Calculus

Problem 33601

Evaluate the limit as $b$ approaches 36: $\lim _{b \rightarrow 36} \frac{36-b}{6-\sqrt{b}}$ .

See Solution

Problem 33602

Find the limit $\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}$ for the series $\sum_{n=1}^{\infty} \frac{(2 n) !}{(n !)(n+1) !}$ . Options: a. 4, b. 2, c. $\infty$ , d. 1, e. 0.

See Solution

Problem 33603

Find the tangent line to $y=4 \sin x$ at $\left(\frac{\pi}{6}, 2\right)$ in the form $y=m x+b$ where $m=$ and $b=$ .

See Solution

Problem 33604

Find the derivative $y'$ of the function $y=4 e^{6}+6 e^{\ln x}$ .

See Solution

Problem 33605

Determine the convergence of these series: I. $\sum_{n=1}^{\infty} \frac{\ln n}{e^{n}+2}$ , II. $\sum_{n=2}^{\infty} \frac{\sin ^{2}(n)}{n+n^{3 / 2}}$ , III. $\sum_{n=1}^{\infty} \frac{1}{n^{7 / 3}}$ . Choose: a. I and III, b. I only, c. None, d. All, e. III only, f. III and II, g. I and II, h. II only.

See Solution

Problem 33606

Find the derivative of the function $f(x)=2 x^{6}-10 x^{10}$ . What is $f^{\prime}(x)=?$

See Solution

Problem 33607

Find the derivative $y' = \frac{d}{dx} \log_{8}(9 - x - x^{5})$ .

See Solution

Problem 33608

Analyze the function $f(x)=\begin{cases} \frac{1}{2} x+10, & x<8 \\ \sqrt{8 x}+6, & x \geq 8 \end{cases}$ .
Part 1: Is $f(x)$ continuous at $x=8$ ? Explain.
Part 2: Is $f(x)$ differentiable at $x=8$ ? Explain.

See Solution

Problem 33609

Find the rate of change of the equation $y = 6x^2 + 1$ .

See Solution

Problem 33610

Determine which series converge:
1. $\sum_{n=1}^{\infty} \frac{\ln n}{e^{n}+2}$
2. $\sum_{n=2}^{\infty} \frac{\sin ^{2}(n)}{n+n^{3 / 2}}$
3. $\sum_{n=1}^{\infty} \frac{1}{n^{7 / 3}}$ Options: a. 3 only, b. None, c. 1 only, d. 1 and 3, e. 1 and 2, f. all, g. 2 only, h. 3 and 2.

See Solution

Problem 33611

Find where the function $f(x)=x e^{x}$ is concave down. Choose from the intervals: $(-1, \infty)$ , $(-\infty, 0)$ , $(-\infty,-2)$ , $(-\infty,-1)$ , or none.

See Solution

Problem 33612

Find the derivative $y'$ of the function $y=2-9 e^{-x^{8}}$ .

See Solution

Problem 33613

Find the derivative $y'$ if $y=4 e^{6}+6 e^{\ln x}$ .

See Solution

Problem 33614

Find where the function $f(x)=x e^{x}$ is concave down. Choose from the given intervals.

See Solution

Problem 33615

Find the derivative $y'$ of the function $y=x^{3} e^{3 \ln x}$ .

See Solution

Problem 33616

Find the derivative $y'$ , if $y=8^{3x+1}$ .

See Solution

Problem 33617

Find $f^{\prime}(0)$ for the function $f(x)=2^{x+2}$ . Options: 4, $8 \ln 2$ , $4 \ln 2$ , none, $\ln 4$ .

See Solution

Problem 33618

Find the derivative $y'$ , given $y=\ln(e^{8x}+9)$ .

See Solution

Problem 33619

Using the Gompertz model, show that $\lim_{N \rightarrow 0} R(N) = 0$ by completing the table for $R(N) = N \ln(1/N)$ .
Fill in: N: 0.1, 0.01, 0.001 R(N): $\square$ , $\square$ , $\square$

See Solution

Problem 33620

Find the intervals where the function $f(x)=x e^{x}$ is concave down. Choices: $(-\infty,-1)$ , $(-\infty,-2)$ , $(-1, \infty)$ , $(-\infty, 0)$ , none.

See Solution

Problem 33621

Find $f^{\prime \prime}(1)$ for $f(x)=x^{3} \ln x$ . Options: 8, 1, 5, 0, none.

See Solution

Problem 33622

Find the derivative $y'=\frac{d}{dx} \ln(e^{8x}+9)$ . Choose the correct option.

See Solution

Problem 33623

Find the intervals where the function $f(x)=x e^{x}$ is concave down. Options: $(-\infty,-2)$ , $(-1, \infty)$ , none, $(-\infty, 0)$ , $(-\infty,-1)$ .

See Solution

Problem 33624

Find the derivative of $y = x^{3} e^{3 \ln x}$ . What is $y'$ ?

See Solution

Problem 33625

Find the derivative $y'$ of the function $y=8^{3x+1}$ . Choices: $(3 \ln 8) 8^{3 x+1}$ , $(\ln 8) 8^{3 x+1}$ , $(8 \ln 3) 8^{3 x+1}$ , $(\ln 3) 8^{3 x+1}$ , none.

See Solution

Problem 33626

Find the derivative of $f(x)=5 x \sin x \cos x$ and evaluate it at the given point. What is $f^{\prime}(x)=?$

See Solution

Problem 33627

Find $f^{\prime \prime}(1)$ for the function $f(x)=x^{3} \ln x$ . Options: 8, 1, 5, 0, none.

See Solution

Problem 33628

Find the derivative $y^{\prime}$ of $y=4 e^{6}+6 e^{\ln x}$ . Choices: $24 e^{5}+6$ , $6 x$ , 6, $24 e^{5}+6 x e^{\ln x}$ , 0.

See Solution

Problem 33629

Find $f^{\prime \prime}(1)$ for the function $f(x)=x^{3} \ln x$ . Options: 8, 5, none, 0, 1.

See Solution

Problem 33630

Find $f^{\prime}(4)$ if $f^{\prime}(x)=5 \sin (x) \cos (x)+5 x \cos (2 x)$ .

See Solution

Problem 33631

Determine the convergence of these series: 1. $\sum_{n=1}^{\infty} \frac{\ln n}{e^{n}+2}$ , 2. $\sum_{n=2}^{\infty} \frac{\sin ^{2}(n)}{n+n^{3 / 2}}$ , 3. $\sum_{n=1}^{\infty} \frac{1}{n^{7 / 3}}$ .

See Solution

Problem 33632

Find the derivative $f^{\prime}(0)$ if $f(x)=2^{x+2}$ .

See Solution

Problem 33633

Find the derivative of the function $f(x)=5 x \sin x \cos x$ : $f^{\prime}(x)=$

See Solution

Problem 33634

Find the derivative $y'$ of the function $y=2-9 e^{-x^{8}}$ . Choose the correct option.

See Solution

Problem 33635

Find the derivative $y'=\frac{d}{dx} \ln(e^{8x}+9)$ and select the correct answer.

See Solution

Problem 33636

Find the derivative $y^{\prime}$ if $y=8^{3x+1}$ . Choices include $(8 \ln 3) 8^{3 x+1}$ , $(3 \ln 8) 8^{3 x+1}$ , etc.

See Solution

Problem 33637

Calculate the average linear momentum $\left\langle\hat{p}_{x}\right\rangle$ for a particle in a box of length $a$ . Specify $n$ .

See Solution

Problem 33638

Find the derivative of the function $f(x)=-3 \sin ^{6}(x)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 33639

Find the derivative of $r(x)=(x+10) \ln (4 x)$ . What is $r^{\prime}(x)=?$

See Solution

Problem 33640

Find the derivative of the function $f(x)=e^{2/x}$ . What is $f'(x)=?$

See Solution

Problem 33641

Find the derivative of $f(x)=x^{2} e^{7 x}$ . What is $f^{\prime}(x)=?$

See Solution

Problem 33642

Find the derivative $y'$ of the function $y=2-9 e^{-x^{8}}$ . Options include: $9 x^{8} e^{-e^{8}}$ , $-72 x^{7} e^{-x^{8}}$ , none, $-9 x^{8} e^{-z^{8}}$ , $72 x^{7} e^{-x^{8}}$ .

See Solution

Problem 33643

Find the derivative of the function $f(x)=3 \ln (6 x)$ . What is $f^{\prime}(x)=?$

See Solution

Problem 33644

Find the derivative $f^{\prime}(x)$ for the function $f(x)=x^{\ln (x)}$ .

See Solution

Problem 33645

Find the derivative of the function $y=2-9 e^{-x^{8}}$ . What is $y^{\prime}$ ?

See Solution

Problem 33646

Find the derivative $y'$ , where $y=\log _{8}(9-x-x^{5})$ . Choices include various fractions.

See Solution

Problem 33647

Find the derivative of the function $f(t)=6 e^{t+3}$ . What is $\frac{d f}{d t}=?$

See Solution

Problem 33648

Find the intervals where the function $f(x)=x e^{x}$ is concave down. Select from the given options.

See Solution

Problem 33649

Find the antiderivative of the function $f(x)=9 x^{8}+10 x^{5}-3 x^{4}-4$ .

See Solution

Problem 33650

Find the function $f(x)$ if $f^{\prime}(x)=4 x+2$ and $f(4)=44$ . What is $f(x)=?$

See Solution

Problem 33651

Find $f\left(\frac{\pi}{4}\right)$ given $f^{\prime \prime}(x)=-16 \sin(4x)$ , $f^{\prime}(0)=-6$ , and $f(0)=-5$ .

See Solution

Problem 33652

Find the position of a particle at time $t=14$ given $a(t)=24t+18$ , $s(0)=13$ , and $v(0)=3$ .

See Solution

Problem 33653

Compute the integral: $\int 4 e^{x} dx = C + \square$

See Solution

Problem 33654

Calculate the indefinite integral and include the constant $C$ : $\int \frac{5}{1+t^{2}} d t=\square$

See Solution

Problem 33655

Find the slope of the curve $f(x)=(x+1)^{\frac{3}{2}}$ at the point where $x=15$ .

See Solution

Problem 33656

Compute the indefinite integral: $\int \frac{28 x^{7}-7 x}{x} d x = C$ .

See Solution

Problem 33657

Find the limit as $x$ approaches 3 for the expression $\frac{\frac{1}{x}-\frac{1}{3}}{x-3}$ .

See Solution

Problem 33658

Evaluate the integral: $\int \frac{7}{(t+9)^{6}} dt$

See Solution

Problem 33659

Integrate the function from -2 to 1.5: $\int_{-2}^{1.5} 24(6 x+4)^{3} d x$ .

See Solution

Problem 33660

Find the interval for a local maximum of the function $h(x)=\frac{1}{2} x^{4}-2 x^{2}+x+4$ . A) (1,2) B) (0,1) C) (3,5) D) (-2,0)

See Solution

Problem 33661

Evaluate the integral from 3 to 7 of (2x + 9): $\int_{3}^{7}(2 x+9) d x=$

See Solution

Problem 33662

Find the integral of $7 x e^{3 x}$ with respect to $x$ .

See Solution

Problem 33663

Evaluate the integral of $6 \sin^5 r \cos r \, dr$ .

See Solution

Problem 33664

Find $y^{\prime}$ if $y=\log _{2}(8 t^{\ln 2})$ . Choices: $\frac{\ln 8}{t}$ , $3+\ln t$ , $\frac{t}{\ln 2}$ , $\frac{1}{t}$ , none.

See Solution

Problem 33665

Find the integral of $\cos^{2} x$ with respect to $x$ : $\int \cos^{2} x \, dx$ .

See Solution

Problem 33666

Calculate the integral: $\int \frac{1+x}{1+\sqrt{x+1}} d x$

See Solution

Problem 33667

Evaluate the limit: $\lim _{\Delta x \rightarrow 0} \frac{\cos \left(\frac{\pi}{1}+\Delta x\right)-\frac{\sqrt{2}}{2}}{\Delta x}=$

See Solution

Problem 33668

Evaluate the limit: $\lim _{\Delta x \rightarrow 0} \frac{\cos \left(\frac{\pi}{4}+\Delta x\right)-\frac{\sqrt{2}}{2}}{\Delta x}.$

See Solution

Problem 33669

Find the limit as $x$ approaches 0 for the expression $x^{2} \sin \frac{1}{x}$ .

See Solution

Problem 33670

Find the rate of change of $f(x)=\tan (x)$ at $x=-\frac{\pi}{4}$ . What is it? Options: a. -0.5, b. 0, c. 2, d. -2.

See Solution

Problem 33671

Approximate the area under the curve using a right Riemann Sum with four subintervals from the given population data.

See Solution

Problem 33672

Find the limit: $\lim _{x \rightarrow 0^{+}} \sqrt{x} e^{\sin (\pi / x)}$

See Solution

Problem 33673

Find the value of $t$ where the local maximum of $f(t)=\int_{0}^{t} \frac{x^{2}+11 x+28}{1+\cos ^{2}(x)} d x$ occurs. $t=$

See Solution

Problem 33674

Find the equation of the tangent line to the function $f(x)=4 \sin (x)$ at the point $\left(\frac{\pi}{6}, 2\right)$ .

See Solution

Problem 33675

Find $F^{\prime}(x)$ for $F(x)=\int_{\ln x}^{e^{x}} 8 \sin t \, dt$ .

See Solution

Problem 33676

Evaluate the integral: $\int_{-5}^{x}(2 t-4) d t$ and express your answer as a function of $x$ .

See Solution

Problem 33677

Is the derivative of $f(x)=e^{\sin (2 x)}$ equal to $f^{\prime}(x)=2 e^{\sin (2 x)} \cos (2 x)$ ? True or False?

See Solution

Problem 33678

Find the tangent line of $f(x)=x^{4}+2 x-1$ at $x=1$ . What is the equation? Choices: a. $y=6 x-4$ , b. $y=6 x+4$ , c. $y=4 x-6$ , d. $y=4 x+6$ .

See Solution

Problem 33679

Find the tangent line and normal line equations for $y=(3+6x)^{2}$ at the point $(2,225)$ .

See Solution

Problem 33680

Find the tangent and normal lines to the curve $y=(3+6x)^{2}$ at the point $(2,225)$ .

See Solution

Problem 33681

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

See Solution

Problem 33682

Find the derivative of $F(x)=\int_{x^{3}}^{x^{5}}(2 t-1)^{3} dt$ using the Fundamental Theorem of Calculus. What is $F'(x)$ ?

See Solution

Problem 33683

Find the tangent line of $f(x)=x^{4}+2x-1$ at $x=1$ . Choose the correct equation: a. $y=6x-4$ , b. $y=6x+4$ , c. $y=4x-6$ , d. $y=4x+6$ .

See Solution

Problem 33684

Find the relative extreme points of $f(x)=x^{3}-3x+8$ and sketch its graph. Identify relative minimum points.

See Solution

Problem 33685

Find the relative extrema of $f(x)=x^{3}-3x+8$ and sketch its graph. Identify min/max points.

See Solution

Problem 33686

Euler-Bernoulli beam problem:
1. Find weak formulation of $E I \frac{\mathrm{d}^{4}}{\mathrm{~d} x^{4}} w=q$ with boundary terms.
2. Define Dirichlet & Neumann conditions and their physical meaning. Write full boundary value problem.
3. State continuity requirements for approximations. Can Lagrange elements be used?

See Solution

Problem 33687

Find the average rate of change of $P(\theta)=\sqrt{4\theta+1}$ over $[0,2]$ . Options: a) 2 b) 0 c) -2 d) 1

See Solution

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Calculus

Problem 33601

Evaluate the limit as bbb approaches 36: lim⁡b→3636−b6−b\lim _{b \rightarrow 36} \frac{36-b}{6-\sqrt{b}}limb→36​6−b​36−b​.

Problem 33602

Find the limit lim⁡n→∞an+1an\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}limn→∞​an​an+1​​ for the series ∑n=1∞(2n)!(n!)(n+1)!\sum_{n=1}^{\infty} \frac{(2 n) !}{(n !)(n+1) !}∑n=1∞​(n!)(n+1)!(2n)!​. Options: a. 4, b. 2, c. ∞\infty∞, d. 1, e. 0.

Problem 33603

Find the tangent line to y=4sin⁡xy=4 \sin xy=4sinx at (π6,2)\left(\frac{\pi}{6}, 2\right)(6π​,2) in the form y=mx+by=m x+by=mx+b where m=m=m= and b=b=b=.

Problem 33604

Find the derivative y′y'y′ of the function y=4e6+6eln⁡xy=4 e^{6}+6 e^{\ln x}y=4e6+6elnx.

Problem 33605

Problem 33606

Find the derivative of the function f(x)=2x6−10x10f(x)=2 x^{6}-10 x^{10}f(x)=2x6−10x10. What is f′(x)=?f^{\prime}(x)=?f′(x)=?

Problem 33607

Find the derivative y′=ddxlog⁡8(9−x−x5)y' = \frac{d}{dx} \log_{8}(9 - x - x^{5})y′=dxd​log8​(9−x−x5).

Problem 33608

Analyze the function f(x)={12x+10,x<88x+6,x≥8f(x)=\begin{cases} \frac{1}{2} x+10, & x<8 \\ \sqrt{8 x}+6, & x \geq 8 \end{cases}f(x)={21​x+10,8x​+6,​x<8x≥8​. Part 1: Is f(x)f(x)f(x) continuous at x=8x=8x=8? Explain. Part 2: Is f(x)f(x)f(x) differentiable at x=8x=8x=8? Explain.

Problem 33609

Find the rate of change of the equation y=6x2+1y = 6x^2 + 1y=6x2+1.

Problem 33610

Problem 33611

Find where the function f(x)=xexf(x)=x e^{x}f(x)=xex is concave down. Choose from the intervals: (−1,∞)(-1, \infty)(−1,∞), (−∞,0)(-\infty, 0)(−∞,0), (−∞,−2)(-\infty,-2)(−∞,−2), (−∞,−1)(-\infty,-1)(−∞,−1), or none.

Problem 33612

Find the derivative y′y'y′ of the function y=2−9e−x8y=2-9 e^{-x^{8}}y=2−9e−x8.

Problem 33613

Find the derivative y′y'y′ if y=4e6+6eln⁡xy=4 e^{6}+6 e^{\ln x}y=4e6+6elnx.

Problem 33614

Find where the function f(x)=xexf(x)=x e^{x}f(x)=xex is concave down. Choose from the given intervals.

Problem 33615

Find the derivative y′y'y′ of the function y=x3e3ln⁡xy=x^{3} e^{3 \ln x}y=x3e3lnx.

Problem 33616

Find the derivative y′y'y′, if y=83x+1y=8^{3x+1}y=83x+1.

Problem 33617

Find f′(0)f^{\prime}(0)f′(0) for the function f(x)=2x+2f(x)=2^{x+2}f(x)=2x+2. Options: 4, 8ln⁡28 \ln 28ln2, 4ln⁡24 \ln 24ln2, none, ln⁡4\ln 4ln4.

Problem 33618

Find the derivative y′y'y′, given y=ln⁡(e8x+9)y=\ln(e^{8x}+9)y=ln(e8x+9).

Problem 33619

Using the Gompertz model, show that lim⁡N→0R(N)=0\lim_{N \rightarrow 0} R(N) = 0limN→0​R(N)=0 by completing the table for R(N)=Nln⁡(1/N)R(N) = N \ln(1/N)R(N)=Nln(1/N). Fill in: N: 0.1, 0.01, 0.001 R(N): □\square□, □\square□, □\square□

Problem 33620

Find the intervals where the function f(x)=xexf(x)=x e^{x}f(x)=xex is concave down. Choices: (−∞,−1)(-\infty,-1)(−∞,−1), (−∞,−2)(-\infty,-2)(−∞,−2), (−1,∞)(-1, \infty)(−1,∞), (−∞,0)(-\infty, 0)(−∞,0), none.

Problem 33621

Find f′′(1)f^{\prime \prime}(1)f′′(1) for f(x)=x3ln⁡xf(x)=x^{3} \ln xf(x)=x3lnx. Options: 8, 1, 5, 0, none.

Problem 33622

Find the derivative y′=ddxln⁡(e8x+9)y'=\frac{d}{dx} \ln(e^{8x}+9)y′=dxd​ln(e8x+9). Choose the correct option.

Problem 33623

Find the intervals where the function f(x)=xexf(x)=x e^{x}f(x)=xex is concave down. Options: (−∞,−2)(-\infty,-2)(−∞,−2), (−1,∞)(-1, \infty)(−1,∞), none, (−∞,0)(-\infty, 0)(−∞,0), (−∞,−1)(-\infty,-1)(−∞,−1).

Problem 33624

Find the derivative of y=x3e3ln⁡xy = x^{3} e^{3 \ln x}y=x3e3lnx. What is y′y'y′?

Problem 33625

Find the derivative y′y'y′ of the function y=83x+1y=8^{3x+1}y=83x+1. Choices: (3ln⁡8)83x+1(3 \ln 8) 8^{3 x+1}(3ln8)83x+1, (ln⁡8)83x+1(\ln 8) 8^{3 x+1}(ln8)83x+1, (8ln⁡3)83x+1(8 \ln 3) 8^{3 x+1}(8ln3)83x+1, (ln⁡3)83x+1(\ln 3) 8^{3 x+1}(ln3)83x+1, none.

Problem 33626

Find the derivative of f(x)=5xsin⁡xcos⁡xf(x)=5 x \sin x \cos xf(x)=5xsinxcosx and evaluate it at the given point. What is f′(x)=?f^{\prime}(x)=?f′(x)=?

Problem 33627

Find f′′(1)f^{\prime \prime}(1)f′′(1) for the function f(x)=x3ln⁡xf(x)=x^{3} \ln xf(x)=x3lnx. Options: 8, 1, 5, 0, none.

Problem 33628

Find the derivative y′y^{\prime}y′ of y=4e6+6eln⁡xy=4 e^{6}+6 e^{\ln x}y=4e6+6elnx. Choices: 24e5+624 e^{5}+624e5+6, 6x6 x6x, 6, 24e5+6xeln⁡x24 e^{5}+6 x e^{\ln x}24e5+6xelnx, 0.

Problem 33629

Find f′′(1)f^{\prime \prime}(1)f′′(1) for the function f(x)=x3ln⁡xf(x)=x^{3} \ln xf(x)=x3lnx. Options: 8, 5, none, 0, 1.

Problem 33630

Find f′(4)f^{\prime}(4)f′(4) if f′(x)=5sin⁡(x)cos⁡(x)+5xcos⁡(2x)f^{\prime}(x)=5 \sin (x) \cos (x)+5 x \cos (2 x)f′(x)=5sin(x)cos(x)+5xcos(2x).

Problem 33631

Problem 33632

Find the derivative f′(0)f^{\prime}(0)f′(0) if f(x)=2x+2f(x)=2^{x+2}f(x)=2x+2.

Problem 33633

Find the derivative of the function f(x)=5xsin⁡xcos⁡xf(x)=5 x \sin x \cos xf(x)=5xsinxcosx: f′(x)=f^{\prime}(x)=f′(x)=

Problem 33634

Find the derivative y′y'y′ of the function y=2−9e−x8y=2-9 e^{-x^{8}}y=2−9e−x8. Choose the correct option.

Problem 33635

Find the derivative y′=ddxln⁡(e8x+9)y'=\frac{d}{dx} \ln(e^{8x}+9)y′=dxd​ln(e8x+9) and select the correct answer.

Problem 33636

Find the derivative y′y^{\prime}y′ if y=83x+1y=8^{3x+1}y=83x+1. Choices include (8ln⁡3)83x+1(8 \ln 3) 8^{3 x+1}(8ln3)83x+1, (3ln⁡8)83x+1(3 \ln 8) 8^{3 x+1}(3ln8)83x+1, etc.

Problem 33637

Calculate the average linear momentum ⟨p^x⟩\left\langle\hat{p}_{x}\right\rangle⟨p^​x​⟩ for a particle in a box of length aaa. Specify nnn.

Problem 33638

Find the derivative of the function f(x)=−3sin⁡6(x)f(x)=-3 \sin ^{6}(x)f(x)=−3sin6(x). What is f′(x)f^{\prime}(x)f′(x)?

Problem 33639

Find the derivative of r(x)=(x+10)ln⁡(4x)r(x)=(x+10) \ln (4 x)r(x)=(x+10)ln(4x). What is r′(x)=?r^{\prime}(x)=?r′(x)=?

Problem 33640

Find the derivative of the function f(x)=e2/xf(x)=e^{2/x}f(x)=e2/x. What is f′(x)=?f'(x)=?f′(x)=?

Problem 33641

Find the derivative of f(x)=x2e7xf(x)=x^{2} e^{7 x}f(x)=x2e7x. What is f′(x)=?f^{\prime}(x)=?f′(x)=?

Evaluate the limit as $b$ approaches 36: $\lim _{b \rightarrow 36} \frac{36-b}{6-\sqrt{b}}$ .

Find the limit $\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}$ for the series $\sum_{n=1}^{\infty} \frac{(2 n) !}{(n !)(n+1) !}$ . Options: a. 4, b. 2, c. $\infty$ , d. 1, e. 0.

Find the tangent line to $y=4 \sin x$ at $\left(\frac{\pi}{6}, 2\right)$ in the form $y=m x+b$ where $m=$ and $b=$ .

Find the derivative $y'$ of the function $y=4 e^{6}+6 e^{\ln x}$ .

Find the derivative of the function $f(x)=2 x^{6}-10 x^{10}$ . What is $f^{\prime}(x)=?$

Find the derivative $y' = \frac{d}{dx} \log_{8}(9 - x - x^{5})$ .

Analyze the function $f(x)=\begin{cases} \frac{1}{2} x+10, & x<8 \\ \sqrt{8 x}+6, & x \geq 8 \end{cases}$ .
Part 1: Is $f(x)$ continuous at $x=8$ ? Explain.
Part 2: Is $f(x)$ differentiable at $x=8$ ? Explain.

Find the rate of change of the equation $y = 6x^2 + 1$ .

Find where the function $f(x)=x e^{x}$ is concave down. Choose from the intervals: $(-1, \infty)$ , $(-\infty, 0)$ , $(-\infty,-2)$ , $(-\infty,-1)$ , or none.

Find the derivative $y'$ of the function $y=2-9 e^{-x^{8}}$ .

Find the derivative $y'$ if $y=4 e^{6}+6 e^{\ln x}$ .

Find where the function $f(x)=x e^{x}$ is concave down. Choose from the given intervals.

Find the derivative $y'$ of the function $y=x^{3} e^{3 \ln x}$ .

Find the derivative $y'$ , if $y=8^{3x+1}$ .

Find $f^{\prime}(0)$ for the function $f(x)=2^{x+2}$ . Options: 4, $8 \ln 2$ , $4 \ln 2$ , none, $\ln 4$ .

Find the derivative $y'$ , given $y=\ln(e^{8x}+9)$ .

Using the Gompertz model, show that $\lim_{N \rightarrow 0} R(N) = 0$ by completing the table for $R(N) = N \ln(1/N)$ .
Fill in: N: 0.1, 0.01, 0.001 R(N): $\square$ , $\square$ , $\square$

Find the intervals where the function $f(x)=x e^{x}$ is concave down. Choices: $(-\infty,-1)$ , $(-\infty,-2)$ , $(-1, \infty)$ , $(-\infty, 0)$ , none.

Find $f^{\prime \prime}(1)$ for $f(x)=x^{3} \ln x$ . Options: 8, 1, 5, 0, none.

Find the derivative $y'=\frac{d}{dx} \ln(e^{8x}+9)$ . Choose the correct option.

Find the intervals where the function $f(x)=x e^{x}$ is concave down. Options: $(-\infty,-2)$ , $(-1, \infty)$ , none, $(-\infty, 0)$ , $(-\infty,-1)$ .

Find the derivative of $y = x^{3} e^{3 \ln x}$ . What is $y'$ ?

Find the derivative $y'$ of the function $y=8^{3x+1}$ . Choices: $(3 \ln 8) 8^{3 x+1}$ , $(\ln 8) 8^{3 x+1}$ , $(8 \ln 3) 8^{3 x+1}$ , $(\ln 3) 8^{3 x+1}$ , none.

Find the derivative of $f(x)=5 x \sin x \cos x$ and evaluate it at the given point. What is $f^{\prime}(x)=?$

Find $f^{\prime \prime}(1)$ for the function $f(x)=x^{3} \ln x$ . Options: 8, 1, 5, 0, none.

Find the derivative $y^{\prime}$ of $y=4 e^{6}+6 e^{\ln x}$ . Choices: $24 e^{5}+6$ , $6 x$ , 6, $24 e^{5}+6 x e^{\ln x}$ , 0.

Find $f^{\prime \prime}(1)$ for the function $f(x)=x^{3} \ln x$ . Options: 8, 5, none, 0, 1.

Find $f^{\prime}(4)$ if $f^{\prime}(x)=5 \sin (x) \cos (x)+5 x \cos (2 x)$ .

Find the derivative $f^{\prime}(0)$ if $f(x)=2^{x+2}$ .

Find the derivative of the function $f(x)=5 x \sin x \cos x$ : $f^{\prime}(x)=$

Find the derivative $y'$ of the function $y=2-9 e^{-x^{8}}$ . Choose the correct option.

Find the derivative $y'=\frac{d}{dx} \ln(e^{8x}+9)$ and select the correct answer.

Find the derivative $y^{\prime}$ if $y=8^{3x+1}$ . Choices include $(8 \ln 3) 8^{3 x+1}$ , $(3 \ln 8) 8^{3 x+1}$ , etc.

Calculate the average linear momentum $\left\langle\hat{p}_{x}\right\rangle$ for a particle in a box of length $a$ . Specify $n$ .

Find the derivative of the function $f(x)=-3 \sin ^{6}(x)$ . What is $f^{\prime}(x)$ ?

Find the derivative of $r(x)=(x+10) \ln (4 x)$ . What is $r^{\prime}(x)=?$

Find the derivative of the function $f(x)=e^{2/x}$ . What is $f'(x)=?$

Find the derivative of $f(x)=x^{2} e^{7 x}$ . What is $f^{\prime}(x)=?$

Find the derivative $y'$ of the function $y=2-9 e^{-x^{8}}$ . Options include: $9 x^{8} e^{-e^{8}}$ , $-72 x^{7} e^{-x^{8}}$ , none, $-9 x^{8} e^{-z^{8}}$ , $72 x^{7} e^{-x^{8}}$ .

Find the derivative of the function $f(x)=3 \ln (6 x)$ . What is $f^{\prime}(x)=?$

Find the derivative $f^{\prime}(x)$ for the function $f(x)=x^{\ln (x)}$ .

Find the derivative of the function $y=2-9 e^{-x^{8}}$ . What is $y^{\prime}$ ?

Find the derivative $y'$ , where $y=\log _{8}(9-x-x^{5})$ . Choices include various fractions.

Find the derivative of the function $f(t)=6 e^{t+3}$ . What is $\frac{d f}{d t}=?$

Find the intervals where the function $f(x)=x e^{x}$ is concave down. Select from the given options.

Find the antiderivative of the function $f(x)=9 x^{8}+10 x^{5}-3 x^{4}-4$ .

Find the function $f(x)$ if $f^{\prime}(x)=4 x+2$ and $f(4)=44$ . What is $f(x)=?$

Find $f\left(\frac{\pi}{4}\right)$ given $f^{\prime \prime}(x)=-16 \sin(4x)$ , $f^{\prime}(0)=-6$ , and $f(0)=-5$ .

Find the position of a particle at time $t=14$ given $a(t)=24t+18$ , $s(0)=13$ , and $v(0)=3$ .

Compute the integral: $\int 4 e^{x} dx = C + \square$

Calculate the indefinite integral and include the constant $C$ : $\int \frac{5}{1+t^{2}} d t=\square$

Find the slope of the curve $f(x)=(x+1)^{\frac{3}{2}}$ at the point where $x=15$ .

Compute the indefinite integral: $\int \frac{28 x^{7}-7 x}{x} d x = C$ .

Find the limit as $x$ approaches 3 for the expression $\frac{\frac{1}{x}-\frac{1}{3}}{x-3}$ .

Evaluate the integral: $\int \frac{7}{(t+9)^{6}} dt$

Integrate the function from -2 to 1.5: $\int_{-2}^{1.5} 24(6 x+4)^{3} d x$ .

Find the interval for a local maximum of the function $h(x)=\frac{1}{2} x^{4}-2 x^{2}+x+4$ . A) (1,2) B) (0,1) C) (3,5) D) (-2,0)

Evaluate the integral from 3 to 7 of (2x + 9): $\int_{3}^{7}(2 x+9) d x=$

Find the integral of $7 x e^{3 x}$ with respect to $x$ .

Evaluate the integral of $6 \sin^5 r \cos r \, dr$ .

Find $y^{\prime}$ if $y=\log _{2}(8 t^{\ln 2})$ . Choices: $\frac{\ln 8}{t}$ , $3+\ln t$ , $\frac{t}{\ln 2}$ , $\frac{1}{t}$ , none.

Find the integral of $\cos^{2} x$ with respect to $x$ : $\int \cos^{2} x \, dx$ .

Calculate the integral: $\int \frac{1+x}{1+\sqrt{x+1}} d x$

Evaluate the limit: $\lim _{\Delta x \rightarrow 0} \frac{\cos \left(\frac{\pi}{1}+\Delta x\right)-\frac{\sqrt{2}}{2}}{\Delta x}=$

Evaluate the limit: $\lim _{\Delta x \rightarrow 0} \frac{\cos \left(\frac{\pi}{4}+\Delta x\right)-\frac{\sqrt{2}}{2}}{\Delta x}.$

Find the limit as $x$ approaches 0 for the expression $x^{2} \sin \frac{1}{x}$ .

Find the rate of change of $f(x)=\tan (x)$ at $x=-\frac{\pi}{4}$ . What is it? Options: a. -0.5, b. 0, c. 2, d. -2.

Find the limit: $\lim _{x \rightarrow 0^{+}} \sqrt{x} e^{\sin (\pi / x)}$

Find the value of $t$ where the local maximum of $f(t)=\int_{0}^{t} \frac{x^{2}+11 x+28}{1+\cos ^{2}(x)} d x$ occurs. $t=$

Find the equation of the tangent line to the function $f(x)=4 \sin (x)$ at the point $\left(\frac{\pi}{6}, 2\right)$ .