Calculus

Problem 31201

Find the limit: $\lim _{x \rightarrow \infty}(7 x-\ln (x))$ . Use l'Hospital's rule if needed.

See Solution

Problem 31202

Find the limit: $\lim _{x \rightarrow 0^{+}} \sin (x) \ln (4 x)$ . Use l'Hospital's Rule if needed.

See Solution

Problem 31203

Find the limit: $\lim _{x \rightarrow \infty}(7 x-\ln (x))$ . Use l'Hospital's rule if needed.

See Solution

Problem 31204

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty}(7 x-\ln (x))$ . Use l'Hospital's rule if needed.

See Solution

Problem 31205

Find the antiderivative $F(x)$ of $f(x)=\frac{9}{x^{2}}-\frac{9}{x^{5}}$ with $F(1)=0$ .

See Solution

Problem 31206

Find the general antiderivative of $f(x)=7 e^{x}+5 \sec ^{2} x$ . Answer: $F(x)=\square$ (use constant $C$ ).

See Solution

Problem 31207

Find the limit: $\lim _{x \rightarrow 0}(1-7 x)^{1 / x}$ . Use I'Hospital's Rule if needed.

See Solution

Problem 31208

Find the normal to the curve $2x - 4y^2 + 3x^2y = 4x^2 + 8$ at point $P(3,2)$ in the form $ax + by + c = 0$ .

See Solution

Problem 31209

An object with mass $m$ dropped from rest has speed $v$ after $t$ seconds given by $v=\frac{m g}{c}\left(1-e^{-c t / m}\right)$ .
(a) Find $\lim _{t \rightarrow \infty} v$ and explain its meaning. (b) Use l'Hospital's Rule to find $\lim _{c \rightarrow 0^{+}} v$ and conclude about falling objects in a vacuum.

See Solution

Problem 31210

Find the average rate of change of $p(t)=50+\frac{2500 t^{2}}{25+t^{2}}$ from $t=2$ to $t=5$ , and the instantaneous rate at $t=3$ .

See Solution

Problem 31211

Find the limit: $\lim _{u \rightarrow \infty} \frac{e^{u / 15}}{u^{3}}$ using l'Hospital's Rule or another method.

See Solution

Problem 31212

Given the curve with parametric equations $x=\sqrt{3} \sin 2t$ , $y=4 \cos^2 t$ , find $k$ in $\frac{\mathrm{d} y}{\mathrm{d} x}=k \sqrt{3} \tan 2t$ . Then, find the tangent line at $t=\frac{\pi}{3}$ in the form $y=ax+b$ .

See Solution

Problem 31213

Find the tangent line equation and $d^{2} y / d x^{2}$ for these curves at specified $t$ values.

See Solution

Problem 31214

Find the limit: $\lim _{u \rightarrow \infty} \frac{e^{u / 15}}{u^{3}}$ .

See Solution

Problem 31215

Solve the equation $-4 y^{\prime}=1+x+y+x y$ and apply the initial condition $y(7)=C$ .

See Solution

Problem 31216

Find the average rate of change (ARoC) of $f(x)=-2 x^{3}$ over these intervals: a. From $x=2$ to $x=2.5$ . b. From $x=2.5$ to $x=3$ . c. From $x=3$ to $x=3.5$ .

See Solution

Problem 31217

Graph the function $f(x)=-0.5 \cdot x^{4}+3 x^{2}$ and find: a. intervals of positive concavity, b. intervals of negative concavity, c. inflection points $(x, y)$ where concavity changes.

See Solution

Problem 31218

Graph $f(x)=(x-2)^{3}+1$ . Find intervals of positive and negative concavity and any inflection points.

See Solution

Problem 31219

Find the volume of the solid formed by rotating the area under $y=\sqrt{x}$ from $x=0$ to $x=3$ around the $x$ axis.

See Solution

Problem 31220

Determine the intervals for graphs A, B, and C where the function is increasing, decreasing, and the rate of change is increasing or decreasing. Use interval notation or write DNE if none exist.

See Solution

Problem 31221

Find the volume of semi-circular cross sections perpendicular to the $x$ -axis for $y=-\frac{1}{2}x+2$ and the axes.

See Solution

Problem 31222

Find the volume of the solid formed by revolving the region bounded by $y=\ln x$ , $x=4$ , and the x-axis around the x-axis. A 8.158 B 9.125 C 10.765 D 11.291 E 15.998

See Solution

Problem 31223

Finde die Ableitung von $f(x)=(x^{2}-1)^{2}$ .

See Solution

Problem 31224

Find the intervals where the function $F(x)=x^{3}-9 x^{2}-81 x+5$ is decreasing.

See Solution

Problem 31225

Bestimmen Sie die zweite Ableitung $f^{\prime\prime}(x)$ aus $f^{\prime}(x)=4x(x^2-1)$ .

See Solution

Problem 31226

A circle in a square has a circumference increasing at 6 in/s. Find how fast the square's perimeter increases.

See Solution

Problem 31227

Find the area of region $R$ under $f(x)=\frac{6}{1+x^{2}}$ and above $y=3$ , volume when rotated about $y=7$ , and volume with semicircle cross sections.

See Solution

Problem 31228

A proton of mass $m=1.67 \times 10^{-27} \text{ kg}$ is in an electric field $E=5000 \text{ N/C}$ . Find the max descent $h_{\text{max}}$ below $0 \text{ m}$ .

See Solution

Problem 31229

A ball is thrown with a velocity of $46 \mathrm{ft} / \mathrm{s}$ . Its height is $y=46t-16t^{2}$ . Find average velocity for $t=2$ over intervals: (i) 0.5s, (ii) 0.1s, (iii) 0.05s, (iv) 0.01s. Estimate instantaneous velocity at $t=2$ .

See Solution

Problem 31230

A circle in a square has a circumference increasing at 6 in/sec. Find the rate of the square's perimeter increase and the area between.
1a. $\frac{24}{\pi}$ in/sec 1b. Area when circle's area is $25\pi$ in².

See Solution

Problem 31231

Differentiate $y = \frac{x^{2} \sqrt{3x-2}}{(x-1)^{2}}$ using logarithmic differentiation.

See Solution

Problem 31232

Calculate the integral from 1 to 2 of the function $3 x^{2}+2 x^{3}$ .

See Solution

Problem 31233

Evaluate the integral $\int_{1}^{2}(3 x^{2}+2 x^{3}) dx$ using limits and compare with FTC part 2 result.

See Solution

Problem 31234

Find the slope of the tangent to $y=x^{2}$ at $\left(\frac{3}{5}, \frac{9}{25}\right)$ and the tangent line equation. Slope: $\square$ .

See Solution

Problem 31235

Find the tangent line equation for $y=x^{2}$ at $x=-0.2$ . (Type an equation.)

See Solution

Problem 31236

Find the point on $y=x^{2}$ where the tangent line is parallel to $x+4y=4$ . Answer as an ordered pair: $\square$ .

See Solution

Problem 31237

Find the slope of the curve $y=x^{3}$ at the point $(-1,-1)$ , where the slope is given by $3x^{2}$ .

See Solution

Problem 31238

Find the derivative of the function $f(x)=(x^{3}+2) \sqrt{x}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 31239

Find $a$ , $f(a)$ , and the slope of $f(x)=x^{2}$ at the point where the tangent line is $y=4x-4$ .

See Solution

Problem 31240

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

See Solution

Problem 31241

A marine manufacturer sells $N(x)$ power boats based on advertising spend $x$ (in \ $k):$ N(x)=960-\frac{3,810}{x}, 5 \leq x \leq 30$.
(A) Find $N'(x)$ .
(B) Calculate $N'(10)$ and $N'(20)$ , then interpret these results.

See Solution

Problem 31242

Find the point on the curve $y=x^{2}$ where the slope is $m=-\frac{1}{5}$ . Answer as an ordered pair.

See Solution

Problem 31243

Evaluate the integral $\int_{-.6180}^{1.6180}\left((x-1)-(x^{2}-2)\right)^{2} dx$ .

See Solution

Problem 31244

Calculate the instantaneous rate of change of $f(x)=3 x^{2}+1$ at $x=-2$ and find the tangent line's equation.

See Solution

Problem 31245

Find the average rate of change of $f(x)=4(2)^{x}$ from $x=0$ to $x=2$ .

See Solution

Problem 31246

Find the sales function $S(t)=0.03 t^{3}+0.8 t^{2}+7 t+6$ .
(A) Compute $S^{\prime}(t)$ . (B) Calculate $S(5)$ and $S^{\prime}(5)$ . (C) Interpret $S(12)=257.04$ and $S^{\prime}(12)=39.16$ .

See Solution

Problem 31247

Find the average rate of change in value per year for the function $V(x)=4500(0.98)^{x}$ between years 5 and 10.

See Solution

Problem 31248

Find how the average rate of change of $f(x)=60(1.5)^{x}$ from Year 3 to Year 6 compares to Year 0 to Year 3.

See Solution

Problem 31249

Find the derivative $S^{\prime}(t)$ , then calculate $S(5)$ and $S^{\prime}(5)$ . Interpret $S(12)=257.04$ and $S^{\prime}(12)=39.16$ .

See Solution

Problem 31250

Invest \ $1 at a continuous interest rate of$ 1\%$. How many years to double the investment? (Round to the nearest hundredth.)

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Problem 31251

Find the derivative $y^{\prime}$ for the function $y=\frac{9}{8 x^{2}}$ .

See Solution

Problem 31252

Find the derivative of the function $f(x)=4x^{2}+7x$ using the limit definition.

See Solution

Problem 31253

Find the derivative of $2 u^{0.1}-5 u^{3.4}$ . What is $\frac{d}{d u}\left(2 u^{0.1}-5 u^{3.4}\right)$ ?

See Solution

Problem 31254

Find the derivative $y^{\prime}$ of the function $y=5 x^{-2}+9 x^{-1}$ .

See Solution

Problem 31255

Find the derivative $G^{\prime}(w)$ of the function $G(w)=\frac{5}{7 w^{5}}+9 w^{1/4}$ .

See Solution

Problem 31256

Compute $\lim _{t \rightarrow+\infty} \frac{1.741t + 29.84}{(1.741 + 1.095 + 1.916)t + (29.84 + 10.65 + 12.36)}$ to two decimal places.

See Solution

Problem 31257

Find the derivative of the function $f(x)=3-2x$ using the limit definition: $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ .

See Solution

Problem 31258

Aloha Company's cost to make $x$ surfboards is $C(x)=-13 x^{2}+500 x+110$ for $0 \leq x \leq 15$ .
(a) Find $C^{\prime}(x)$ .
(b) What is the cost change rate when producing 12 surfboards? \$ per surfboard.

See Solution

Problem 31259

Find the derivative $G^{\prime}(w)$ for the function $G(w)=\frac{7}{3 w^{6}}+9 \sqrt{w}$ .

See Solution

Problem 31260

Find the limit of $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=-5 x^{2}-1$ .

See Solution

Problem 31261

Find the growth rate of the population $P=f(t)=3 t^{2}+2 t+1$ at $t=11$ minutes (bacteria/min).

See Solution

Problem 31262

Find the tangent line equation for $f(x)=\frac{2}{3 x}$ at the point $\left(1, \frac{2}{3}\right)$ .

See Solution

Problem 31263

Find the derivative $h^{\prime}(t)$ for the function $h(t)=\frac{4}{t^{1/2}}-\frac{9}{t^{2/3}}$ .

See Solution

Problem 31264

Find the average rate of change of $f(x)=\frac{x^{2}-16}{x-4}$ on the interval $[13,14]$ .

See Solution

Problem 31265

Find the tangent line equation for $f(x)=7x-4x^2$ at the point (-1, -11). What is $y=$ ?

See Solution

Problem 31266

Find the derivative $f^{\prime}(x)$ for the function $f(x)=(6x-8)^{2}$ .

See Solution

Problem 31267

A store's profit function is $P(x)=-x^{2}+35x-210$ . Find the average profit change when price increases from \$10 to \$11 and \$30 to \$31.

See Solution

Problem 31268

Find $\lim_{x \rightarrow +\infty} S(x)$ for the model $S(x)=573-33 e^{-0.0131 x}$ where $x$ is household income in thousands.

See Solution

Problem 31269

Find the derivative of the function $f(x)=\frac{1}{x-7}$ . What is $f^{\prime}(x)=?$

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Problem 31270

Given $f(x)=x^{2}-3x$ , find the average rate of change from $x=6$ to $7$ , $6$ to $6.5$ , and $6$ to $6.1$ . Also, find the instantaneous rate of change at $x=6$ .

See Solution

Problem 31271

Find the average velocity of a ball with height $s=f(t)=96t-16t^2$ over intervals $[2,3]$ , $[2,2.5]$ , $[2,2.1]$ . Also, find instantaneous velocities at $t=2$ and $t=5$ , and when it hits the ground.

See Solution

Problem 31272

Find the average rate of change of the unit price $p=f(x)=-0.1 x^{2}-x+40$ for 4900-4950 and 4900-4910 tents. What is the rate at 4900?

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Problem 31273

Arthur lance une balle à $26,5 \mathrm{~m/s}$ sur la planète $X$ . Si elle retombe après 24,0 s, quelle est l'accélération gravitationnelle? $\left(2.21 \mathrm{~m/s}^{2}\right)$

See Solution

Problem 31274

Find the derivative $h^{\prime}(t)$ for the function $h(t)=\frac{5}{t^{4/5}}-\frac{5}{t^{2/7}}$ .

See Solution

Problem 31275

Find $\lim _{t \rightarrow+\infty}[p(t)-q(t)]$ where $p(t)=100(1-\frac{12,800}{t^{4.48}})$ and $q(t)=100(1-\frac{5.27 \times 10^{17}}{t^{12}})$ .

See Solution

Problem 31276

Find the velocity function $v=f'(x)$ , velocity at $x=0$ and $x=4$ , and when $v=0$ for $f(x)=195x-13x^2$ .

See Solution

Problem 31277

Find the limits as $t$ approaches infinity for $I(t)=t^{2}+3.5t+48$ and $\frac{I(t)}{E(t)}$ where $E(t)=0.1t^{2}-1.6t+14$ .

See Solution

Problem 31278

Find $\lim_{t \rightarrow +\infty} p(t)$ where $p(t) = 100\left(1-\frac{11,200}{t^{4.54}}\right)$ for $t \geq 8.5$ . What does this mean?

See Solution

Problem 31279

Find the velocity function $v=f'(x)$ for $f(x)=195x-13x^2$ , then calculate $v$ at $x=0$ and $x=4$ , and find when $v=0$ .

See Solution

Problem 31280

Find the velocity function $v=f'(x)$ for $f(x)=66x-11x^2$ , then calculate $v$ at $x=0$ and $x=1$ , and find when $v=0$ .

See Solution

Problem 31281

An object moves along the $y$ -axis with position $f(x)=x^{3}-15 x^{2}+72 x$ . Find $v=f^{\prime}(x)$ , $v(1)$ , and when $v=0$ .

See Solution

Problem 31282

Differentiate the following functions: 11. $f(x)=e^{x}$ , 13. $g(x)=e^{2x}$ .

See Solution

Problem 31283

Find the slope of the tangent line to $y=x^{2}+2$ at $x=3$ .

See Solution

Problem 31284

Find the slope of the tangent line to the curve $y=x^{2}-1$ at $x=1$ .

See Solution

Problem 31285

Find the current $i$ at $t=10$ s for the charge function $q=10-10 e^{-0.1 t}$ . Round to 2 decimal places.

See Solution

Problem 31286

Given $y=2x-\sin(kx)$ and $x=2kt$ , find $\frac{dy}{dx}=\sqrt[A]{ }$ in terms of $k$ and $x$ .

See Solution

Problem 31287

Find the current $i$ at $t=3$ seconds for $q=\frac{7}{4} t+\frac{7}{40} e^{-10 t}$ . Round to 2 decimal places.

See Solution

Problem 31288

Given $y=4 x+\cos (k x)$ and $x=6 k t$ , find $\frac{d y}{d x}$ in terms of $k$ and $x$ .

See Solution

Problem 31289

Sales function is $S(t)=0.03 t^{3}+0.2 t^{2}+4 t+3$ . Find $S^{\prime}(t)$ , $S(7)$ , $S^{\prime}(7)$ , and interpret $S(11)$ and $S^{\prime}(11)$ .

See Solution

Problem 31290

Find the current $i$ in a circuit at $t=2$ s, given $q=\frac{1}{3} \sqrt{t^{3}}-\frac{1}{8} t^{2}$ . Answer to 2 decimal places.

See Solution

Problem 31291

Given $y=4x+\cos(kx)$ and $x=6kt$ , find: (a) $\frac{dy}{dx}$ ; (b) $\frac{dx}{dt}$ ; (c) For $k=2$ , at $t=\pi$ , find $\frac{dy}{dt}$ .

See Solution

Problem 31292

Find the current $i$ at $t=2 \mathrm{s}$ given $q=\frac{1}{3} \sqrt{t^{3}}-\frac{1}{8} t^{2}$ , where $i=\frac{d q}{d t}$ .

See Solution

Problem 31293

Given $y=2 x-\sin (k x)$ and $x=2 k t$ , find: (a) $\frac{d y}{d x}=$ (in terms of $k$ and $x$ ) (b) $\frac{d x}{d t}=$ (in terms of $k$ and/or $t$ ) (c) For $k=1$ , at $t=\frac{\pi}{2}$ , find $\frac{d y}{d t}=$ (1 decimal place or integer)

See Solution

Problem 31294

Given $y=3x-\sec(kx)$ and $x=4kt$ , find:
(a) $\frac{dy}{dx}=3-k\sec(kx)\tan(kx)$
(b) $\frac{dx}{dt}=4k$
(c) For $k=2$ , at $t=\pi$ , find $\frac{dy}{dt}$ (round to 1 decimal place).

See Solution

Problem 31295

Find the average velocity of an object with position $s(t)=-16 t^{2}+100 t$ between $t=0.5$ and $t=2$ . Sketch the secant line.

See Solution

Problem 31296

Find the velocity of a saber saw blade given $y=2 \sin (100 t)$ at $t=0.03$ seconds. Use $v=\frac{d y}{d t}$ .

See Solution

Problem 31297

Find the derivative $f^{\prime}(x)$ for $f(x)=x^{2}+5x-6$ and calculate $f^{\prime}(1), f^{\prime}(2), f^{\prime}(3)$ .

See Solution

Problem 31298

Find the derivative of the relation $x^{2}+4 y^{2}+20 y=15 x+4 x y$ using implicit differentiation.

See Solution

Problem 31299

Find $f^{\prime}(x)$ for $f(x)=7 x^{3}$ , then calculate $f^{\prime}(1)$ , $f^{\prime}(2)$ , and $f^{\prime}(3)$ .

See Solution

Problem 31300

Find the blade velocity at $t=0.03$ seconds for $y=2 \sin (100 t)$ . Options: 198, -198, -1.98, 1.98 cm/s.

See Solution

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Calculus

Problem 31201

Find the limit: lim⁡x→∞(7x−ln⁡(x))\lim _{x \rightarrow \infty}(7 x-\ln (x))x→∞lim​(7x−ln(x)). Use l'Hospital's rule if needed.

Problem 31202

Find the limit: lim⁡x→0+sin⁡(x)ln⁡(4x)\lim _{x \rightarrow 0^{+}} \sin (x) \ln (4 x)x→0+lim​sin(x)ln(4x). Use l'Hospital's Rule if needed.

Problem 31203

Find the limit: lim⁡x→∞(7x−ln⁡(x))\lim _{x \rightarrow \infty}(7 x-\ln (x))x→∞lim​(7x−ln(x)). Use l'Hospital's rule if needed.

Problem 31204

Find the limit as xxx approaches infinity: lim⁡x→∞(7x−ln⁡(x))\lim _{x \rightarrow \infty}(7 x-\ln (x))limx→∞​(7x−ln(x)). Use l'Hospital's rule if needed.

Problem 31205

Find the antiderivative F(x)F(x)F(x) of f(x)=9x2−9x5f(x)=\frac{9}{x^{2}}-\frac{9}{x^{5}}f(x)=x29​−x59​ with F(1)=0F(1)=0F(1)=0.

Problem 31206

Find the general antiderivative of f(x)=7ex+5sec⁡2xf(x)=7 e^{x}+5 \sec ^{2} xf(x)=7ex+5sec2x. Answer: F(x)=□F(x)=\squareF(x)=□ (use constant CCC).

Problem 31207

Find the limit: lim⁡x→0(1−7x)1/x\lim _{x \rightarrow 0}(1-7 x)^{1 / x}limx→0​(1−7x)1/x. Use I'Hospital's Rule if needed.

Problem 31208

Find the normal to the curve 2x−4y2+3x2y=4x2+82x - 4y^2 + 3x^2y = 4x^2 + 82x−4y2+3x2y=4x2+8 at point P(3,2)P(3,2)P(3,2) in the form ax+by+c=0ax + by + c = 0ax+by+c=0.

Problem 31209

Problem 31210

Find the average rate of change of p(t)=50+2500t225+t2p(t)=50+\frac{2500 t^{2}}{25+t^{2}}p(t)=50+25+t22500t2​ from t=2t=2t=2 to t=5t=5t=5, and the instantaneous rate at t=3t=3t=3.

Problem 31211

Find the limit: lim⁡u→∞eu/15u3\lim _{u \rightarrow \infty} \frac{e^{u / 15}}{u^{3}}u→∞lim​u3eu/15​ using l'Hospital's Rule or another method.

Problem 31212

Problem 31213

Find the tangent line equation and d2y/dx2d^{2} y / d x^{2}d2y/dx2 for these curves at specified ttt values.

Problem 31214

Find the limit: lim⁡u→∞eu/15u3\lim _{u \rightarrow \infty} \frac{e^{u / 15}}{u^{3}}limu→∞​u3eu/15​.

Problem 31215

Solve the equation −4y′=1+x+y+xy-4 y^{\prime}=1+x+y+x y−4y′=1+x+y+xy and apply the initial condition y(7)=Cy(7)=Cy(7)=C.

Problem 31216

Find the average rate of change (ARoC) of f(x)=−2x3f(x)=-2 x^{3}f(x)=−2x3 over these intervals: a. From x=2x=2x=2 to x=2.5x=2.5x=2.5. b. From x=2.5x=2.5x=2.5 to x=3x=3x=3. c. From x=3x=3x=3 to x=3.5x=3.5x=3.5.

Problem 31217

Graph the function f(x)=−0.5⋅x4+3x2f(x)=-0.5 \cdot x^{4}+3 x^{2}f(x)=−0.5⋅x4+3x2 and find: a. intervals of positive concavity, b. intervals of negative concavity, c. inflection points (x,y)(x, y)(x,y) where concavity changes.

Problem 31218

Graph f(x)=(x−2)3+1f(x)=(x-2)^{3}+1f(x)=(x−2)3+1. Find intervals of positive and negative concavity and any inflection points.

Problem 31219

Find the volume of the solid formed by rotating the area under y=xy=\sqrt{x}y=x​ from x=0x=0x=0 to x=3x=3x=3 around the xxx axis.

Problem 31220

Determine the intervals for graphs A, B, and C where the function is increasing, decreasing, and the rate of change is increasing or decreasing. Use interval notation or write DNE if none exist.

Problem 31221

Find the volume of semi-circular cross sections perpendicular to the xxx-axis for y=−12x+2y=-\frac{1}{2}x+2y=−21​x+2 and the axes.

Problem 31222

Find the volume of the solid formed by revolving the region bounded by y=ln⁡xy=\ln xy=lnx, x=4x=4x=4, and the x-axis around the x-axis. A 8.158 B 9.125 C 10.765 D 11.291 E 15.998

Problem 31223

Finde die Ableitung von f(x)=(x2−1)2f(x)=(x^{2}-1)^{2}f(x)=(x2−1)2.

Problem 31224

Find the intervals where the function F(x)=x3−9x2−81x+5F(x)=x^{3}-9 x^{2}-81 x+5F(x)=x3−9x2−81x+5 is decreasing.

Problem 31225

Bestimmen Sie die zweite Ableitung f′′(x)f^{\prime\prime}(x)f′′(x) aus f′(x)=4x(x2−1)f^{\prime}(x)=4x(x^2-1)f′(x)=4x(x2−1).

Problem 31226

A circle in a square has a circumference increasing at 6 in/s. Find how fast the square's perimeter increases.

Problem 31227

Find the area of region RRR under f(x)=61+x2f(x)=\frac{6}{1+x^{2}}f(x)=1+x26​ and above y=3y=3y=3, volume when rotated about y=7y=7y=7, and volume with semicircle cross sections.

Problem 31228

A proton of mass m=1.67×10−27 kgm=1.67 \times 10^{-27} \text{ kg}m=1.67×10−27 kg is in an electric field E=5000 N/CE=5000 \text{ N/C}E=5000 N/C. Find the max descent hmaxh_{\text{max}}hmax​ below 0 m0 \text{ m}0 m.

Problem 31229

A ball is thrown with a velocity of 46ft/s46 \mathrm{ft} / \mathrm{s}46ft/s. Its height is y=46t−16t2y=46t-16t^{2}y=46t−16t2. Find average velocity for t=2t=2t=2 over intervals: (i) 0.5s, (ii) 0.1s, (iii) 0.05s, (iv) 0.01s. Estimate instantaneous velocity at t=2t=2t=2.

Problem 31230

A circle in a square has a circumference increasing at 6 in/sec. Find the rate of the square's perimeter increase and the area between. 1a. 24π\frac{24}{\pi}π24​ in/sec 1b. Area when circle's area is 25π25\pi25π in².

Problem 31231

Differentiate y=x23x−2(x−1)2y = \frac{x^{2} \sqrt{3x-2}}{(x-1)^{2}}y=(x−1)2x23x−2​​ using logarithmic differentiation.

Problem 31232

Calculate the integral from 1 to 2 of the function 3x2+2x33 x^{2}+2 x^{3}3x2+2x3.

Problem 31233

Evaluate the integral ∫12(3x2+2x3)dx\int_{1}^{2}(3 x^{2}+2 x^{3}) dx∫12​(3x2+2x3)dx using limits and compare with FTC part 2 result.

Problem 31234

Find the slope of the tangent to y=x2y=x^{2}y=x2 at (35,925)\left(\frac{3}{5}, \frac{9}{25}\right)(53​,259​) and the tangent line equation. Slope: □\square□.

Problem 31235

Find the tangent line equation for y=x2y=x^{2}y=x2 at x=−0.2x=-0.2x=−0.2. (Type an equation.)

Problem 31236

Find the point on y=x2y=x^{2}y=x2 where the tangent line is parallel to x+4y=4x+4y=4x+4y=4. Answer as an ordered pair: □\square□.

Problem 31237

Find the slope of the curve y=x3y=x^{3}y=x3 at the point (−1,−1)(-1,-1)(−1,−1), where the slope is given by 3x23x^{2}3x2.

Problem 31238

Find the derivative of the function f(x)=(x3+2)xf(x)=(x^{3}+2) \sqrt{x}f(x)=(x3+2)x​. What is f′(x)f^{\prime}(x)f′(x)?

Problem 31239

Find aaa, f(a)f(a)f(a), and the slope of f(x)=x2f(x)=x^{2}f(x)=x2 at the point where the tangent line is y=4x−4y=4x-4y=4x−4.

Problem 31240

Find the derivative of the function f(x)=(x3+2x)xf(x)=(x^{3}+2x) \sqrt{x}f(x)=(x3+2x)x​. What is f′(x)f'(x)f′(x)?

Problem 31241

Find the limit: $\lim _{x \rightarrow \infty}(7 x-\ln (x))$ . Use l'Hospital's rule if needed.

Find the limit: $\lim _{x \rightarrow 0^{+}} \sin (x) \ln (4 x)$ . Use l'Hospital's Rule if needed.

Find the limit: $\lim _{x \rightarrow \infty}(7 x-\ln (x))$ . Use l'Hospital's rule if needed.

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty}(7 x-\ln (x))$ . Use l'Hospital's rule if needed.

Find the antiderivative $F(x)$ of $f(x)=\frac{9}{x^{2}}-\frac{9}{x^{5}}$ with $F(1)=0$ .

Find the general antiderivative of $f(x)=7 e^{x}+5 \sec ^{2} x$ . Answer: $F(x)=\square$ (use constant $C$ ).

Find the limit: $\lim _{x \rightarrow 0}(1-7 x)^{1 / x}$ . Use I'Hospital's Rule if needed.

Find the normal to the curve $2x - 4y^2 + 3x^2y = 4x^2 + 8$ at point $P(3,2)$ in the form $ax + by + c = 0$ .

Find the average rate of change of $p(t)=50+\frac{2500 t^{2}}{25+t^{2}}$ from $t=2$ to $t=5$ , and the instantaneous rate at $t=3$ .

Find the limit: $\lim _{u \rightarrow \infty} \frac{e^{u / 15}}{u^{3}}$ using l'Hospital's Rule or another method.

Find the tangent line equation and $d^{2} y / d x^{2}$ for these curves at specified $t$ values.

Find the limit: $\lim _{u \rightarrow \infty} \frac{e^{u / 15}}{u^{3}}$ .

Solve the equation $-4 y^{\prime}=1+x+y+x y$ and apply the initial condition $y(7)=C$ .

Find the average rate of change (ARoC) of $f(x)=-2 x^{3}$ over these intervals: a. From $x=2$ to $x=2.5$ . b. From $x=2.5$ to $x=3$ . c. From $x=3$ to $x=3.5$ .

Graph the function $f(x)=-0.5 \cdot x^{4}+3 x^{2}$ and find: a. intervals of positive concavity, b. intervals of negative concavity, c. inflection points $(x, y)$ where concavity changes.

Graph $f(x)=(x-2)^{3}+1$ . Find intervals of positive and negative concavity and any inflection points.

Find the volume of the solid formed by rotating the area under $y=\sqrt{x}$ from $x=0$ to $x=3$ around the $x$ axis.

Find the volume of semi-circular cross sections perpendicular to the $x$ -axis for $y=-\frac{1}{2}x+2$ and the axes.

Find the volume of the solid formed by revolving the region bounded by $y=\ln x$ , $x=4$ , and the x-axis around the x-axis. A 8.158 B 9.125 C 10.765 D 11.291 E 15.998

Finde die Ableitung von $f(x)=(x^{2}-1)^{2}$ .

Find the intervals where the function $F(x)=x^{3}-9 x^{2}-81 x+5$ is decreasing.

Bestimmen Sie die zweite Ableitung $f^{\prime\prime}(x)$ aus $f^{\prime}(x)=4x(x^2-1)$ .

Find the area of region $R$ under $f(x)=\frac{6}{1+x^{2}}$ and above $y=3$ , volume when rotated about $y=7$ , and volume with semicircle cross sections.

A proton of mass $m=1.67 \times 10^{-27} \text{ kg}$ is in an electric field $E=5000 \text{ N/C}$ . Find the max descent $h_{\text{max}}$ below $0 \text{ m}$ .

A ball is thrown with a velocity of $46 \mathrm{ft} / \mathrm{s}$ . Its height is $y=46t-16t^{2}$ . Find average velocity for $t=2$ over intervals: (i) 0.5s, (ii) 0.1s, (iii) 0.05s, (iv) 0.01s. Estimate instantaneous velocity at $t=2$ .

A circle in a square has a circumference increasing at 6 in/sec. Find the rate of the square's perimeter increase and the area between.
1a. $\frac{24}{\pi}$ in/sec 1b. Area when circle's area is $25\pi$ in².

Differentiate $y = \frac{x^{2} \sqrt{3x-2}}{(x-1)^{2}}$ using logarithmic differentiation.

Calculate the integral from 1 to 2 of the function $3 x^{2}+2 x^{3}$ .

Evaluate the integral $\int_{1}^{2}(3 x^{2}+2 x^{3}) dx$ using limits and compare with FTC part 2 result.

Find the slope of the tangent to $y=x^{2}$ at $\left(\frac{3}{5}, \frac{9}{25}\right)$ and the tangent line equation. Slope: $\square$ .

Find the tangent line equation for $y=x^{2}$ at $x=-0.2$ . (Type an equation.)

Find the point on $y=x^{2}$ where the tangent line is parallel to $x+4y=4$ . Answer as an ordered pair: $\square$ .

Find the slope of the curve $y=x^{3}$ at the point $(-1,-1)$ , where the slope is given by $3x^{2}$ .

Find the derivative of the function $f(x)=(x^{3}+2) \sqrt{x}$ . What is $f^{\prime}(x)$ ?

Find $a$ , $f(a)$ , and the slope of $f(x)=x^{2}$ at the point where the tangent line is $y=4x-4$ .

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

A marine manufacturer sells $N(x)$ power boats based on advertising spend $x$ (in \ $k):$ N(x)=960-\frac{3,810}{x}, 5 \leq x \leq 30$.
(A) Find $N'(x)$ .
(B) Calculate $N'(10)$ and $N'(20)$ , then interpret these results.

Find the point on the curve $y=x^{2}$ where the slope is $m=-\frac{1}{5}$ . Answer as an ordered pair.

Evaluate the integral $\int_{-.6180}^{1.6180}\left((x-1)-(x^{2}-2)\right)^{2} dx$ .

Calculate the instantaneous rate of change of $f(x)=3 x^{2}+1$ at $x=-2$ and find the tangent line's equation.

Find the average rate of change of $f(x)=4(2)^{x}$ from $x=0$ to $x=2$ .

Find the average rate of change in value per year for the function $V(x)=4500(0.98)^{x}$ between years 5 and 10.

Find how the average rate of change of $f(x)=60(1.5)^{x}$ from Year 3 to Year 6 compares to Year 0 to Year 3.

Find the derivative $S^{\prime}(t)$ , then calculate $S(5)$ and $S^{\prime}(5)$ . Interpret $S(12)=257.04$ and $S^{\prime}(12)=39.16$ .

Invest \ $1 at a continuous interest rate of$ 1\%$. How many years to double the investment? (Round to the nearest hundredth.)

Find the derivative $y^{\prime}$ for the function $y=\frac{9}{8 x^{2}}$ .

Find the derivative of the function $f(x)=4x^{2}+7x$ using the limit definition.

Find the derivative of $2 u^{0.1}-5 u^{3.4}$ . What is $\frac{d}{d u}\left(2 u^{0.1}-5 u^{3.4}\right)$ ?

Find the derivative $y^{\prime}$ of the function $y=5 x^{-2}+9 x^{-1}$ .

Find the derivative $G^{\prime}(w)$ of the function $G(w)=\frac{5}{7 w^{5}}+9 w^{1/4}$ .

Compute $\lim _{t \rightarrow+\infty} \frac{1.741t + 29.84}{(1.741 + 1.095 + 1.916)t + (29.84 + 10.65 + 12.36)}$ to two decimal places.

Find the derivative of the function $f(x)=3-2x$ using the limit definition: $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ .

Aloha Company's cost to make $x$ surfboards is $C(x)=-13 x^{2}+500 x+110$ for $0 \leq x \leq 15$ .
(a) Find $C^{\prime}(x)$ .
(b) What is the cost change rate when producing 12 surfboards? \$ per surfboard.

Find the derivative $G^{\prime}(w)$ for the function $G(w)=\frac{7}{3 w^{6}}+9 \sqrt{w}$ .

Find the limit of $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=-5 x^{2}-1$ .

Find the growth rate of the population $P=f(t)=3 t^{2}+2 t+1$ at $t=11$ minutes (bacteria/min).

Find the tangent line equation for $f(x)=\frac{2}{3 x}$ at the point $\left(1, \frac{2}{3}\right)$ .

Find the derivative $h^{\prime}(t)$ for the function $h(t)=\frac{4}{t^{1/2}}-\frac{9}{t^{2/3}}$ .

Find the average rate of change of $f(x)=\frac{x^{2}-16}{x-4}$ on the interval $[13,14]$ .

Find the tangent line equation for $f(x)=7x-4x^2$ at the point (-1, -11). What is $y=$ ?

Find the derivative $f^{\prime}(x)$ for the function $f(x)=(6x-8)^{2}$ .

A store's profit function is $P(x)=-x^{2}+35x-210$ . Find the average profit change when price increases from \$10 to \$11 and \$30 to \$31.

Find $\lim_{x \rightarrow +\infty} S(x)$ for the model $S(x)=573-33 e^{-0.0131 x}$ where $x$ is household income in thousands.

Find the derivative of the function $f(x)=\frac{1}{x-7}$ . What is $f^{\prime}(x)=?$

Given $f(x)=x^{2}-3x$ , find the average rate of change from $x=6$ to $7$ , $6$ to $6.5$ , and $6$ to $6.1$ . Also, find the instantaneous rate of change at $x=6$ .

Find the average velocity of a ball with height $s=f(t)=96t-16t^2$ over intervals $[2,3]$ , $[2,2.5]$ , $[2,2.1]$ . Also, find instantaneous velocities at $t=2$ and $t=5$ , and when it hits the ground.

Find the average rate of change of the unit price $p=f(x)=-0.1 x^{2}-x+40$ for 4900-4950 and 4900-4910 tents. What is the rate at 4900?

Arthur lance une balle à $26,5 \mathrm{~m/s}$ sur la planète $X$ . Si elle retombe après 24,0 s, quelle est l'accélération gravitationnelle? $\left(2.21 \mathrm{~m/s}^{2}\right)$

Find the derivative $h^{\prime}(t)$ for the function $h(t)=\frac{5}{t^{4/5}}-\frac{5}{t^{2/7}}$ .