Calculus

Problem 2201

Find the slope of the tangent line for $y=x^{4}-2 x^{3}+8$ at $x=2$ . What is the equation of the line?

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Problem 2202

Find the slope and equation of the tangent line for $y=x^{4}-13 x^{2}+36$ at $x=2$ . Simplify your answer.

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Problem 2203

Find the slope of the tangent line for $y=11 x^{1/2}+x^{3/2}$ at $x=4$ . Simplify your answer.

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Problem 2204

Find values of $x$ where the tangent line of $f(x)=2 x^{3}+39 x^{2}+240 x+6$ is horizontal. $x=$

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Problem 2205

Find $x$ where the tangent line to $f(x) = 2x^3 + 39x^2 + 240x + 6$ is horizontal.

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Problem 2206

1. (a) Calculate the 1st order partial derivatives of $f(x, y, z)=x^{3} \sqrt{y}+4 z^{3} y^{2}-x y z+x^{2}-\sqrt[3]{z^{5}}$ . (b) Water drips from an inverted cone at $0.8 \mathrm{~cm}^{3} \mathrm{~s}^{-1}$ . (i) Show $V=\frac{4}{27} \pi h^{3}$ for $r=12 \mathrm{~cm}$ , $h=18 \mathrm{~cm}$ . (ii) Find the height decrease rate when $h=6 \mathrm{~cm}$ .

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Problem 2207

A parachutist drops a camera from 50 m while descending at 10 m/s. Find (a) time to hit ground and (b) impact velocity.

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Problem 2208

A rocket accelerates at $+29.4 \mathrm{~m} / \mathrm{s}^{2}$ for $4.0 \mathrm{~s}$ . Find (a) max height, (b) time to max height, (c) position and velocity graphs.

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Problem 2209

A ball is thrown up at $19.6 \mathrm{~m/s}$ . Find its velocity and height at 1, 2, 3, and 4 seconds. Also, graph position and velocity vs. time.

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Problem 2210

Find the derivative $D^{\prime}(3)$ for the demand function $D(p)=-3 p^{2}+3 p+7$ .

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Problem 2211

Find the marginal profit for the function $P(x)=0.08 x^{2}-4 x+4 x^{0.6}-5700$ at $x=500, 1000, 5000, 10000$ .

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Problem 2212

Find the derivative of the function: $y=x^{3}-7 x^{2}+15 x+9$ .

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Problem 2213

Find $f^{\prime}(3)$ for the function $f(x)=x^{4}+6x^{3}-2x-2$ .

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Problem 2214

Calculate the average rate of change of $y=x^{3}+x^{2}-8 x-7$ from $x=0$ to $x=2$ .

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Problem 2215

Calculate the integral $\int_{-5}^{10} \frac{\ln x}{25 x^{2}} d x$ .

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Problem 2216

Find the derivative of the function $f(x)=\frac{5+2 x^{2}}{1-2 x^{2}}$ , denoted as $f^{\prime}(x)$ .

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Problem 2217

Evaluate the integral $\int_{-1}^{1} \frac{\cos x}{1+e^{1 / x}} d x$ .

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Problem 2218

Evaluate the integral $\int^{1} \frac{\cos x}{1+e^{1 / x}} d x$ .

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Problem 2219

Find the derivative $y^{\prime}$ for the function $y=\frac{6}{5 x^{4}}$ .

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Problem 2220

Differentiate the function $\frac{2 x^{3}}{7}-\frac{4}{9 x^{3}}$ with respect to $x$ .

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Problem 2221

Calculate the average rate of change of $f(x)=\sqrt{3x-2}$ from $x=2$ to $x=6$ .

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Problem 2222

Find the average rate of change of employees (in thousands) with respect to revenue (in billions) for these intervals:
a. From \$11B to \$16B: 147K to 191K.
b. From \$16B to \$27B: 191K to 346K.
c. From \$11B to \$27B: 147K to 346K.

See Solution

Problem 2223

Sales function is $S(t)=0.02 t^{3}+0.8 t^{2}+9 t+2$ . Find $S^{\prime}(t)$ , $S(2)$ , $S^{\prime}(2)$ , and interpret $S(13)$ and $S^{\prime}(13)$ .

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Problem 2224

Find the marginal cost from the total cost function $C(x)=100+12x+0.1x^{2}+0.001x^{3}$ for $x=0$ , $x=10$ , and $x=30$ .

See Solution

Problem 2225

Find the marginal cost for producing 1500 and 2000 shirts given the cost function $C(x)=1500+1000 x^{9}$ , where $x$ is in hundreds.

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Problem 2226

Given the sales function $S(t)=0.02 t^{3}+0.7 t^{2}+7 t+9$ , find: (A) $S^{\prime}(t)$ , (B) $S(4)$ and $S^{\prime}(4)$ , (C) Interpret $S(8)=120.04$ and $S^{\prime}(8)=22.04$ .

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Problem 2227

Find the marginal profit from selling $x$ smartwatches given $P(x)=0.03x^{2}-4x+3x^{0.8}-5000$ for $x=100$ .

See Solution

Problem 2228

Find the marginal revenue function for $R(x)=0.016x^{2}+1.034x+11.338$ and calculate it at $x=10$ .

See Solution

Problem 2229

Find the speed $v$ that minimizes energy $E(v)=2.73 v^{3} \frac{14}{v-7}$ for a fish swimming against a $7 \mathrm{mi/h}$ current.

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Problem 2230

Find the difference quotient $\frac{f(a+h)-f(a)}{h}$ for: (a) $f(x)=\sqrt{x^{2}-x}$ , (b) $g(x)=2x^{2}-4x+5$ . Also, given the line $y=-\frac{6}{7}x+\frac{3}{7}$ , find the parallel and perpendicular lines through $(3,8)$ . Rate class difficulty from 1 to 10.

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Problem 2231

Order the values $I$ , $R_{n}$ , and $L_{n}$ for a decreasing function $f$ on $[a, b]$ and for an increasing function. Explain.

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Problem 2232

Complete the table and find the limit: $\lim _{x \rightarrow 1} f(x)$ for $f(x)=\frac{x^{3}-1}{x^{2}-1}$ .

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Problem 2233

Find the tangent slope at (5,0) for $f(x)=3(5-x)^{2}$ and at (0,0) for $f(\theta)=4 \sin \theta - \theta$ .

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Problem 2234

Find the absolute minimum and maximum of the function $f(x)=2.5 x^{4}+3 x^{3}-2.6 x^{2}-5.1 x-5.6$ on $[-0.5,1.2]$ .

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Problem 2235

Determine the slope of $f(t)=\frac{3}{4} e^{t}$ at the point $(0, \frac{3}{4})$ .

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Problem 2236

Find the derivative of $g(t)=t^{2}-\frac{4}{t^{3}}$ .

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Problem 2237

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

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Problem 2238

Find the derivative of these functions and note restrictions for the second:
1. $f(x)=\frac{x}{x}$
2. $f(x)=\frac{x^{3}-3 x^{2}+4}{x^{2}}$

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Problem 2239

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

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Problem 2240

Find the derivative of the function $y=x(x^{2}+1)$ .

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Problem 2241

Find the derivative of $f(x)=\sqrt{x}-6\sqrt[3]{x}$ .

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Problem 2242

Sketch a function $g$ such that $g(0)=1$ , $g'(0)=-2$ , $g'(1)=0$ , and $g'(2)=4$ .

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Problem 2243

Find the derivative of $h(s)=s^{4/5}-s^{2/3}$ .

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Problem 2244

Given the limits, determine which statements about function $f$ are true: A. removable discontinuity at $x=1$ ; B. differentiable at $x=1$ ; C. $f(1)=2$ , $f'(1)=3$ .

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Problem 2245

Find the derivative of $f(x) = 6 \sqrt{x} + 5 \cos x$ .

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Problem 2246

Find the derivative of $f(x)=x^{-2}-2 e^{x}$ .

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Problem 2247

Find the tangent line for $f(x)=x^{4}-x$ at the given points and confirm using a graphing utility. Points: $(-1,2)$ , $(1,2)$ , $(0,1)$ , $(\pi, \frac{1}{2} e^{\pi})$ .

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Problem 2248

Find points where the following functions have a horizontal tangent line:
61. $y=x^{4}-2 x^{2}+3$
62. $y=x^{3}+x$
63. $y=\frac{1}{x^{2}}$
64. $y=x^{2}+9$
65. $y=x+\sin x, \quad 0 \leq x<2 \pi$
66. $y=\sqrt{3} x+2 \cos x, \quad 0 \leq x<2 \pi$
67. $y=-4 x+e^{x}$
68. $y=x+4 e^{x}$

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Problem 2249

Find the horizontal tangent line for the function $y=\frac{1}{x^{2}}$ .

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Problem 2250

Find the horizontal tangent line for $y=x+\sin x$ in the interval $0 \leq x < 2\pi$ .

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Problem 2251

Find values of $a$ and $b$ so that the piecewise function $f(x)=\begin{cases}a x^{3}, & x \leq 2 \\ x^{2}+b, & x>2\end{cases}$ is differentiable everywhere.

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Problem 2252

Find the derivative or integral of the function $f(x)=\frac{3}{(x+2)^{2}}$ .

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Problem 2253

Graph the function $f(x)=\left\{\begin{array}{lr}|x| & x<2 \\ -\frac{1}{2} x+3 & 2 \leq x<6 \\ -\frac{1}{2} x & x \geq 6\end{array}\right.$ and find: a) Where is $f$ continuous? b) $\lim _{h \rightarrow 0} \frac{f(1+h)-f(1)}{h}$ c) $f^{\prime}(4)=1$ d) $\lim _{h \rightarrow 0^{-}} \frac{f(6+h)-f(6)}{h}$ e) $\lim _{h \rightarrow 0^{+}} \frac{f(6+h)-f(6)}{h}$ f) Where is $f$ differentiable? $x=0 \quad x=1 \quad x=2 \quad x=3 \quad x=4$

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Problem 2254

Find the derivative of the constant function $y=7$ using the definition of a derivative. What is the result?

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Problem 2255

Find the derivatives using the definition: a) $f(x)=x^{3}$ , b) $g(x)=x^{2}$ . Also, state the Constant Rule: $\frac{d}{d x}[c]=$ .

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Problem 2256

Find the derivative of $f(x) = x^{3} 3x$ .

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Problem 2257

Find the derivative of $f(x)=x^{3}$ .

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Problem 2258

Find the tangent line equation to $y=x^{2}$ at $x=-2$ . What do we need to write the line's equation?

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Problem 2259

Find the average velocity of $s=4\sin(\pi t) + 4\cos(\pi t)$ over intervals and estimate instantaneous velocity at $t=1$ .

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Problem 2260

Find constants $a$ and $b$ such that the derivative of $\frac{3x^2 - 4x - 4}{4x + 1}$ equals $\frac{12x^2 + ax + b}{(4x + 1)^2}$ .

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Problem 2261

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

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Problem 2262

Find the derivative of $x^{p}$ and choose the correct answer: $p \cdot x^{p+1}$ , $p \cdot x^{p-1}$ , $x^{p-1}$ , $x^{p+1}$ , or None.

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Problem 2263

Find $h'(4)$ for $h(x)=f(x) \times g(x)$ using $f(4)=-5$ , $f'(4)=2$ , $g(4)=3$ , $g'(4)=-3$ .

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Problem 2264

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

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Problem 2265

Find $h^{\prime}(0)$ for $h(x)=\frac{f(x)}{g(x)}$ given $f(0)=-6$ , $f^{\prime}(0)=5$ , $g(0)=2$ , $g^{\prime}(0)=-2$ .

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Problem 2266

Find $f^{\prime}(4)$ if $f(x)=-2h(x)+\frac{4x}{h(x)}$ , with $h(4)=5$ and $h^{\prime}(4)=-2$ .

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Problem 2267

Find the limit: $\lim _{x \rightarrow 2} \frac{x^{2}-6 x+4}{x-2}$ .

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Problem 2268

Find $f^{\prime}(8)$ if $f(x)=2x \cdot h(x)$ , $h(8)=-1$ , and $h^{\prime}(8)=-4$ . Options: -65, -66, 62, -62, -48.

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Problem 2269

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

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Problem 2270

Find the derivative of the function $f(x)=-\frac{4 x+5}{2 x-10}$ . Choices include various forms of $f^{\prime}(x)$ .

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Problem 2271

Find the derivative of $h(t)=\frac{3 t^{3}+5 t^{2}-\sqrt{t}}{t}$ .

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Problem 2272

Find the derivative of $r=\sqrt[4]{\theta^{3}}-\frac{5}{2 \theta^{3}}-\frac{\theta^{4}}{2}$ .

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Problem 2273

Find the derivative of $G(x)=(-5 x^{3}+2 x^{2})^{5}$ . What is $G'(x)$ ?

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Problem 2274

Find $g^{\prime}(1)$ for the function $g(x)=\frac{1}{5 x^{2}+2 x}$ . Choices: $\frac{49}{12}$ , $\frac{1}{49}$ , $\frac{12}{49}$ .

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Problem 2275

Find the slope of the tangent line for $f(x)=\frac{2 x}{5 x^{2}+8}$ at $x=1$ . Options: $\frac{6}{169}$ , $\frac{16}{169}$ , $\frac{21}{169}$ , $\frac{46}{169}$ , $-\frac{8}{169}$ , None.

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Problem 2276

Find the limit: $\lim _{x \rightarrow 0} f(x)$ for the function $f(x)=\frac{2|x|}{x}$ .

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Problem 2277

What is the slope of the secant line from $(a, f(a))$ to $(b, f(b))$ if $f(x)$ is constant on $[a, b]$ ? Explain.

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Problem 2278

Find the limit: $\lim _{x \rightarrow 2^{-}} \frac{x-2}{3|x-2|}$ . What is the value?

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Problem 2279

Find $f^{\prime}(4)$ if $f(x)=-2h(x)+\frac{4x}{h(x)}$ , with $h(4)=5$ and $h^{\prime}(4)=-2$ .

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Problem 2280

Find $f^{\prime}(4)$ if $f(x)=-2h(x)+\frac{4x}{h(x)}$ , with $h(4)=5$ and $h^{\prime}(4)=-2$ .

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Problem 2281

Find $f^{\prime}(4)$ if $f(x)=-2h(x)+\frac{4x}{h(x)}$ , $h(4)=5$ , and $h^{\prime}(4)=-2$ .

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Problem 2282

Find the limit: $\lim _{x \rightarrow 0^{-}} \frac{x}{2|x|}$ . What is the value?

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Problem 2283

Find where the tangent line is horizontal or vertical for $f(x)=3 x+3 \sin x$ in $0 \leq x \leq 2 \pi$ .

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Problem 2284

Evaluate the limit: $\lim _{x \rightarrow 2} \frac{\sqrt{x+4}-2}{x-2}=$ . Justify your answer.

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Problem 2285

Find the limit: $\lim _{x \rightarrow 0}\left(\frac{\sec (x)-1}{x \sec (x)}\right)$ .

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Problem 2286

Find the limit: $\lim _{x \rightarrow 0}\left(\frac{\sec (x)-1}{x \sec (x)}\right)=$

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Problem 2287

Evaluate the limit:
$\lim _{x \rightarrow 2} \frac{x^{2}-8 x+62}{x-2}$
Is it DNE or a real number? Justify your answer.

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Problem 2288

Find the tangent line equation for $y=g(x)$ at $x=3$ where $g(3)=-4$ and $g'(3)=5$ . Answer: $y=5(x-3)-4$

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Problem 2289

Find the average and instantaneous rate of change of $C(x)=8000+6x+0.15x^2$ at $x=101$ and $x=103$ , and at $x=100$ .

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Problem 2290

Differentiate $z=\frac{A}{y^{14}}+B e^{y}$ with respect to $y$ . What is $z^{\prime}=?$

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Problem 2291

A ball is thrown with a velocity of $40 \mathrm{ft/s}$ . Its height is $s(t)=40t-16t^2$ . Find the velocity at $t=1$ second.

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Problem 2292

Determine if the limit $\lim _{x \rightarrow 0+} \frac{3+2 x}{1+f\left(\frac{1}{x}\right)}$ exists, given $\lim _{x \rightarrow-\infty} f(x)=4$ and $\lim _{x \rightarrow \infty} f(x)=8$ .

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Problem 2293

Find the tangent line to $y=x \sqrt{x}$ parallel to $y=9+3x$ . What is the equation? $y=$

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Problem 2294

Evaluate the limit: $\lim _{x \rightarrow 8} \frac{64-x^{2}}{x-8}$ using algebraic transformation and continuity. Enter "DNE" if it doesn't exist.

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Problem 2295

Find $h'(3)$ for $h(x)=\frac{f(x)}{g(x)}$ using given values of $f$ and $g$ .

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Problem 2296

Find $h^{\prime}(3)$ for $h(x)=\frac{f(x)}{g(x)}$ given $f(3)=-6$ , $f^{\prime}(3)=2$ , $g(3)=5$ , $g^{\prime}(3)=-5$ .

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Problem 2297

Find the derivative of $f(x)=3 x^{4} \cot (x)$ .

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Problem 2298

Differentiate $y=\frac{1}{p+k e^{p}}$ with respect to $p$ . Find $y^{\prime}(p)=$

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Problem 2299

Find $f^{\prime}(-8)$ if $f(x)=2x \cdot h(x)$ , with $h(-8)=-4$ and $h^{\prime}(-8)=3$ .

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Problem 2300

Find $f^{\prime}(1)$ if $f(x)=-3 \cdot h(x)-\frac{4 x}{h(x)}$ , with $h(1)=4$ and $h^{\prime}(1)=-3$ .

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Calculus

Problem 2201

Find the slope of the tangent line for y=x4−2x3+8y=x^{4}-2 x^{3}+8y=x4−2x3+8 at x=2x=2x=2. What is the equation of the line?

Problem 2202

Find the slope and equation of the tangent line for y=x4−13x2+36y=x^{4}-13 x^{2}+36y=x4−13x2+36 at x=2x=2x=2. Simplify your answer.

Problem 2203

Find the slope of the tangent line for y=11x1/2+x3/2y=11 x^{1/2}+x^{3/2}y=11x1/2+x3/2 at x=4x=4x=4. Simplify your answer.

Problem 2204

Find values of xxx where the tangent line of f(x)=2x3+39x2+240x+6f(x)=2 x^{3}+39 x^{2}+240 x+6f(x)=2x3+39x2+240x+6 is horizontal. x=x= x=

Problem 2205

Find xxx where the tangent line to f(x)=2x3+39x2+240x+6f(x) = 2x^3 + 39x^2 + 240x + 6f(x)=2x3+39x2+240x+6 is horizontal.

Problem 2206

Problem 2207

A parachutist drops a camera from 50 m while descending at 10 m/s. Find (a) time to hit ground and (b) impact velocity.

Problem 2208

A rocket accelerates at +29.4 m/s2+29.4 \mathrm{~m} / \mathrm{s}^{2}+29.4 m/s2 for 4.0 s4.0 \mathrm{~s}4.0 s. Find (a) max height, (b) time to max height, (c) position and velocity graphs.

Problem 2209

A ball is thrown up at 19.6 m/s19.6 \mathrm{~m/s}19.6 m/s. Find its velocity and height at 1, 2, 3, and 4 seconds. Also, graph position and velocity vs. time.

Problem 2210

Find the derivative D′(3)D^{\prime}(3)D′(3) for the demand function D(p)=−3p2+3p+7D(p)=-3 p^{2}+3 p+7D(p)=−3p2+3p+7.

Problem 2211

Find the marginal profit for the function P(x)=0.08x2−4x+4x0.6−5700P(x)=0.08 x^{2}-4 x+4 x^{0.6}-5700P(x)=0.08x2−4x+4x0.6−5700 at x=500,1000,5000,10000x=500, 1000, 5000, 10000x=500,1000,5000,10000.

Problem 2212

Find the derivative of the function: y=x3−7x2+15x+9y=x^{3}-7 x^{2}+15 x+9y=x3−7x2+15x+9.

Problem 2213

Find f′(3)f^{\prime}(3)f′(3) for the function f(x)=x4+6x3−2x−2f(x)=x^{4}+6x^{3}-2x-2f(x)=x4+6x3−2x−2.

Problem 2214

Calculate the average rate of change of y=x3+x2−8x−7y=x^{3}+x^{2}-8 x-7y=x3+x2−8x−7 from x=0x=0x=0 to x=2x=2x=2.

Problem 2215

Calculate the integral ∫−510ln⁡x25x2dx\int_{-5}^{10} \frac{\ln x}{25 x^{2}} d x∫−510​25x2lnx​dx.

Problem 2216

Find the derivative of the function f(x)=5+2x21−2x2f(x)=\frac{5+2 x^{2}}{1-2 x^{2}}f(x)=1−2x25+2x2​, denoted as f′(x)f^{\prime}(x)f′(x).

Problem 2217

Evaluate the integral ∫−11cos⁡x1+e1/xdx\int_{-1}^{1} \frac{\cos x}{1+e^{1 / x}} d x∫−11​1+e1/xcosx​dx.

Problem 2218

Evaluate the integral ∫1cos⁡x1+e1/xdx\int^{1} \frac{\cos x}{1+e^{1 / x}} d x∫11+e1/xcosx​dx.

Problem 2219

Find the derivative y′y^{\prime}y′ for the function y=65x4y=\frac{6}{5 x^{4}}y=5x46​.

Problem 2220

Differentiate the function 2x37−49x3\frac{2 x^{3}}{7}-\frac{4}{9 x^{3}}72x3​−9x34​ with respect to xxx.

Problem 2221

Calculate the average rate of change of f(x)=3x−2f(x)=\sqrt{3x-2}f(x)=3x−2​ from x=2x=2x=2 to x=6x=6x=6.

Problem 2222

Find the average rate of change of employees (in thousands) with respect to revenue (in billions) for these intervals:a. From \$11B to \$16B: 147K to 191K.b. From \$16B to \$27B: 191K to 346K.c. From \$11B to \$27B: 147K to 346K.

Problem 2223

Sales function is S(t)=0.02t3+0.8t2+9t+2S(t)=0.02 t^{3}+0.8 t^{2}+9 t+2S(t)=0.02t3+0.8t2+9t+2. Find S′(t)S^{\prime}(t)S′(t), S(2)S(2)S(2), S′(2)S^{\prime}(2)S′(2), and interpret S(13)S(13)S(13) and S′(13)S^{\prime}(13)S′(13).

Problem 2224

Find the marginal cost from the total cost function C(x)=100+12x+0.1x2+0.001x3C(x)=100+12x+0.1x^{2}+0.001x^{3}C(x)=100+12x+0.1x2+0.001x3 for x=0x=0x=0, x=10x=10x=10, and x=30x=30x=30.

Problem 2225

Find the marginal cost for producing 1500 and 2000 shirts given the cost function C(x)=1500+1000x9C(x)=1500+1000 x^{9}C(x)=1500+1000x9, where xxx is in hundreds.

Problem 2226

Given the sales function S(t)=0.02t3+0.7t2+7t+9S(t)=0.02 t^{3}+0.7 t^{2}+7 t+9S(t)=0.02t3+0.7t2+7t+9, find: (A) S′(t)S^{\prime}(t)S′(t), (B) S(4)S(4)S(4) and S′(4)S^{\prime}(4)S′(4), (C) Interpret S(8)=120.04S(8)=120.04S(8)=120.04 and S′(8)=22.04S^{\prime}(8)=22.04S′(8)=22.04.

Problem 2227

Find the marginal profit from selling xxx smartwatches given P(x)=0.03x2−4x+3x0.8−5000P(x)=0.03x^{2}-4x+3x^{0.8}-5000P(x)=0.03x2−4x+3x0.8−5000 for x=100x=100x=100.

Problem 2228

Find the marginal revenue function for R(x)=0.016x2+1.034x+11.338R(x)=0.016x^{2}+1.034x+11.338R(x)=0.016x2+1.034x+11.338 and calculate it at x=10x=10x=10.

Problem 2229

Find the speed vvv that minimizes energy E(v)=2.73v314v−7E(v)=2.73 v^{3} \frac{14}{v-7}E(v)=2.73v3v−714​ for a fish swimming against a 7mi/h7 \mathrm{mi/h}7mi/h current.

Problem 2230

Problem 2231

Order the values III, RnR_{n}Rn​, and LnL_{n}Ln​ for a decreasing function fff on [a,b][a, b][a,b] and for an increasing function. Explain.

Problem 2232

Complete the table and find the limit: lim⁡x→1f(x)\lim _{x \rightarrow 1} f(x)limx→1​f(x) for f(x)=x3−1x2−1f(x)=\frac{x^{3}-1}{x^{2}-1}f(x)=x2−1x3−1​.

Problem 2233

Find the tangent slope at (5,0) for f(x)=3(5−x)2f(x)=3(5-x)^{2}f(x)=3(5−x)2 and at (0,0) for f(θ)=4sin⁡θ−θf(\theta)=4 \sin \theta - \thetaf(θ)=4sinθ−θ.

Problem 2234

Find the absolute minimum and maximum of the function f(x)=2.5x4+3x3−2.6x2−5.1x−5.6f(x)=2.5 x^{4}+3 x^{3}-2.6 x^{2}-5.1 x-5.6f(x)=2.5x4+3x3−2.6x2−5.1x−5.6 on [−0.5,1.2][-0.5,1.2][−0.5,1.2].

Problem 2235

Determine the slope of f(t)=34etf(t)=\frac{3}{4} e^{t}f(t)=43​et at the point (0,34)(0, \frac{3}{4})(0,43​).

Problem 2236

Find the derivative of g(t)=t2−4t3g(t)=t^{2}-\frac{4}{t^{3}}g(t)=t2−t34​.

Problem 2237

Find the derivative of f(x)=4x3+3x2xf(x)=\frac{4x^{3}+3x^{2}}{x}f(x)=x4x3+3x2​.

Problem 2238

Find the derivative of these functions and note restrictions for the second:1. f(x)=xxf(x)=\frac{x}{x}f(x)=xx​2. f(x)=x3−3x2+4x2f(x)=\frac{x^{3}-3 x^{2}+4}{x^{2}}f(x)=x2x3−3x2+4​

Problem 2239

Find the derivative of the function f(x)=x3−3x2+4x2f(x)=\frac{x^{3}-3x^{2}+4}{x^{2}}f(x)=x2x3−3x2+4​.

Problem 2240

Find the derivative of the function y=x(x2+1)y=x(x^{2}+1)y=x(x2+1).

Problem 2241

Find the slope of the tangent line for $y=x^{4}-2 x^{3}+8$ at $x=2$ . What is the equation of the line?

Find the slope and equation of the tangent line for $y=x^{4}-13 x^{2}+36$ at $x=2$ . Simplify your answer.

Find the slope of the tangent line for $y=11 x^{1/2}+x^{3/2}$ at $x=4$ . Simplify your answer.

Find values of $x$ where the tangent line of $f(x)=2 x^{3}+39 x^{2}+240 x+6$ is horizontal. $x=$

Find $x$ where the tangent line to $f(x) = 2x^3 + 39x^2 + 240x + 6$ is horizontal.

A rocket accelerates at $+29.4 \mathrm{~m} / \mathrm{s}^{2}$ for $4.0 \mathrm{~s}$ . Find (a) max height, (b) time to max height, (c) position and velocity graphs.

A ball is thrown up at $19.6 \mathrm{~m/s}$ . Find its velocity and height at 1, 2, 3, and 4 seconds. Also, graph position and velocity vs. time.

Find the derivative $D^{\prime}(3)$ for the demand function $D(p)=-3 p^{2}+3 p+7$ .

Find the marginal profit for the function $P(x)=0.08 x^{2}-4 x+4 x^{0.6}-5700$ at $x=500, 1000, 5000, 10000$ .

Find the derivative of the function: $y=x^{3}-7 x^{2}+15 x+9$ .

Find $f^{\prime}(3)$ for the function $f(x)=x^{4}+6x^{3}-2x-2$ .

Calculate the average rate of change of $y=x^{3}+x^{2}-8 x-7$ from $x=0$ to $x=2$ .

Calculate the integral $\int_{-5}^{10} \frac{\ln x}{25 x^{2}} d x$ .

Find the derivative of the function $f(x)=\frac{5+2 x^{2}}{1-2 x^{2}}$ , denoted as $f^{\prime}(x)$ .

Evaluate the integral $\int_{-1}^{1} \frac{\cos x}{1+e^{1 / x}} d x$ .

Evaluate the integral $\int^{1} \frac{\cos x}{1+e^{1 / x}} d x$ .

Find the derivative $y^{\prime}$ for the function $y=\frac{6}{5 x^{4}}$ .

Differentiate the function $\frac{2 x^{3}}{7}-\frac{4}{9 x^{3}}$ with respect to $x$ .

Calculate the average rate of change of $f(x)=\sqrt{3x-2}$ from $x=2$ to $x=6$ .

Find the average rate of change of employees (in thousands) with respect to revenue (in billions) for these intervals:
a. From \$11B to \$16B: 147K to 191K.
b. From \$16B to \$27B: 191K to 346K.
c. From \$11B to \$27B: 147K to 346K.

Sales function is $S(t)=0.02 t^{3}+0.8 t^{2}+9 t+2$ . Find $S^{\prime}(t)$ , $S(2)$ , $S^{\prime}(2)$ , and interpret $S(13)$ and $S^{\prime}(13)$ .

Find the marginal cost from the total cost function $C(x)=100+12x+0.1x^{2}+0.001x^{3}$ for $x=0$ , $x=10$ , and $x=30$ .

Find the marginal cost for producing 1500 and 2000 shirts given the cost function $C(x)=1500+1000 x^{9}$ , where $x$ is in hundreds.

Given the sales function $S(t)=0.02 t^{3}+0.7 t^{2}+7 t+9$ , find: (A) $S^{\prime}(t)$ , (B) $S(4)$ and $S^{\prime}(4)$ , (C) Interpret $S(8)=120.04$ and $S^{\prime}(8)=22.04$ .

Find the marginal profit from selling $x$ smartwatches given $P(x)=0.03x^{2}-4x+3x^{0.8}-5000$ for $x=100$ .

Find the marginal revenue function for $R(x)=0.016x^{2}+1.034x+11.338$ and calculate it at $x=10$ .

Find the speed $v$ that minimizes energy $E(v)=2.73 v^{3} \frac{14}{v-7}$ for a fish swimming against a $7 \mathrm{mi/h}$ current.

Order the values $I$ , $R_{n}$ , and $L_{n}$ for a decreasing function $f$ on $[a, b]$ and for an increasing function. Explain.

Complete the table and find the limit: $\lim _{x \rightarrow 1} f(x)$ for $f(x)=\frac{x^{3}-1}{x^{2}-1}$ .

Find the tangent slope at (5,0) for $f(x)=3(5-x)^{2}$ and at (0,0) for $f(\theta)=4 \sin \theta - \theta$ .

Find the absolute minimum and maximum of the function $f(x)=2.5 x^{4}+3 x^{3}-2.6 x^{2}-5.1 x-5.6$ on $[-0.5,1.2]$ .

Determine the slope of $f(t)=\frac{3}{4} e^{t}$ at the point $(0, \frac{3}{4})$ .

Find the derivative of $g(t)=t^{2}-\frac{4}{t^{3}}$ .

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

Find the derivative of these functions and note restrictions for the second:
1. $f(x)=\frac{x}{x}$
2. $f(x)=\frac{x^{3}-3 x^{2}+4}{x^{2}}$

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

Find the derivative of the function $y=x(x^{2}+1)$ .

Find the derivative of $f(x)=\sqrt{x}-6\sqrt[3]{x}$ .

Sketch a function $g$ such that $g(0)=1$ , $g'(0)=-2$ , $g'(1)=0$ , and $g'(2)=4$ .

Find the derivative of $h(s)=s^{4/5}-s^{2/3}$ .

Given the limits, determine which statements about function $f$ are true: A. removable discontinuity at $x=1$ ; B. differentiable at $x=1$ ; C. $f(1)=2$ , $f'(1)=3$ .

Find the derivative of $f(x) = 6 \sqrt{x} + 5 \cos x$ .

Find the derivative of $f(x)=x^{-2}-2 e^{x}$ .

Find the tangent line for $f(x)=x^{4}-x$ at the given points and confirm using a graphing utility. Points: $(-1,2)$ , $(1,2)$ , $(0,1)$ , $(\pi, \frac{1}{2} e^{\pi})$ .

Find the horizontal tangent line for the function $y=\frac{1}{x^{2}}$ .

Find the horizontal tangent line for $y=x+\sin x$ in the interval $0 \leq x < 2\pi$ .

Find values of $a$ and $b$ so that the piecewise function $f(x)=\begin{cases}a x^{3}, & x \leq 2 \\ x^{2}+b, & x>2\end{cases}$ is differentiable everywhere.

Find the derivative or integral of the function $f(x)=\frac{3}{(x+2)^{2}}$ .

Find the derivative of the constant function $y=7$ using the definition of a derivative. What is the result?

Find the derivatives using the definition: a) $f(x)=x^{3}$ , b) $g(x)=x^{2}$ . Also, state the Constant Rule: $\frac{d}{d x}[c]=$ .

Find the derivative of $f(x) = x^{3} 3x$ .

Find the derivative of $f(x)=x^{3}$ .

Find the tangent line equation to $y=x^{2}$ at $x=-2$ . What do we need to write the line's equation?

Find the average velocity of $s=4\sin(\pi t) + 4\cos(\pi t)$ over intervals and estimate instantaneous velocity at $t=1$ .

Find constants $a$ and $b$ such that the derivative of $\frac{3x^2 - 4x - 4}{4x + 1}$ equals $\frac{12x^2 + ax + b}{(4x + 1)^2}$ .

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

Find the derivative of $x^{p}$ and choose the correct answer: $p \cdot x^{p+1}$ , $p \cdot x^{p-1}$ , $x^{p-1}$ , $x^{p+1}$ , or None.

Find $h'(4)$ for $h(x)=f(x) \times g(x)$ using $f(4)=-5$ , $f'(4)=2$ , $g(4)=3$ , $g'(4)=-3$ .

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

Find $h^{\prime}(0)$ for $h(x)=\frac{f(x)}{g(x)}$ given $f(0)=-6$ , $f^{\prime}(0)=5$ , $g(0)=2$ , $g^{\prime}(0)=-2$ .

Find $f^{\prime}(4)$ if $f(x)=-2h(x)+\frac{4x}{h(x)}$ , with $h(4)=5$ and $h^{\prime}(4)=-2$ .

Find the limit: $\lim _{x \rightarrow 2} \frac{x^{2}-6 x+4}{x-2}$ .

Find $f^{\prime}(8)$ if $f(x)=2x \cdot h(x)$ , $h(8)=-1$ , and $h^{\prime}(8)=-4$ . Options: -65, -66, 62, -62, -48.

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

Find the derivative of the function $f(x)=-\frac{4 x+5}{2 x-10}$ . Choices include various forms of $f^{\prime}(x)$ .

Find the derivative of $h(t)=\frac{3 t^{3}+5 t^{2}-\sqrt{t}}{t}$ .

Find the derivative of $r=\sqrt[4]{\theta^{3}}-\frac{5}{2 \theta^{3}}-\frac{\theta^{4}}{2}$ .

Find the derivative of $G(x)=(-5 x^{3}+2 x^{2})^{5}$ . What is $G'(x)$ ?

Find $g^{\prime}(1)$ for the function $g(x)=\frac{1}{5 x^{2}+2 x}$ . Choices: $\frac{49}{12}$ , $\frac{1}{49}$ , $\frac{12}{49}$ .