Calculus

Problem 8201

Find the derivative of $G(t)=\frac{1-5 t}{3+t}$ and state the domains of $G$ and $G'(t)$ in interval notation.

See Solution

Problem 8202

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

See Solution

Problem 8203

Find the derivative of $s$ with respect to $x$ from the equation $\sqrt{x+s}=\frac{4}{x}+\frac{4}{s}$ .

See Solution

Problem 8204

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

See Solution

Problem 8205

Find $G^{\prime}(\pi)$ if $G(x)=\cos(f(x))$ using $f(\pi)=\frac{3\pi}{2}$ and $f^{\prime}(\pi)=2$ .

See Solution

Problem 8206

Find the derivative $\left.y^{\prime}\right|_{x=1}$ for $y=f\left(x^{2}+2\right)$ using the given values of $f(x)$ .

See Solution

Problem 8207

Find the derivative of $f(x)=x+\sqrt{x}+1$ using its definition. What are the domains of $f(x)$ and $f'(x)$ ?

See Solution

Problem 8208

Find $h^{\prime}(2)$ if $h(x)=f(3x)$ using given values of $f(x)$ and its derivatives.

See Solution

Problem 8209

Differentiate $h=\frac{5 t-2 t}{3 h^{2}+5 t}$ with respect to $t$ .

See Solution

Problem 8210

Find the average rate of change for $f(x)=\frac{1}{x}$ using the difference quotient on $[0.5,0.51]$ and $[3,3.01]$ .

See Solution

Problem 8211

Derive the equation $y^{2}=\frac{x^{2}}{x-4}$ using implicit differentiation.

See Solution

Problem 8212

Find the derivative of $y^{2}=\frac{x^{2}}{x y-4}$ using implicit differentiation.

See Solution

Problem 8213

Find the limits of the function as $x \rightarrow \infty$ and $x \rightarrow -\infty$ : $\frac{(x-1)(x^{2}+x)}{(x+2)^{2}(x-3)}$

See Solution

Problem 8214

Determine if the integral $\int_{0}^{1} \frac{\cos ^{2} x}{\sqrt{x}} d x$ converges or diverges by analyzing behavior near $x=0$ .

See Solution

Problem 8215

Prove the Quotient Rule for $\frac{d}{d x}\left(\frac{f(x)}{x}\right)=\frac{x f^{\prime}(x)-f(x)}{x^{2}}$ using limits.

See Solution

Problem 8216

Analyze the integral $\int_{0}^{1} \frac{\cos ^{2} x}{\sqrt{x}} d x$ for convergence or divergence near $x=0$ .

See Solution

Problem 8217

Evaluate the integral $\int_{0}^{1} \frac{e^{-x}}{x} dx$ for convergence by analyzing the behavior near $x=0$ using Taylor series.

See Solution

Problem 8218

Find the leading term of the integrand as $x \rightarrow+\infty$ to determine if $\int_{1}^{+\infty} \frac{\sqrt[3]{x+3}}{x^{3}} dx$ converges or diverges.

See Solution

Problem 8219

Identify all improper integrals from the following list:
1. $\int_{-1}^{1} \ln(1+x^{2}) \, dx$
2. $\int_{\pi/4}^{3\pi/4} \sec \theta \, d\theta$
3. $\int_{0}^{\infty} t^{x-1} e^{-t} \, dt$
4. $\int_{-1,000,000}^{1,000,000} e^{-x^{2}/2} \, dx$
5. $\int_{0}^{1} \frac{dx}{x-1}$
6. $\int_{-\pi/4}^{\pi/4} \sec \theta \, d\theta$
7. $\int_{0}^{1} \frac{dx}{\sqrt{x}-10}$
8. $\int_{0}^{1} \frac{dx}{x^{2}}$
9. $\int_{1}^{2} \sqrt{2-x} \, dx$
10. $\int_{-\infty}^{\infty} e^{-x^{2}/2} \, dx$

See Solution

Problem 8220

Trouver la dérivée de $f(x)=\left(4 x^{8}+6\right)^{4}\left(12 x^{6}+5\right)^{3}$ . $f^{\prime}(x)=$

See Solution

Problem 8221

Find $\frac{dy}{dx}$ for $e^{2xy} = \sin(y^7)$ , expressing it in terms of $x$ and $y$ .

See Solution

Problem 8222

Find the marginal productivity of the function $p(x, y)=2400 x^{\frac{4}{5}} y^{\frac{1}{5}}$ . Calculate $\frac{\partial p}{\partial x}$ .

See Solution

Problem 8223

Find the leading term of $\frac{\sqrt[3]{x+3}}{x^{3}}$ as $x \to +\infty$ to check if the integral converges or diverges:
$\int_{1}^{+\infty} \frac{\sqrt[3]{x+3}}{x^{3}} dx$

See Solution

Problem 8224

Find the derivative of $f(x)=\csc (x) e^{x}$ . What is $f^{\prime}(x)$ ? Options: a) $-\csc (x) \tan (x)$ b) $\csc (x) e^{x}-e^{x} \csc (x) \cot (x)$ c) $-e^{x} \csc (x) \cot (x)$ d) $\sec (x) \tan (x)+e^{x}$ e) None of the above

See Solution

Problem 8225

Find the leading term of $\frac{\sqrt[3]{x+3}}{x^{3}}$ as $x \rightarrow+\infty$ to check integral convergence: $\int_{1}^{+\infty} \frac{\sqrt[3]{x+3}}{x^{3}} dx$ .

See Solution

Problem 8226

Find the derivative $f^{\prime}(x)$ of the function $f(x)=x^{-5}$ . Choose the correct option from: a) $-6 x^{-6}$ , b) $x^{-6}$ , c) $-5 x^{-6}$ , d) $5 x^{-6}$ , e) None.

See Solution

Problem 8227

Calculez la dérivée de $f(x)=\frac{(10 x^{8}+7)^{4}}{(7 x^{8}-6)^{4}}$ . $f^{\prime}(x)=$

See Solution

Problem 8228

Evaluate the integral $\int_{0}^{1} \frac{e^{-x}}{x} dx$ for convergence using Taylor series near $x=0$ .

See Solution

Problem 8229

Find the derivative $f^{\prime}(x)$ of $f(x)=\tan(t) e^{x}$ . Options: a) $\sec^{2}(t) e^{x}$ b) $\tan(t) e^{x}+e^{x} \sec^{2}(t)$ c) $e^{x} \tan(t)$ d) $\sec^{2}(t)+e^{x}$ e) None.

See Solution

Problem 8230

Find the derivative $g^{\prime}(x)$ of $g(x)=\frac{x^{4}}{6-x}$ . Choose from the options provided.

See Solution

Problem 8231

Find the leading term of the integrand as $x \to +\infty$ to check if the integral $\int_{1}^{+\infty} \frac{\sqrt[3]{x+3}}{x^{3}} dx$ converges or diverges.

See Solution

Problem 8232

Find $g^{\prime}(x)$ if $g(x)=(x^{3}+1)(x^{3}-1)$ . Options: a) $6 x^{5}-1$ b) $6 x^{5}$ c) $6 x^{3}$ d) $6 x^{5}+6 x$ e) $6 x^{5}+6 x-1$ f) None.

See Solution

Problem 8233

Calculer la dérivée de $f(x)=5^{9 x^{6}+5 x^{5}+12 x}$ . Quelle est $f^{\prime}(x)$ ?

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Problem 8234

Show that $E=-\nabla V$ leads to LaPlace's equation, $\nabla^{2} V=0$ , when charge density is $0$ .

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Problem 8235

Find $f^{\prime}(x)$ for $f(x)=\frac{6-x}{x-6}$ . Choose from: a) $\frac{6 x-6}{(x-6)^{2}}$ , b) -1, c) 0, d) $\frac{6 x}{(x-6)^{2}}$ , e) $\frac{6 x+1}{(x-6)^{2}}$ , f) None.

See Solution

Problem 8236

Calculer la dérivée de $f(x)=48 \sqrt{11 x^{8}+7 x^{4}-4 x^{3}+8 x}$ . Trouvez $f^{\prime}(x)$ .

See Solution

Problem 8237

Find the rate of change of infected people at $t=10$ weeks for $N(t)=300 t^{\frac{3}{2}} e^{-(t-10)}$ . Options: a) 142 b) -260 c) -283 d) -2000 e) -3307 f) -8064 g) None.

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Problem 8238

Evaluate the integral $\int_{0}^{1} \frac{e^{-x}}{x} dx$ . Analyze the behavior near $x=0$ to check for convergence or divergence.

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Problem 8239

Calculer la dérivée de $f(x)=x e^{3 x+20}$ . Quelle est $f^{\prime}(x)$ ?

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Problem 8240

A 12-lb weight is attached to a spring and dashpot. Find its position function $x(t)$ after being pulled down 1 ft and released. Also, determine the frequency, amplitude, and phase angle.

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Problem 8241

A carbon-14 sample is now 52.0 grams after 17,190 years. Find the original sample size, knowing the half-life is 5,730 years. Use three significant figures.

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Problem 8242

Determine the true statement about $f(x)=|x-4|$ at $x=4$ and $x=-4$ from the options provided.

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Problem 8243

Determine the true statement about a function with corners at $x=-1$ and $x=1$ regarding differentiability and continuity.

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Problem 8244

Find the value of $f(1)$ for $f(x)=\frac{x^{2}-1}{x-1}$ to make it continuous at $x=1$ .

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Problem 8245

For the function $f(x)=|x-4|$ , which statement is true about its continuity and differentiability at $x=4$ ?

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Problem 8246

Find $f^{\prime \prime}(1)$ for $f(x)=x^{4}+x^{2}+1$ .

See Solution

Problem 8247

Find the gradient function $\frac{d y}{d x}$ for: a) $y=(4 x-5)^{2}$ , b) $y=\frac{1}{5-2 x}$ .

See Solution

Problem 8248

Find the derivative $f^{\prime}(c)$ for $f(x)=x-\frac{1}{x}$ at $c=1$ .

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Problem 8249

Find the arc length $s$ of the curve $\vec{r}(t)=\langle 3 t, 5 \cos (4 t), 5 \sin (4 t)\rangle$ for $-1 \leq t \leq 3$ .

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Problem 8250

Show that the derivative of $f^3$ is $3 f^2 f'$ .

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Problem 8251

Find the tangent line to the curve $y=3x^2$ at $x=3$ . Choose the correct equation from the options given.

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Problem 8252

Find the vertical asymptotes of the function $v(x)=\frac{1}{x^{2}+1}$ . What are the equations?

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Problem 8253

A 12-lb weight stretches a spring 6 in. and has 3 lb resistance. Find position $x(t)$ and motion properties after 1 ft pull.

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Problem 8254

Find the derivative $\left.\frac{d z}{d x}\right|_{x=1} ^{n}$ for $z=\frac{1}{x^{3}+1}$ .

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Problem 8255

Find the derivative of $y=7 \sin^{-1}(x)$ at $x=-\frac{3}{4}$ . Calculate $y^{\prime}\left(-\frac{3}{4}\right)=$

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Problem 8256

Evaluate the limit: $\lim _{x \rightarrow 3} \frac{x^{2}-8 x+15}{x^{2}-x-6}$ .

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Problem 8257

Find $g^{\prime}(1)$ where $f(x)=\sin x + 2x + 1$ and $g$ is the inverse of $f$ .

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Problem 8258

Find the position vector $\vec{r}(t)$ for a particle with acceleration $\vec{a}(t)=\langle 4 t, 6 \sin (t), \cos (2 t)\rangle$ , initial velocity $\vec{v}(0)=\langle-1,-2,1\rangle$ , and initial position $\vec{r}(0)=\langle-2,0,1\rangle$ .

See Solution

Problem 8259

Evaluate the limit as $x$ approaches 0: $\lim_{x \rightarrow 0} \frac{6 \cos(5x) - 5 e^{-2x} - 1}{5x - \sin(3x)}$

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Problem 8260

Find the derivative of $f(t)=(2t+1)(t^{2}-2)$ using the product or quotient rule, then simplify and calculate the derivative.

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Problem 8261

Find the derivative $\frac{d y}{d x}$ for $y=(4+x^{3} \ln x)^{7}$ .

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Problem 8262

Find $\left.\frac{d p}{d h}\right|_{h=4}$ for the function $p=7 e^{h-2}$ .

See Solution

Problem 8263

Find the derivative of $f(s) = s^{1/4} + s^{1/3}$ .

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Problem 8264

Find the limits of the function as $x \rightarrow \infty$ and $x \rightarrow -\infty$ : $\frac{3-\left(1 / x^{2}\right)}{(-5 / x)+\left(7 / x^{2}\right)}$

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Problem 8265

Find the derivative of the integral: $\frac{d}{d x}\left(\int_{-3}^{\sqrt{x}} t \sin \left(t^{2}\right) d t\right)$ .

See Solution

Problem 8266

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ for $y=\frac{(5 x^{2}-3)^{9/10}}{(6 x^{2}-6 x+1)^{7/6}}$ using logarithmic differentiation.

See Solution

Problem 8267

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ using logarithmic differentiation for $y=\frac{(5 x^{2}-3)^{9/10}}{(6 x^{2}-6 x+1)^{7/6}}$ .

See Solution

Problem 8268

Find values of $c$ such that $\lim_{x \to 2} f(x)$ exists for the piecewise function: $f(x)=\{3 x^{2}+c x \text{ if } x<2, 1+\frac{3 x}{c} \text{ if } x>2\}$ .

See Solution

Problem 8269

Find $\frac{d y}{d x}$ using logarithmic differentiation for $y=\frac{(5 x^{2}-3)^{9/10}}{(6 x^{2}-6 x+1)^{7/6}}$ .

See Solution

Problem 8270

Find the integral $I=\int \frac{1}{x^{2}+17} \mathrm{~d} x$ .

See Solution

Problem 8271

Find the derivative of $y=6 \cos^{-1}(7x)$ at $x=-\frac{1}{8}$ . Compute $y^{\prime}\left(-\frac{1}{8}\right)=$

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Problem 8272

Find the derivative of $y=5 \sec^{-1}(x+7)$ . What is $y'$ ?

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Problem 8273

Find the integral $I=\int \frac{1}{x^{2}+17} \mathrm{~d} x$ .

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Problem 8274

Find the integral $I=\int(7 x+8) e^{7 x} \mathrm{~d} x$ .

See Solution

Problem 8275

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $3 x^{2}-8 x y-y^{3}=10$ .

See Solution

Problem 8276

Find the integral $I=\int(7 x+8) e^{7 x} \mathrm{~d} x$ .

See Solution

Problem 8277

Calculate the average rate of change of $f(x)=x^{3}-2 x^{2}+3 x$ between $x=1$ and $x=2$ .

See Solution

Problem 8278

Find the integral $I=\int(7 x+8) e^{7 x} d x$ .

See Solution

Problem 8279

Find the integral $I=\int \frac{6 x^{2}}{\left(2 x^{3}+5\right)^{4}} \mathrm{~d} x$ .

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Problem 8280

Find the derivative of $y=\frac{\cos^{-1}(12x)}{\sin^{-1}(12x)}$ . What is $y'$ ?

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Problem 8281

Differentiate $p(x)=(x+1) f(x)+x g(x)$ to find $p^{\prime}(x)$ . Then, compute $p^{\prime}(-1)$ using $f(-1)=6$ , $f^{\prime}(-1)=3$ , $g(-1)=4$ , $g^{\prime}(-1)=4$ .

See Solution

Problem 8282

Find the critical number $A$ for the function $f(x)=9(x-5)^{2/3}$ in the intervals $(-\infty, A)$ and $(A, \infty)$ .

See Solution

Problem 8283

Find the derivative of $f(x) = \ln \left(\frac{(-2 x^{3}+4)^{4}}{(2 x-1)^{5}}\right)$ .

See Solution

Problem 8284

Find the derivative of $y=\frac{\cos^{-1}(12x)}{\sin^{-1}(12x)}$ .

See Solution

Problem 8285

Solve the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}=8 x(x^{2}-1) \sqrt{y}$ with the initial condition $y(1)=1$ .

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Problem 8286

Find the derivative of $f(x)=\frac{x}{5x+2}$ using the limit definition.
(a) Simplify $\frac{f(x+h)-f(x)}{h}$ to the form $\frac{A}{(5(x+h)+2)(5x+2)}$ . Find $A=$ .
(b) Compute $f^{\prime}(x)=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=$ .

See Solution

Problem 8287

Solve the differential equation $\frac{d y}{d x}+6 \frac{y}{x}=\frac{8 x^{2}+3 x-4}{x^{5}}$ with initial condition $y(1)=0$ .

See Solution

Problem 8288

Find the first three non-zero terms of the MacLaurin series for $f(x)=\frac{1-e^{-x}}{x}$ .

See Solution

Problem 8289

Find the interval where the quartic polynomial $f(x) = \frac{x^{4}}{2} + 1.5 x^{3} - 7.5 x^{2}$ is concave down. $<x<$

See Solution

Problem 8290

Find the interval where the quartic polynomial $f(x) = \frac{x^{4}}{2} + 2.5 x^{3} - 18 x^{2}$ is concave down.

See Solution

Problem 8291

Find the critical numbers of the function $F(x)=x^{4/5}(x-9)^{2}$ .

See Solution

Problem 8292

Find the first three non-zero terms of $f(x) = \frac{1 - e^{-x}}{x}$ using its MacLaurin series.

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Problem 8293

Find the first three non-zero terms of $f(x)=(1-e^{-x})/x$ using the MacLaurin series and approximate $I=\int_{0}^{b}(1-e^{-x})/x \, dx$ as a function of $b$ .

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Problem 8294

Find the interval where the quartic polynomial $f(x)=-\frac{x^{4}}{2}-1.5 x^{3}+21 x^{2}$ is concave up. $<x<$

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Problem 8295

Find the value of $f^{\prime}(1)$ for the function $f(x)=\frac{\ln (x)}{x^{6}}$ .

See Solution

Problem 8296

Find the tangent line equation for the curve $x^{3}+y^{3}=3xy$ at the point $\left(\frac{48}{65}, \frac{12}{65}\right)$ .

See Solution

Problem 8297

Find the domain of $f(x, y)=\cosh^{-1}(y-x^2)$ , compute $f_x$ , $f_y$ for $f(x, y)=\sinh(x^2 y+y^3$ , and find $\frac{dz}{dt}$ for $z=xy^2+3\cos y$ with $x=e^t$ , $y=e^{-2t}$ .

See Solution

Problem 8298

Find the derivative $\frac{d y}{d x}$ for $y \cos(y+x+x^{2})=x^{3}$ at point $P=\left(0, \frac{\pi}{2}\right)$ .

See Solution

Problem 8299

Calculate the integral $I=\int_{-1}^{\infty}\left(\frac{x^{4}}{1+x^{6}}\right)^{2} d x$ .

See Solution

Problem 8300

Find constants $a$ and $b$ such that $\lim _{x \rightarrow 0} \frac{\sin (a x)+b x}{x^{3}}=-\frac{4}{3}$ .

See Solution

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Calculus

Problem 8201

Find the derivative of G(t)=1−5t3+tG(t)=\frac{1-5 t}{3+t}G(t)=3+t1−5t​ and state the domains of GGG and G′(t)G'(t)G′(t) in interval notation.

Problem 8202

Find the limit as xxx approaches 1 for the expression x2−1x−1\frac{x^{2}-1}{x-1}x−1x2−1​.

Problem 8203

Find the derivative of sss with respect to xxx from the equation x+s=4x+4s\sqrt{x+s}=\frac{4}{x}+\frac{4}{s}x+s​=x4​+s4​.

Problem 8204

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

Problem 8205

Find G′(π)G^{\prime}(\pi)G′(π) if G(x)=cos⁡(f(x))G(x)=\cos(f(x))G(x)=cos(f(x)) using f(π)=3π2f(\pi)=\frac{3\pi}{2}f(π)=23π​ and f′(π)=2f^{\prime}(\pi)=2f′(π)=2.

Problem 8206

Find the derivative y′∣x=1\left.y^{\prime}\right|_{x=1}y′∣x=1​ for y=f(x2+2)y=f\left(x^{2}+2\right)y=f(x2+2) using the given values of f(x)f(x)f(x).

Problem 8207

Find the derivative of f(x)=x+x+1f(x)=x+\sqrt{x}+1f(x)=x+x​+1 using its definition. What are the domains of f(x)f(x)f(x) and f′(x)f'(x)f′(x)?

Problem 8208

Find h′(2)h^{\prime}(2)h′(2) if h(x)=f(3x)h(x)=f(3x)h(x)=f(3x) using given values of f(x)f(x)f(x) and its derivatives.

Problem 8209

Differentiate h=5t−2t3h2+5th=\frac{5 t-2 t}{3 h^{2}+5 t}h=3h2+5t5t−2t​ with respect to ttt.

Problem 8210

Find the average rate of change for f(x)=1xf(x)=\frac{1}{x}f(x)=x1​ using the difference quotient on [0.5,0.51][0.5,0.51][0.5,0.51] and [3,3.01][3,3.01][3,3.01].

Problem 8211

Derive the equation y2=x2x−4y^{2}=\frac{x^{2}}{x-4}y2=x−4x2​ using implicit differentiation.

Problem 8212

Find the derivative of y2=x2xy−4y^{2}=\frac{x^{2}}{x y-4}y2=xy−4x2​ using implicit differentiation.

Problem 8213

Find the limits of the function as x→∞x \rightarrow \inftyx→∞ and x→−∞x \rightarrow -\inftyx→−∞: (x−1)(x2+x)(x+2)2(x−3)\frac{(x-1)(x^{2}+x)}{(x+2)^{2}(x-3)}(x+2)2(x−3)(x−1)(x2+x)​

Problem 8214

Determine if the integral ∫01cos⁡2xxdx\int_{0}^{1} \frac{\cos ^{2} x}{\sqrt{x}} d x∫01​x​cos2x​dx converges or diverges by analyzing behavior near x=0x=0x=0.

Problem 8215

Prove the Quotient Rule for ddx(f(x)x)=xf′(x)−f(x)x2\frac{d}{d x}\left(\frac{f(x)}{x}\right)=\frac{x f^{\prime}(x)-f(x)}{x^{2}}dxd​(xf(x)​)=x2xf′(x)−f(x)​ using limits.

Problem 8216

Analyze the integral ∫01cos⁡2xxdx\int_{0}^{1} \frac{\cos ^{2} x}{\sqrt{x}} d x∫01​x​cos2x​dx for convergence or divergence near x=0x=0x=0.

Problem 8217

Evaluate the integral ∫01e−xxdx\int_{0}^{1} \frac{e^{-x}}{x} dx∫01​xe−x​dx for convergence by analyzing the behavior near x=0x=0x=0 using Taylor series.

Problem 8218

Find the leading term of the integrand as x→+∞x \rightarrow+\inftyx→+∞ to determine if ∫1+∞x+33x3dx\int_{1}^{+\infty} \frac{\sqrt[3]{x+3}}{x^{3}} dx∫1+∞​x33x+3​​dx converges or diverges.

Problem 8219

Problem 8220

Trouver la dérivée de f(x)=(4x8+6)4(12x6+5)3f(x)=\left(4 x^{8}+6\right)^{4}\left(12 x^{6}+5\right)^{3}f(x)=(4x8+6)4(12x6+5)3. f′(x)=f^{\prime}(x)=f′(x)=

Problem 8221

Find dydx\frac{dy}{dx}dxdy​ for e2xy=sin⁡(y7)e^{2xy} = \sin(y^7)e2xy=sin(y7), expressing it in terms of xxx and yyy.

Problem 8222

Find the marginal productivity of the function p(x,y)=2400x45y15p(x, y)=2400 x^{\frac{4}{5}} y^{\frac{1}{5}}p(x,y)=2400x54​y51​. Calculate ∂p∂x\frac{\partial p}{\partial x}∂x∂p​.

Problem 8223

Find the leading term of x+33x3\frac{\sqrt[3]{x+3}}{x^{3}}x33x+3​​ as x→+∞x \to +\inftyx→+∞ to check if the integral converges or diverges: ∫1+∞x+33x3dx\int_{1}^{+\infty} \frac{\sqrt[3]{x+3}}{x^{3}} dx∫1+∞​x33x+3​​dx

Problem 8224

Problem 8225

Find the leading term of x+33x3\frac{\sqrt[3]{x+3}}{x^{3}}x33x+3​​ as x→+∞x \rightarrow+\inftyx→+∞ to check integral convergence: ∫1+∞x+33x3dx\int_{1}^{+\infty} \frac{\sqrt[3]{x+3}}{x^{3}} dx∫1+∞​x33x+3​​dx.

Problem 8226

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=x−5f(x)=x^{-5}f(x)=x−5. Choose the correct option from: a) −6x−6-6 x^{-6}−6x−6, b) x−6x^{-6}x−6, c) −5x−6-5 x^{-6}−5x−6, d) 5x−65 x^{-6}5x−6, e) None.

Problem 8227

Calculez la dérivée de f(x)=(10x8+7)4(7x8−6)4f(x)=\frac{(10 x^{8}+7)^{4}}{(7 x^{8}-6)^{4}}f(x)=(7x8−6)4(10x8+7)4​. f′(x)=f^{\prime}(x)=f′(x)=

Problem 8228

Evaluate the integral ∫01e−xxdx\int_{0}^{1} \frac{e^{-x}}{x} dx∫01​xe−x​dx for convergence using Taylor series near x=0x=0x=0.

Problem 8229

Problem 8230

Find the derivative g′(x)g^{\prime}(x)g′(x) of g(x)=x46−xg(x)=\frac{x^{4}}{6-x}g(x)=6−xx4​. Choose from the options provided.

Problem 8231

Find the leading term of the integrand as x→+∞x \to +\inftyx→+∞ to check if the integral ∫1+∞x+33x3dx\int_{1}^{+\infty} \frac{\sqrt[3]{x+3}}{x^{3}} dx∫1+∞​x33x+3​​dx converges or diverges.

Problem 8232

Find g′(x)g^{\prime}(x)g′(x) if g(x)=(x3+1)(x3−1)g(x)=(x^{3}+1)(x^{3}-1)g(x)=(x3+1)(x3−1). Options: a) 6x5−16 x^{5}-16x5−1 b) 6x56 x^{5}6x5 c) 6x36 x^{3}6x3 d) 6x5+6x6 x^{5}+6 x6x5+6x e) 6x5+6x−16 x^{5}+6 x-16x5+6x−1 f) None.

Problem 8233

Calculer la dérivée de f(x)=59x6+5x5+12xf(x)=5^{9 x^{6}+5 x^{5}+12 x}f(x)=59x6+5x5+12x. Quelle est f′(x)f^{\prime}(x)f′(x) ?

Problem 8234

Show that E=−∇VE=-\nabla VE=−∇V leads to LaPlace's equation, ∇2V=0\nabla^{2} V=0∇2V=0, when charge density is 000.

Problem 8235

Find f′(x)f^{\prime}(x)f′(x) for f(x)=6−xx−6f(x)=\frac{6-x}{x-6}f(x)=x−66−x​. Choose from: a) 6x−6(x−6)2\frac{6 x-6}{(x-6)^{2}}(x−6)26x−6​, b) -1, c) 0, d) 6x(x−6)2\frac{6 x}{(x-6)^{2}}(x−6)26x​, e) 6x+1(x−6)2\frac{6 x+1}{(x-6)^{2}}(x−6)26x+1​, f) None.

Problem 8236

Calculer la dérivée de f(x)=4811x8+7x4−4x3+8xf(x)=48 \sqrt{11 x^{8}+7 x^{4}-4 x^{3}+8 x}f(x)=4811x8+7x4−4x3+8x​. Trouvez f′(x)f^{\prime}(x)f′(x).

Problem 8237

Find the rate of change of infected people at t=10t=10t=10 weeks for N(t)=300t32e−(t−10)N(t)=300 t^{\frac{3}{2}} e^{-(t-10)}N(t)=300t23​e−(t−10). Options: a) 142 b) -260 c) -283 d) -2000 e) -3307 f) -8064 g) None.

Problem 8238

Evaluate the integral ∫01e−xxdx\int_{0}^{1} \frac{e^{-x}}{x} dx∫01​xe−x​dx. Analyze the behavior near x=0x=0x=0 to check for convergence or divergence.

Problem 8239

Calculer la dérivée de f(x)=xe3x+20f(x)=x e^{3 x+20}f(x)=xe3x+20. Quelle est f′(x)f^{\prime}(x)f′(x)?

Problem 8240

A 12-lb weight is attached to a spring and dashpot. Find its position function x(t)x(t)x(t) after being pulled down 1 ft and released. Also, determine the frequency, amplitude, and phase angle.

Problem 8241

A carbon-14 sample is now 52.0 grams after 17,190 years. Find the original sample size, knowing the half-life is 5,730 years. Use three significant figures.

Find the derivative of $G(t)=\frac{1-5 t}{3+t}$ and state the domains of $G$ and $G'(t)$ in interval notation.

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

Find the derivative of $s$ with respect to $x$ from the equation $\sqrt{x+s}=\frac{4}{x}+\frac{4}{s}$ .

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

Find $G^{\prime}(\pi)$ if $G(x)=\cos(f(x))$ using $f(\pi)=\frac{3\pi}{2}$ and $f^{\prime}(\pi)=2$ .

Find the derivative $\left.y^{\prime}\right|_{x=1}$ for $y=f\left(x^{2}+2\right)$ using the given values of $f(x)$ .

Find the derivative of $f(x)=x+\sqrt{x}+1$ using its definition. What are the domains of $f(x)$ and $f'(x)$ ?

Find $h^{\prime}(2)$ if $h(x)=f(3x)$ using given values of $f(x)$ and its derivatives.

Differentiate $h=\frac{5 t-2 t}{3 h^{2}+5 t}$ with respect to $t$ .

Find the average rate of change for $f(x)=\frac{1}{x}$ using the difference quotient on $[0.5,0.51]$ and $[3,3.01]$ .

Derive the equation $y^{2}=\frac{x^{2}}{x-4}$ using implicit differentiation.

Find the derivative of $y^{2}=\frac{x^{2}}{x y-4}$ using implicit differentiation.

Find the limits of the function as $x \rightarrow \infty$ and $x \rightarrow -\infty$ : $\frac{(x-1)(x^{2}+x)}{(x+2)^{2}(x-3)}$

Determine if the integral $\int_{0}^{1} \frac{\cos ^{2} x}{\sqrt{x}} d x$ converges or diverges by analyzing behavior near $x=0$ .

Prove the Quotient Rule for $\frac{d}{d x}\left(\frac{f(x)}{x}\right)=\frac{x f^{\prime}(x)-f(x)}{x^{2}}$ using limits.

Analyze the integral $\int_{0}^{1} \frac{\cos ^{2} x}{\sqrt{x}} d x$ for convergence or divergence near $x=0$ .

Evaluate the integral $\int_{0}^{1} \frac{e^{-x}}{x} dx$ for convergence by analyzing the behavior near $x=0$ using Taylor series.

Find the leading term of the integrand as $x \rightarrow+\infty$ to determine if $\int_{1}^{+\infty} \frac{\sqrt[3]{x+3}}{x^{3}} dx$ converges or diverges.

Trouver la dérivée de $f(x)=\left(4 x^{8}+6\right)^{4}\left(12 x^{6}+5\right)^{3}$ . $f^{\prime}(x)=$

Find $\frac{dy}{dx}$ for $e^{2xy} = \sin(y^7)$ , expressing it in terms of $x$ and $y$ .

Find the marginal productivity of the function $p(x, y)=2400 x^{\frac{4}{5}} y^{\frac{1}{5}}$ . Calculate $\frac{\partial p}{\partial x}$ .

Find the leading term of $\frac{\sqrt[3]{x+3}}{x^{3}}$ as $x \to +\infty$ to check if the integral converges or diverges:
$\int_{1}^{+\infty} \frac{\sqrt[3]{x+3}}{x^{3}} dx$

Find the leading term of $\frac{\sqrt[3]{x+3}}{x^{3}}$ as $x \rightarrow+\infty$ to check integral convergence: $\int_{1}^{+\infty} \frac{\sqrt[3]{x+3}}{x^{3}} dx$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=x^{-5}$ . Choose the correct option from: a) $-6 x^{-6}$ , b) $x^{-6}$ , c) $-5 x^{-6}$ , d) $5 x^{-6}$ , e) None.

Calculez la dérivée de $f(x)=\frac{(10 x^{8}+7)^{4}}{(7 x^{8}-6)^{4}}$ . $f^{\prime}(x)=$

Evaluate the integral $\int_{0}^{1} \frac{e^{-x}}{x} dx$ for convergence using Taylor series near $x=0$ .

Find the derivative $g^{\prime}(x)$ of $g(x)=\frac{x^{4}}{6-x}$ . Choose from the options provided.

Find the leading term of the integrand as $x \to +\infty$ to check if the integral $\int_{1}^{+\infty} \frac{\sqrt[3]{x+3}}{x^{3}} dx$ converges or diverges.

Find $g^{\prime}(x)$ if $g(x)=(x^{3}+1)(x^{3}-1)$ . Options: a) $6 x^{5}-1$ b) $6 x^{5}$ c) $6 x^{3}$ d) $6 x^{5}+6 x$ e) $6 x^{5}+6 x-1$ f) None.

Calculer la dérivée de $f(x)=5^{9 x^{6}+5 x^{5}+12 x}$ . Quelle est $f^{\prime}(x)$ ?

Show that $E=-\nabla V$ leads to LaPlace's equation, $\nabla^{2} V=0$ , when charge density is $0$ .

Find $f^{\prime}(x)$ for $f(x)=\frac{6-x}{x-6}$ . Choose from: a) $\frac{6 x-6}{(x-6)^{2}}$ , b) -1, c) 0, d) $\frac{6 x}{(x-6)^{2}}$ , e) $\frac{6 x+1}{(x-6)^{2}}$ , f) None.

Calculer la dérivée de $f(x)=48 \sqrt{11 x^{8}+7 x^{4}-4 x^{3}+8 x}$ . Trouvez $f^{\prime}(x)$ .

Find the rate of change of infected people at $t=10$ weeks for $N(t)=300 t^{\frac{3}{2}} e^{-(t-10)}$ . Options: a) 142 b) -260 c) -283 d) -2000 e) -3307 f) -8064 g) None.

Evaluate the integral $\int_{0}^{1} \frac{e^{-x}}{x} dx$ . Analyze the behavior near $x=0$ to check for convergence or divergence.

Calculer la dérivée de $f(x)=x e^{3 x+20}$ . Quelle est $f^{\prime}(x)$ ?

A 12-lb weight is attached to a spring and dashpot. Find its position function $x(t)$ after being pulled down 1 ft and released. Also, determine the frequency, amplitude, and phase angle.

Determine the true statement about $f(x)=|x-4|$ at $x=4$ and $x=-4$ from the options provided.

Determine the true statement about a function with corners at $x=-1$ and $x=1$ regarding differentiability and continuity.

Find the value of $f(1)$ for $f(x)=\frac{x^{2}-1}{x-1}$ to make it continuous at $x=1$ .

For the function $f(x)=|x-4|$ , which statement is true about its continuity and differentiability at $x=4$ ?

Find $f^{\prime \prime}(1)$ for $f(x)=x^{4}+x^{2}+1$ .

Find the gradient function $\frac{d y}{d x}$ for: a) $y=(4 x-5)^{2}$ , b) $y=\frac{1}{5-2 x}$ .

Find the derivative $f^{\prime}(c)$ for $f(x)=x-\frac{1}{x}$ at $c=1$ .

Find the arc length $s$ of the curve $\vec{r}(t)=\langle 3 t, 5 \cos (4 t), 5 \sin (4 t)\rangle$ for $-1 \leq t \leq 3$ .

Show that the derivative of $f^3$ is $3 f^2 f'$ .

Find the tangent line to the curve $y=3x^2$ at $x=3$ . Choose the correct equation from the options given.

Find the vertical asymptotes of the function $v(x)=\frac{1}{x^{2}+1}$ . What are the equations?

A 12-lb weight stretches a spring 6 in. and has 3 lb resistance. Find position $x(t)$ and motion properties after 1 ft pull.

Find the derivative $\left.\frac{d z}{d x}\right|_{x=1} ^{n}$ for $z=\frac{1}{x^{3}+1}$ .

Find the derivative of $y=7 \sin^{-1}(x)$ at $x=-\frac{3}{4}$ . Calculate $y^{\prime}\left(-\frac{3}{4}\right)=$

Evaluate the limit: $\lim _{x \rightarrow 3} \frac{x^{2}-8 x+15}{x^{2}-x-6}$ .

Find $g^{\prime}(1)$ where $f(x)=\sin x + 2x + 1$ and $g$ is the inverse of $f$ .

Evaluate the limit as $x$ approaches 0: $\lim_{x \rightarrow 0} \frac{6 \cos(5x) - 5 e^{-2x} - 1}{5x - \sin(3x)}$

Find the derivative of $f(t)=(2t+1)(t^{2}-2)$ using the product or quotient rule, then simplify and calculate the derivative.

Find the derivative $\frac{d y}{d x}$ for $y=(4+x^{3} \ln x)^{7}$ .

Find $\left.\frac{d p}{d h}\right|_{h=4}$ for the function $p=7 e^{h-2}$ .

Find the derivative of $f(s) = s^{1/4} + s^{1/3}$ .

Find the limits of the function as $x \rightarrow \infty$ and $x \rightarrow -\infty$ : $\frac{3-\left(1 / x^{2}\right)}{(-5 / x)+\left(7 / x^{2}\right)}$

Find the derivative of the integral: $\frac{d}{d x}\left(\int_{-3}^{\sqrt{x}} t \sin \left(t^{2}\right) d t\right)$ .

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ for $y=\frac{(5 x^{2}-3)^{9/10}}{(6 x^{2}-6 x+1)^{7/6}}$ using logarithmic differentiation.

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ using logarithmic differentiation for $y=\frac{(5 x^{2}-3)^{9/10}}{(6 x^{2}-6 x+1)^{7/6}}$ .

Find values of $c$ such that $\lim_{x \to 2} f(x)$ exists for the piecewise function: $f(x)=\{3 x^{2}+c x \text{ if } x<2, 1+\frac{3 x}{c} \text{ if } x>2\}$ .

Find $\frac{d y}{d x}$ using logarithmic differentiation for $y=\frac{(5 x^{2}-3)^{9/10}}{(6 x^{2}-6 x+1)^{7/6}}$ .

Find the integral $I=\int \frac{1}{x^{2}+17} \mathrm{~d} x$ .

Find the derivative of $y=6 \cos^{-1}(7x)$ at $x=-\frac{1}{8}$ . Compute $y^{\prime}\left(-\frac{1}{8}\right)=$

Find the derivative of $y=5 \sec^{-1}(x+7)$ . What is $y'$ ?

Find the integral $I=\int \frac{1}{x^{2}+17} \mathrm{~d} x$ .

Find the integral $I=\int(7 x+8) e^{7 x} \mathrm{~d} x$ .