Calculus

Problem 16201

Given the function $g(t)=t \sqrt{7-t}$ for $t<6$ , find $g^{\prime}(t)$ , then determine where $g^{\prime}(c)=0$ or is undefined. List critical numbers.

See Solution

Problem 16202

Given the function $h(x)=\sin ^{2}(x)+\cos (x)$ for $0<x<2 \pi$ , find $h^{\prime}(x)$ and the critical numbers where $h^{\prime}(c)=0$ .

See Solution

Problem 16203

Find $x$ values for local minima of $f(x)$ given its derivative $f^{\prime}(x)$ with points $(0,4)$ , $(0,3.2)$ , and zeros at $x=-2$ , $x=4$ .

See Solution

Problem 16204

Find the area under the normal curve between $z_{1}=-1.56$ and $z_{2}=1.56$ . Round to four decimal places.

See Solution

Problem 16205

Find critical numbers of the function $g(x)=x^{7}-7 x^{5}$ . Enter answers as a comma-separated list.

See Solution

Problem 16206

Find the absolute extrema of $f(x)=x^{3}-\frac{3}{2} x^{2}$ on $[-5,4]$ . Minimum $(x, y)=(\square)$ , Maximum $(x, y)=(\square)$ .

See Solution

Problem 16207

Find the derivative $f \prime(3)$ for the function $f(x)=3 \sqrt[3]{x}+3$ .

See Solution

Problem 16208

Find the derivative $f^{\prime}(2)$ for the function $f(x)=(a x+b)^{3}$ .

See Solution

Problem 16209

Find the derivative $f \prime(-1)$ for the function $f(x)=(x^{2}-x+1)(x^{3}+1)$ .

See Solution

Problem 16210

Find the derivative $f^{\prime}(-1)$ for the function $f(x)=(3x+1)^{3}$ .

See Solution

Problem 16211

Evaluate the integral $\int_{0}^{3 a} (x^{3}-9 a x^{2}+18 a^{2} x) d x$ and $\int_{3 a}^{6 a} (x^{3}-9 a x^{2}+18 a^{2} x) d x$ .

See Solution

Problem 16212

Find the derivative $f'(a)$ for the function $f(x)=\sqrt[n]{x-a+1}$ .

See Solution

Problem 16213

Calculate the area under the standard normal curve from $z_{1}=-1.645$ to $z_{2}=1.645$ . Round to four decimal places.

See Solution

Problem 16214

Given the function $y=x^{2}-8 \ln (x)$ on $[1,6]$ , find the critical number and evaluate $y$ at endpoints.

See Solution

Problem 16215

Find the second-order partial derivatives of $f(x, y)=\ln(5+x^{2}y^{2})$ and show $f_{xy} = f_{yx}$ .

See Solution

Problem 16216

Find the derivative $f \prime(2)$ for the function $f(x)=2(1-x)^{4}$ .

See Solution

Problem 16217

Find the derivative $f^{\prime}(2)$ for the function $f(x)=(x^{2}+x)(x-2)$ .

See Solution

Problem 16218

Dada la función de demanda $p=-0.05 x+600$ , ¿cuál es el ingreso marginal para 5000 dispositivos vendidos?

See Solution

Problem 16219

Find the critical numbers of the function $h(x)=\sin ^{2}(x)+\cos (x)$ for $0<x<2\pi$ where $h^{\prime}(c)=0$ .

See Solution

Problem 16220

Find the derivative $f \prime(3)$ for the function $f(x)=(x-3)(x^{3}-2)$ .

See Solution

Problem 16221

Find the derivative $f^{\prime}(1)$ for the function $f(x)=\sqrt{2x-1}$ .

See Solution

Problem 16222

Find the second partial derivatives of the function $f(x, y) = 5x\sqrt{y} + 5y\sqrt{x}$ .

See Solution

Problem 16223

Find the derivative $f'(1)$ for the function $f(x)=2(3x^{2}+x-3)^{2}$ .

See Solution

Problem 16224

Find the derivative $g'(0)$ for the function $g(x)=(x^{2}+7x-1)^{4}$ .

See Solution

Problem 16225

Find the second-order partial derivatives of $f(x, y)=5 x \sqrt{y}+5 y \sqrt{x}$ and show $f_{xy}=f_{yx}$ .

See Solution

Problem 16226

Find the derivative of $f(x)=\frac{1}{x}+\sqrt{x^{3}}$ using the correct derivative rules.

See Solution

Problem 16227

Find the derivative of $f(x)=\frac{1}{x}+\sqrt{x^{3}}$ using the correct derivative rules.

See Solution

Problem 16228

Find the tangent line equation to $y=\sqrt{5x-4}$ at $x=4$ .

See Solution

Problem 16229

Find the derivative of $f(x)=2 x^{2}-x^{3}$ using first principles, then verify with the power rule.

See Solution

Problem 16230

Find the second derivative of the following functions: a) $y=x^{5}\left(2-\frac{3}{x^{2}}\right)$ b) $y=\sqrt{e^{2-3 x^{3}}}$ c) $y=\left(x+x^{3} \ln (6 x)\right)^{2}$ d) $y=x e^{4 x}$

See Solution

Problem 16231

Find the derivative of $g(x)=(3 x^{2}-5 x)(\frac{1}{x^{2}}-\sqrt{x})$ without simplifying.

See Solution

Problem 16232

Find a function $f(x)$ such that $f \prime(x)=x^{3}+x^{2}-3$ . Explain why there are multiple solutions.

See Solution

Problem 16233

Find the area under the curve $g(x)=2x^{2}-x-1$ on the interval $[3,5]$ using 4 rectangles.

See Solution

Problem 16234

Find the second derivative of these functions:
f) $y=\frac{x-1}{(x+1)^{2}}$
g) $y=e^{(2 x+7)^{2}}$

See Solution

Problem 16235

Bestimmen Sie die Ableitung von $f(x)$ an den gegebenen Stellen: a) $x_{0}=2$ , b) $x_{0}=-3$ , c) $x_{0}=4$ , d) $x_{0}=-2$ , e) $x_{0}=-1$ , f) $x_{0}=9$ .

See Solution

Problem 16236

Find when the acceleration of the particle given by $s(t)=t^{3}-12 t^{2}+45 t+3$ is zero.

See Solution

Problem 16237

An object at $180^{\circ} \mathrm{F}$ cools in water at $50^{\circ} \mathrm{F}$ . Find $F(t)$ with cooling constant $k=-0.4$ .

See Solution

Problem 16238

Find points on the curve $y=(x-1)(x^{2}-4)$ where the tangent is parallel to $y=-4x$ .

See Solution

Problem 16239

Find the population change rate at $x=15$ months for $p(x)=3x^2+5xe^{2x}+5000$ and the change during the 6th month.

See Solution

Problem 16240

Find the derivative of $h(x)=\frac{\left(-2 x+3 x^{2}\right)^{3}}{\sqrt{4 x+1}}$ using the chain and product rules.

See Solution

Problem 16241

Find the derivative $F^{\prime}(x)$ where $F(x)=\int_{2}^{x^{2}} \frac{1}{t^{4}} dt$ .

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Problem 16242

Find the time for $61 \%$ of a nucleus with a half-life of $32 \mathrm{~min}$ to decompose. Options: a) $23 \mathrm{~min}$ b) $21 \mathrm{~min}$ c) $58 \mathrm{~min}$ d) $51 \mathrm{~min}$ e) $43 \mathrm{~min}$ .

See Solution

Problem 16243

Find the second derivative for the following functions: a) $y=x^{5}\left(2-\frac{3}{x^{2}}\right)$ b) $y=\sqrt{e^{2-3 x^{3}}}$ c) $y=\left(x+x^{3} \ln (6 x)\right)^{2}$ d) $y=x e^{4 x}$ e) $y=4 x^{3}-3 x$ f) $y=\frac{x-1}{(x+1)^{2}}$ g) $y=e^{(2 x+7)^{2}}$

See Solution

Problem 16244

Find the derivative $F^{\prime}(x)$ where $F(x)=\int_{2}^{x^{2}} \frac{1}{t^{4}} dt$ .

See Solution

Problem 16245

Evaluate the integral $\int_{2}^{7} 3 \, dx$ using the limit definition. Find width $\Delta x = \frac{5}{n}$ .

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Problem 16246

Find the function $f$ such that $f^{\prime \prime}(x)=x^{-2}$ for $x>0$ , with conditions $f(1)=0$ and $f(6)=0$ .

See Solution

Problem 16247

Solve $\frac{d y}{d x}=2 x-1$ given $y(1) = 1$ .

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Problem 16248

Solve the differential equation $f'(x)=4x$ with the initial condition $f(0)=5$ . Find $f(x)$ .

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Problem 16249

Find the function $y$ such that $\frac{d y}{d x}=2(x-1)$ and $y(1)=-2$ .

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Problem 16250

Solve the differential equation $f'(s) = 14s - 12s^3$ with initial condition $f(3) = 5$ . Find $f(s) = \square$ .

See Solution

Problem 16251

Find the function $f$ given that $f''(x)=10+6x+24x^2$ , with conditions $f(0)=2$ and $f(1)=12$ .

See Solution

Problem 16252

Find the second derivative of these functions:
b) $y=\sqrt{e^{2-3 x^{3}}$ c) $y=\left(x+x^{3} \ln (6 x)\right)^{2}$ d) $y=x e^{4 x}$

See Solution

Problem 16253

Find the particle's position $s(t)$ given $a(t)=t^{2}-7t+4$ , $s(0)=0$ , and $s(1)=20$ .

See Solution

Problem 16254

Find the general antiderivative of $g(t)=\frac{6+t+t^{2}}{\sqrt{t}}$ and verify by differentiation: $G(t)=\square$ .

See Solution

Problem 16255

Bestimmen Sie den Gesamtinhalt A zwischen dem Graphen von $f(x)=x^{4}-5 x^{2}+4$ und der $x$ -Achse im Intervall $I=[-2 ; 1]$ .

See Solution

Problem 16256

Find the elasticity $E(70)$ for the demand equation $1 p + 3 x^{2} - 150 = 0$ .

See Solution

Problem 16257

Find the function $f$ such that $f''(\theta)=\sin(\theta)+\cos(\theta)$ with $f(0)=5$ and $f'(0)=4$ .

See Solution

Problem 16258

Evaluate the integral $\int_{4}^{8}\left(\frac{1}{2} x^{3}-2 x+9\right) d x$ using given values.

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Problem 16259

Find lower and upper estimates of $\int_{0}^{10} f(x) dx$ for a decreasing function $f$ using given values.

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Problem 16260

Find the area under $f(x)=x^{2}+\sqrt{1+2 x}$ from $x=4$ to $x=6$ using right endpoints as a limit: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n}(\square)$ .

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Problem 16261

Calculate the left Riemann sum for $f(x)=10-x^{2}$ on $[0,1]$ with $n=4$ . Sketch the curve and rectangles. Is it an underestimate or overestimate?

See Solution

Problem 16262

Find the population change rate at $x=15$ for $p(x)=3x^2+5xe^{2x}+5000$ and the change during the 16th month.

See Solution

Problem 16263

Set up the integral for the area under $f(x) = 4 - 2x$ from $x = 0$ to its x-intercept. Do not evaluate.

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Problem 16264

Bestimme die ersten und zweiten Ableitungen der Funktionen $f(x)=(x+3) \cdot x^{2}$ , $f(x)=(2-3 x)^{4}$ und $f(x)=x \cdot \sin (x)$ .

See Solution

Problem 16265

Find the initial mass of a radioactive substance that is $3 \mathrm{~kg}$ today, decaying at $1 \%$ per day for 6 days.

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Problem 16266

Express the area under $y=x^{3}$ from 0 to 1 as a limit using right endpoints: $A=\lim_{n \to \infty} \sum_{i=1}^{n} f(x_{i}) \Delta x$ . Evaluate using the sum of cubes: $1^{3}+2^{3}+\cdots+n^{3}=\left[\frac{n(n+1)}{2}\right]^{2}$ .

See Solution

Problem 16267

Evaluate the integral $\int_{6}^{7}\left(\frac{2}{x^{2}}-1\right) d x$ and verify using a graphing utility.

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Problem 16268

Estimate the area under $f(x)=\frac{1}{x}$ from $x=3$ to $x=6$ using 3 rectangles and the midpoint rule. Show your work.

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Problem 16269

Find the mass after 3 days of a 15g sample growing at $9\%$ per day. Round to the nearest tenth.

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Problem 16270

Find the limit using L'Hôpital's Rule: $\lim _{x \rightarrow \infty} \frac{\ln (2 x)}{x^{1 / 2}} =$

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Problem 16271

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \tan ^{-1}\left(\frac{x-x^{2}-2}{x^{2}+3}\right)$ . Answer with a number, $-\infty$ , $+\infty$ , or DNE.

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Problem 16272

Evaluate the limit using L'Hôpital's Rule: $\lim _{x \rightarrow 0} \frac{5^{x}-6^{x}}{x}=$

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Problem 16273

Find the general antiderivative of $f(x)=\frac{4}{1+x^{2}}-\sqrt[5]{x^{4}}$ .

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Problem 16274

Approximate the root of $f(x)=x^{3}-8$ using 5 iterations of Newton's Method, starting with $x_{0}=2.0$ , to 5 decimal places.

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Problem 16275

Find the function $f$ where $f''(x)=\sqrt[3]{x}-\cos x$ , with conditions $f(0)=2$ and $f(1)=2$ .

See Solution

Problem 16276

Calculate the left Riemann sum for $f(x)=10-x^{2}$ on $[0,1]$ with $n=4$ . Sketch the curve and rectangles. Is it an underestimate or overestimate?

See Solution

Problem 16277

Find the long-term number of people who will hear the rumor modeled by $N(t)=\frac{360}{1+39 e^{-0.5 t}}$ .

See Solution

Problem 16278

Find the derivative of the function $r(t)=3^{2 \sqrt{t}}$ .

See Solution

Problem 16279

Find the derivative of the function using logarithmic differentiation: $y=x^{2 \sin (x)}$ .

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Problem 16280

Find the area under $f(x)=x+\ln x$ from $x=3$ to $x=8$ as a limit without evaluating it. Area = $\int_{3}^{8} f(x) dx$ .

See Solution

Problem 16281

Find the half-life of a radioactive material decaying at $26\%$ per minute using the formula $A(t)=A_{0} e^{r t}$ .

See Solution

Problem 16282

Find the limit expression for the area under $f(x)=x+\ln x$ from $x=3$ to $x=8$ . Do not evaluate.

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Problem 16283

Find a region with area equal to $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{5}{n} \sqrt{1+\frac{5 i}{n}}$ .

See Solution

Problem 16284

Montrez que la fonction $f$ avec $f(x)=\frac{\sqrt{x+1}-1}{\tan x}$ et $f(0)=\frac{1}{2}$ est continue en 0.

See Solution

Problem 16285

Wyznacz pochodną funkcji $f(x)=3 \ln x-2 \cos x+x^{3}+4 \sqrt[7]{x^{5}}$ .

See Solution

Problem 16286

How long does it take for a bacteria population to double with a growth rate of $8\%$ per hour? Round to the nearest hundredth.

See Solution

Problem 16287

Calculate the area under the standard normal curve for $z_{1}=-1.55$ and $z_{2}=1.55$ , rounding to four decimal places.

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Problem 16288

Find the hourly growth rate of a bacteria population that grows from 2500 to 3305 in 6 hours using exponential growth.

See Solution

Problem 16289

Find the half-life of a substance with a decay rate of $4.4\%$ per day using the continuous exponential decay model.

See Solution

Problem 16290

How long does it take for a bacteria population to double with a growth rate of $2.1\%$ per hour? Round to the nearest hundredth.

See Solution

Problem 16291

Calculate the integral: $\int \frac{\tan^{-1}(9x)}{1+81x^{2}} \, dx$ .

See Solution

Problem 16292

Find where the function $f(x)=\sqrt[3]{x^{2}+4 x}$ is increasing or decreasing. Use the second derivative test for local extrema.

See Solution

Problem 16293

Find the minimum sunlight hours from the function $D(t)=12+5.2 \cos \left(\frac{2 \pi}{365} t\right)$ .

See Solution

Problem 16294

Find the derivative $f^{\prime}(3)$ for the function $f(x)=2x^{3}+x$ .

See Solution

Problem 16295

Find the derivative of $f(x)=2x^2+x+5+3x^{-1}+x^{-2}$ .

See Solution

Problem 16296

Evaluate the integral: $\int_{0}^{1} \frac{x \, dx}{(x^{2}+1)^{3}}$

See Solution

Problem 16297

Calculate the integral $\int_{0}^{3} \frac{d x}{\sqrt{9-x^{2}}}$ .

See Solution

Problem 16298

Evaluate the integral $\int_{0}^{\pi} \sin ^{6} \frac{x}{2} d x$ .

See Solution

Problem 16299

Find the derivative of $f(x)=(x^{4}-3)(2 x^{2}+x-5)$ .

See Solution

Problem 16300

Evaluate the integral: $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\cos x}{\sin ^{2} x} d x$

See Solution

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Calculus

Problem 16201

Given the function g(t)=t7−tg(t)=t \sqrt{7-t}g(t)=t7−t​ for t<6t<6t<6, find g′(t)g^{\prime}(t)g′(t), then determine where g′(c)=0g^{\prime}(c)=0g′(c)=0 or is undefined. List critical numbers.

Problem 16202

Given the function h(x)=sin⁡2(x)+cos⁡(x)h(x)=\sin ^{2}(x)+\cos (x)h(x)=sin2(x)+cos(x) for 0<x<2π0<x<2 \pi0<x<2π, find h′(x)h^{\prime}(x)h′(x) and the critical numbers where h′(c)=0h^{\prime}(c)=0h′(c)=0.

Problem 16203

Find xxx values for local minima of f(x)f(x)f(x) given its derivative f′(x)f^{\prime}(x)f′(x) with points (0,4)(0,4)(0,4), (0,3.2)(0,3.2)(0,3.2), and zeros at x=−2x=-2x=−2, x=4x=4x=4.

Problem 16204

Find the area under the normal curve between z1=−1.56z_{1}=-1.56z1​=−1.56 and z2=1.56z_{2}=1.56z2​=1.56. Round to four decimal places.

Problem 16205

Find critical numbers of the function g(x)=x7−7x5g(x)=x^{7}-7 x^{5}g(x)=x7−7x5. Enter answers as a comma-separated list.

Problem 16206

Find the absolute extrema of f(x)=x3−32x2f(x)=x^{3}-\frac{3}{2} x^{2}f(x)=x3−23​x2 on [−5,4][-5,4][−5,4]. Minimum (x,y)=(□)(x, y)=(\square)(x,y)=(□), Maximum (x,y)=(□)(x, y)=(\square)(x,y)=(□).

Problem 16207

Find the derivative f′(3)f \prime(3)f′(3) for the function f(x)=3x3+3f(x)=3 \sqrt[3]{x}+3f(x)=33x​+3.

Problem 16208

Find the derivative f′(2)f^{\prime}(2)f′(2) for the function f(x)=(ax+b)3f(x)=(a x+b)^{3}f(x)=(ax+b)3.

Problem 16209

Find the derivative f′(−1)f \prime(-1)f′(−1) for the function f(x)=(x2−x+1)(x3+1)f(x)=(x^{2}-x+1)(x^{3}+1)f(x)=(x2−x+1)(x3+1).

Problem 16210

Find the derivative f′(−1)f^{\prime}(-1)f′(−1) for the function f(x)=(3x+1)3f(x)=(3x+1)^{3}f(x)=(3x+1)3.

Problem 16211

Evaluate the integral ∫03a(x3−9ax2+18a2x)dx\int_{0}^{3 a} (x^{3}-9 a x^{2}+18 a^{2} x) d x∫03a​(x3−9ax2+18a2x)dx and ∫3a6a(x3−9ax2+18a2x)dx\int_{3 a}^{6 a} (x^{3}-9 a x^{2}+18 a^{2} x) d x∫3a6a​(x3−9ax2+18a2x)dx.

Problem 16212

Find the derivative f′(a)f'(a)f′(a) for the function f(x)=x−a+1nf(x)=\sqrt[n]{x-a+1}f(x)=nx−a+1​.

Problem 16213

Calculate the area under the standard normal curve from z1=−1.645z_{1}=-1.645z1​=−1.645 to z2=1.645z_{2}=1.645z2​=1.645. Round to four decimal places.

Problem 16214

Given the function y=x2−8ln⁡(x)y=x^{2}-8 \ln (x)y=x2−8ln(x) on [1,6][1,6][1,6], find the critical number and evaluate yyy at endpoints.

Problem 16215

Find the second-order partial derivatives of f(x,y)=ln⁡(5+x2y2)f(x, y)=\ln(5+x^{2}y^{2})f(x,y)=ln(5+x2y2) and show fxy=fyxf_{xy} = f_{yx}fxy​=fyx​.

Problem 16216

Find the derivative f′(2)f \prime(2)f′(2) for the function f(x)=2(1−x)4f(x)=2(1-x)^{4}f(x)=2(1−x)4.

Problem 16217

Find the derivative f′(2)f^{\prime}(2)f′(2) for the function f(x)=(x2+x)(x−2)f(x)=(x^{2}+x)(x-2)f(x)=(x2+x)(x−2).

Problem 16218

Dada la función de demanda p=−0.05x+600p=-0.05 x+600p=−0.05x+600, ¿cuál es el ingreso marginal para 5000 dispositivos vendidos?

Problem 16219

Find the critical numbers of the function h(x)=sin⁡2(x)+cos⁡(x)h(x)=\sin ^{2}(x)+\cos (x)h(x)=sin2(x)+cos(x) for 0<x<2π0<x<2\pi0<x<2π where h′(c)=0h^{\prime}(c)=0h′(c)=0.

Problem 16220

Find the derivative f′(3)f \prime(3)f′(3) for the function f(x)=(x−3)(x3−2)f(x)=(x-3)(x^{3}-2)f(x)=(x−3)(x3−2).

Problem 16221

Find the derivative f′(1)f^{\prime}(1)f′(1) for the function f(x)=2x−1f(x)=\sqrt{2x-1}f(x)=2x−1​.

Problem 16222

Find the second partial derivatives of the function f(x,y)=5xy+5yxf(x, y) = 5x\sqrt{y} + 5y\sqrt{x}f(x,y)=5xy​+5yx​.

Problem 16223

Find the derivative f′(1)f'(1)f′(1) for the function f(x)=2(3x2+x−3)2f(x)=2(3x^{2}+x-3)^{2}f(x)=2(3x2+x−3)2.

Problem 16224

Find the derivative g′(0)g'(0)g′(0) for the function g(x)=(x2+7x−1)4g(x)=(x^{2}+7x-1)^{4}g(x)=(x2+7x−1)4.

Problem 16225

Find the second-order partial derivatives of f(x,y)=5xy+5yxf(x, y)=5 x \sqrt{y}+5 y \sqrt{x}f(x,y)=5xy​+5yx​ and show fxy=fyxf_{xy}=f_{yx}fxy​=fyx​.

Problem 16226

Find the derivative of f(x)=1x+x3f(x)=\frac{1}{x}+\sqrt{x^{3}}f(x)=x1​+x3​ using the correct derivative rules.

Problem 16227

Find the derivative of f(x)=1x+x3f(x)=\frac{1}{x}+\sqrt{x^{3}}f(x)=x1​+x3​ using the correct derivative rules.

Problem 16228

Find the tangent line equation to y=5x−4y=\sqrt{5x-4}y=5x−4​ at x=4x=4x=4.

Problem 16229

Find the derivative of f(x)=2x2−x3f(x)=2 x^{2}-x^{3}f(x)=2x2−x3 using first principles, then verify with the power rule.

Problem 16230

Find the second derivative of the following functions: a) y=x5(2−3x2)y=x^{5}\left(2-\frac{3}{x^{2}}\right)y=x5(2−x23​) b) y=e2−3x3y=\sqrt{e^{2-3 x^{3}}}y=e2−3x3​ c) y=(x+x3ln⁡(6x))2y=\left(x+x^{3} \ln (6 x)\right)^{2}y=(x+x3ln(6x))2 d) y=xe4xy=x e^{4 x}y=xe4x

Problem 16231

Find the derivative of g(x)=(3x2−5x)(1x2−x)g(x)=(3 x^{2}-5 x)(\frac{1}{x^{2}}-\sqrt{x})g(x)=(3x2−5x)(x21​−x​) without simplifying.

Problem 16232

Find a function f(x)f(x)f(x) such that f′(x)=x3+x2−3f \prime(x)=x^{3}+x^{2}-3f′(x)=x3+x2−3. Explain why there are multiple solutions.

Problem 16233

Find the area under the curve g(x)=2x2−x−1g(x)=2x^{2}-x-1g(x)=2x2−x−1 on the interval [3,5][3,5][3,5] using 4 rectangles.

Problem 16234

Find the second derivative of these functions: f) y=x−1(x+1)2y=\frac{x-1}{(x+1)^{2}}y=(x+1)2x−1​ g) y=e(2x+7)2y=e^{(2 x+7)^{2}}y=e(2x+7)2

Problem 16235

Bestimmen Sie die Ableitung von f(x)f(x)f(x) an den gegebenen Stellen: a) x0=2x_{0}=2x0​=2, b) x0=−3x_{0}=-3x0​=−3, c) x0=4x_{0}=4x0​=4, d) x0=−2x_{0}=-2x0​=−2, e) x0=−1x_{0}=-1x0​=−1, f) x0=9x_{0}=9x0​=9.

Problem 16236

Find when the acceleration of the particle given by s(t)=t3−12t2+45t+3s(t)=t^{3}-12 t^{2}+45 t+3s(t)=t3−12t2+45t+3 is zero.

Problem 16237

An object at 180∘F180^{\circ} \mathrm{F}180∘F cools in water at 50∘F50^{\circ} \mathrm{F}50∘F. Find F(t)F(t)F(t) with cooling constant k=−0.4k=-0.4k=−0.4.

Problem 16238

Find points on the curve y=(x−1)(x2−4)y=(x-1)(x^{2}-4)y=(x−1)(x2−4) where the tangent is parallel to y=−4xy=-4xy=−4x.

Problem 16239

Find the population change rate at x=15x=15x=15 months for p(x)=3x2+5xe2x+5000p(x)=3x^2+5xe^{2x}+5000p(x)=3x2+5xe2x+5000 and the change during the 6th month.

Problem 16240

Given the function $g(t)=t \sqrt{7-t}$ for $t<6$ , find $g^{\prime}(t)$ , then determine where $g^{\prime}(c)=0$ or is undefined. List critical numbers.

Given the function $h(x)=\sin ^{2}(x)+\cos (x)$ for $0<x<2 \pi$ , find $h^{\prime}(x)$ and the critical numbers where $h^{\prime}(c)=0$ .

Find $x$ values for local minima of $f(x)$ given its derivative $f^{\prime}(x)$ with points $(0,4)$ , $(0,3.2)$ , and zeros at $x=-2$ , $x=4$ .

Find the area under the normal curve between $z_{1}=-1.56$ and $z_{2}=1.56$ . Round to four decimal places.

Find critical numbers of the function $g(x)=x^{7}-7 x^{5}$ . Enter answers as a comma-separated list.

Find the absolute extrema of $f(x)=x^{3}-\frac{3}{2} x^{2}$ on $[-5,4]$ . Minimum $(x, y)=(\square)$ , Maximum $(x, y)=(\square)$ .

Find the derivative $f \prime(3)$ for the function $f(x)=3 \sqrt[3]{x}+3$ .

Find the derivative $f^{\prime}(2)$ for the function $f(x)=(a x+b)^{3}$ .

Find the derivative $f \prime(-1)$ for the function $f(x)=(x^{2}-x+1)(x^{3}+1)$ .

Find the derivative $f^{\prime}(-1)$ for the function $f(x)=(3x+1)^{3}$ .

Evaluate the integral $\int_{0}^{3 a} (x^{3}-9 a x^{2}+18 a^{2} x) d x$ and $\int_{3 a}^{6 a} (x^{3}-9 a x^{2}+18 a^{2} x) d x$ .

Find the derivative $f'(a)$ for the function $f(x)=\sqrt[n]{x-a+1}$ .

Calculate the area under the standard normal curve from $z_{1}=-1.645$ to $z_{2}=1.645$ . Round to four decimal places.

Given the function $y=x^{2}-8 \ln (x)$ on $[1,6]$ , find the critical number and evaluate $y$ at endpoints.

Find the second-order partial derivatives of $f(x, y)=\ln(5+x^{2}y^{2})$ and show $f_{xy} = f_{yx}$ .

Find the derivative $f \prime(2)$ for the function $f(x)=2(1-x)^{4}$ .

Find the derivative $f^{\prime}(2)$ for the function $f(x)=(x^{2}+x)(x-2)$ .

Dada la función de demanda $p=-0.05 x+600$ , ¿cuál es el ingreso marginal para 5000 dispositivos vendidos?

Find the critical numbers of the function $h(x)=\sin ^{2}(x)+\cos (x)$ for $0<x<2\pi$ where $h^{\prime}(c)=0$ .

Find the derivative $f \prime(3)$ for the function $f(x)=(x-3)(x^{3}-2)$ .

Find the derivative $f^{\prime}(1)$ for the function $f(x)=\sqrt{2x-1}$ .

Find the second partial derivatives of the function $f(x, y) = 5x\sqrt{y} + 5y\sqrt{x}$ .

Find the derivative $f'(1)$ for the function $f(x)=2(3x^{2}+x-3)^{2}$ .

Find the derivative $g'(0)$ for the function $g(x)=(x^{2}+7x-1)^{4}$ .

Find the second-order partial derivatives of $f(x, y)=5 x \sqrt{y}+5 y \sqrt{x}$ and show $f_{xy}=f_{yx}$ .

Find the derivative of $f(x)=\frac{1}{x}+\sqrt{x^{3}}$ using the correct derivative rules.

Find the derivative of $f(x)=\frac{1}{x}+\sqrt{x^{3}}$ using the correct derivative rules.

Find the tangent line equation to $y=\sqrt{5x-4}$ at $x=4$ .

Find the derivative of $f(x)=2 x^{2}-x^{3}$ using first principles, then verify with the power rule.

Find the second derivative of the following functions: a) $y=x^{5}\left(2-\frac{3}{x^{2}}\right)$ b) $y=\sqrt{e^{2-3 x^{3}}}$ c) $y=\left(x+x^{3} \ln (6 x)\right)^{2}$ d) $y=x e^{4 x}$

Find the derivative of $g(x)=(3 x^{2}-5 x)(\frac{1}{x^{2}}-\sqrt{x})$ without simplifying.

Find a function $f(x)$ such that $f \prime(x)=x^{3}+x^{2}-3$ . Explain why there are multiple solutions.

Find the area under the curve $g(x)=2x^{2}-x-1$ on the interval $[3,5]$ using 4 rectangles.

Find the second derivative of these functions:
f) $y=\frac{x-1}{(x+1)^{2}}$
g) $y=e^{(2 x+7)^{2}}$

Bestimmen Sie die Ableitung von $f(x)$ an den gegebenen Stellen: a) $x_{0}=2$ , b) $x_{0}=-3$ , c) $x_{0}=4$ , d) $x_{0}=-2$ , e) $x_{0}=-1$ , f) $x_{0}=9$ .

Find when the acceleration of the particle given by $s(t)=t^{3}-12 t^{2}+45 t+3$ is zero.

An object at $180^{\circ} \mathrm{F}$ cools in water at $50^{\circ} \mathrm{F}$ . Find $F(t)$ with cooling constant $k=-0.4$ .

Find points on the curve $y=(x-1)(x^{2}-4)$ where the tangent is parallel to $y=-4x$ .

Find the population change rate at $x=15$ months for $p(x)=3x^2+5xe^{2x}+5000$ and the change during the 6th month.

Find the derivative of $h(x)=\frac{\left(-2 x+3 x^{2}\right)^{3}}{\sqrt{4 x+1}}$ using the chain and product rules.

Find the derivative $F^{\prime}(x)$ where $F(x)=\int_{2}^{x^{2}} \frac{1}{t^{4}} dt$ .

Find the time for $61 \%$ of a nucleus with a half-life of $32 \mathrm{~min}$ to decompose. Options: a) $23 \mathrm{~min}$ b) $21 \mathrm{~min}$ c) $58 \mathrm{~min}$ d) $51 \mathrm{~min}$ e) $43 \mathrm{~min}$ .

Find the derivative $F^{\prime}(x)$ where $F(x)=\int_{2}^{x^{2}} \frac{1}{t^{4}} dt$ .

Evaluate the integral $\int_{2}^{7} 3 \, dx$ using the limit definition. Find width $\Delta x = \frac{5}{n}$ .

Find the function $f$ such that $f^{\prime \prime}(x)=x^{-2}$ for $x>0$ , with conditions $f(1)=0$ and $f(6)=0$ .

Solve $\frac{d y}{d x}=2 x-1$ given $y(1) = 1$ .

Solve the differential equation $f'(x)=4x$ with the initial condition $f(0)=5$ . Find $f(x)$ .

Find the function $y$ such that $\frac{d y}{d x}=2(x-1)$ and $y(1)=-2$ .

Solve the differential equation $f'(s) = 14s - 12s^3$ with initial condition $f(3) = 5$ . Find $f(s) = \square$ .

Find the function $f$ given that $f''(x)=10+6x+24x^2$ , with conditions $f(0)=2$ and $f(1)=12$ .

Find the second derivative of these functions:
b) $y=\sqrt{e^{2-3 x^{3}}$ c) $y=\left(x+x^{3} \ln (6 x)\right)^{2}$ d) $y=x e^{4 x}$

Find the particle's position $s(t)$ given $a(t)=t^{2}-7t+4$ , $s(0)=0$ , and $s(1)=20$ .

Find the general antiderivative of $g(t)=\frac{6+t+t^{2}}{\sqrt{t}}$ and verify by differentiation: $G(t)=\square$ .

Bestimmen Sie den Gesamtinhalt A zwischen dem Graphen von $f(x)=x^{4}-5 x^{2}+4$ und der $x$ -Achse im Intervall $I=[-2 ; 1]$ .

Find the elasticity $E(70)$ for the demand equation $1 p + 3 x^{2} - 150 = 0$ .

Find the function $f$ such that $f''(\theta)=\sin(\theta)+\cos(\theta)$ with $f(0)=5$ and $f'(0)=4$ .

Evaluate the integral $\int_{4}^{8}\left(\frac{1}{2} x^{3}-2 x+9\right) d x$ using given values.

Find lower and upper estimates of $\int_{0}^{10} f(x) dx$ for a decreasing function $f$ using given values.

Find the area under $f(x)=x^{2}+\sqrt{1+2 x}$ from $x=4$ to $x=6$ using right endpoints as a limit: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n}(\square)$ .

Calculate the left Riemann sum for $f(x)=10-x^{2}$ on $[0,1]$ with $n=4$ . Sketch the curve and rectangles. Is it an underestimate or overestimate?

Find the population change rate at $x=15$ for $p(x)=3x^2+5xe^{2x}+5000$ and the change during the 16th month.

Set up the integral for the area under $f(x) = 4 - 2x$ from $x = 0$ to its x-intercept. Do not evaluate.

Bestimme die ersten und zweiten Ableitungen der Funktionen $f(x)=(x+3) \cdot x^{2}$ , $f(x)=(2-3 x)^{4}$ und $f(x)=x \cdot \sin (x)$ .

Find the initial mass of a radioactive substance that is $3 \mathrm{~kg}$ today, decaying at $1 \%$ per day for 6 days.

Evaluate the integral $\int_{6}^{7}\left(\frac{2}{x^{2}}-1\right) d x$ and verify using a graphing utility.

Estimate the area under $f(x)=\frac{1}{x}$ from $x=3$ to $x=6$ using 3 rectangles and the midpoint rule. Show your work.

Find the mass after 3 days of a 15g sample growing at $9\%$ per day. Round to the nearest tenth.

Find the limit using L'Hôpital's Rule: $\lim _{x \rightarrow \infty} \frac{\ln (2 x)}{x^{1 / 2}} =$

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \tan ^{-1}\left(\frac{x-x^{2}-2}{x^{2}+3}\right)$ . Answer with a number, $-\infty$ , $+\infty$ , or DNE.

Evaluate the limit using L'Hôpital's Rule: $\lim _{x \rightarrow 0} \frac{5^{x}-6^{x}}{x}=$

Find the general antiderivative of $f(x)=\frac{4}{1+x^{2}}-\sqrt[5]{x^{4}}$ .

Approximate the root of $f(x)=x^{3}-8$ using 5 iterations of Newton's Method, starting with $x_{0}=2.0$ , to 5 decimal places.