9+2x \text{ if } x>4 \}$ .

Estimate the max error in the volume of a sphere with radius error of $0.5 \mathrm{in}$ using $V(r)=\frac{4}{3} \pi r^{3}$ .

Estimate the derivative of $f(x)=\frac{1}{2} x^{3}$ at $x=-2$ . Choose from: $-6$ , $\frac{1}{6}$ , $-\frac{1}{6}$ , 6.

Is the function $f(x)=\sqrt[3]{x}$ differentiable at $x=-27$ ? If not, explain why.

Find the derivative of $y=\arctan(5x^2-1)$ . What is $\frac{dy}{dx}$ ?

Find the derivative of $y=\sin(3x^2-4x+1)$ . What is $\frac{dy}{dx}$ ?

Find the linear approximation of $f(x)=\sqrt[5]{x}$ at $x=-1$ . Which formula represents $L(x)$ ?

Find the tangent line equation for $f(x)=\frac{1}{x^{5}}$ at $x=1$ . Options: $y=\frac{1}{5} x+\frac{4}{5}$ , $y=-5 x-4$ , $y=-5 x+6$ , $y=5 x-4$ .

Find the error in the linear approximation of $f(x)=\sqrt[3]{x}$ near $x=-1$ for estimating $\sqrt[3]{-0.75}$ .

Determine if the function $f(x)$ is continuous at $x=-3$ :
$f(x)=\left\{\begin{array}{ll} 14-3 x^{2}, & x>-3 \\ -13-x, & x<-3 \end{array}\right.$

Find the derivative of $y=e^{5 x} \cos (3 x)$ . What is $\frac{d y}{d x}$ ?

Check if the function $f(x)$ is continuous at $x=-2$ , where $f(x)=\begin{cases} 9-x^{2}, & x<-2 \\ 3-x, & x \geq-2 \end{cases}$ .

Find the derivative of $f(x)=\sqrt[3]{x^{6}-64}$ . What is $f'(x)$ ?

Find the derivative of $f(x)=-12 \sin x+x^{7} \cos x$ .

Use Newton's method to solve $-x^{3}-3x+65=0$ and find the third iteration value. Verify your answer.

Check if the function $f(x)$ is continuous at $x=3$ , where $f(x)=5+x^{2}$ for $x<3$ and $f(x)=5+3x$ for $x>3$ .

Find the area between $f(x)=\cos x$ and $g(x)=2-\cos x$ from $x=0$ to $x=2\pi$ .

Find the limit as $y$ approaches $\frac{\pi}{2}$ for the expression $\frac{\sin y}{\cos y - 1}$ .

Evaluate the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{x^{6}+4 x}}{5 x+x^{3}}$ and determine if it equals $\infty$ , $-\infty$ , or a specific value.

Calculate the area between the $x$ -axis and $f(x)=-x^{2}+4$ from $x=-1$ to $x=3$ . Options: $\frac{34}{3}$ , $\frac{28}{3}$ , $\frac{29}{3}$ , $\frac{20}{3}$ .

Calculate the indefinite integral: $\int(6 \csc x \cot x - \frac{25}{x^{4}}) \, dx$ .

Find the total area between $f(x)=3x^{3}-x^{2}-10x$ and $g(x)=-x^{2}+2x$ . Options: 24, 32, 12, or 8 units².

Evaluate the integral $\int_{0}^{\frac{3 \pi}{4}} 15 \tan ^{9} x \sec ^{2} x \, dx$ . What is its value?

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

Evaluate the integral: $\int_{0}^{\frac{3 \pi}{4}} 15 \tan ^{9} x \sec ^{2} x d x$ . What is its value?

Determine if the limit is $\infty$ , $-\infty$ , or a specific value: $\lim _{x \rightarrow-\infty}\left(1+x-6 x^{5}\right)$

Determine if the limit $\lim _{x \rightarrow-\infty} \frac{\sqrt{9 x^{6}+9 x^{5}}}{4-3 x^{3}}$ equals $\infty$ , $-\infty$ , or a specific value.

Find the limit as $x$ approaches $-\infty$ : $\lim _{x \rightarrow-\infty} \frac{\sqrt{4 x^{2}-4 x}}{6 x^{2}+5 x^{3}}$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{2 x-e^{x}}{4-2 x}$ ; is it $\infty$ , $-\infty$ , or a specific value?

Find the average value and values of $c$ from the mean value theorem for $f(x)=3x^{2}+2x$ on $[0,2]$ .

Approximate the area between $f(x)=x^{2}+1$ and the $x$ -axis on $[1,4]$ using a Riemann sum with 6 rectangles.

Find the limit: $\lim _{x \rightarrow \infty} \frac{2 x-e^{x}}{4-2 x}$ and determine if it's $\infty$ , $-\infty$ , or a specific value.

Determine the limit as $x$ approaches $-\infty$ for $\frac{\sqrt{9 x^{6}+9 x^{3}}}{x^{2}-2 x^{3}}$ . Is it $\infty$ , $-\infty$ , or a specific value?

Approximate the area between $y=-\frac{1}{4} x^{2}+9$ , the $x$ -axis, $x=-4$ , and $x=0$ using rectangles.

Calculate the average value of $f(x)=e^{2 x}+3$ on $[0, \ln 3]$ .

Calculate the indefinite integral: $\int \frac{12 x^{9}-15 x^{5}+27 x^{4}}{3 x^{5}} d x$ .

Find the derivative of the functions $f(x) = 1 - x^{2}$ and $f(x) = 2x^{2} + 1$ using $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ .