Calculus

Problem 28201

Find two positive numbers that add up to 20 and have the maximum product.

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Problem 28202

Find the derivative $f^{\prime}(x)$ of $f(x)=\frac{2}{5 x+5}$ using the limit definition.

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Problem 28203

Find when the instantaneous rate of change (IRoc) of $N=200(2)^{0.2 t}$ is least, and estimate the IRoc at that time.

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Problem 28204

Prove if the soccer ball reaches its highest point at $t=1.5$ s using the height function $h(t)=-4.9 t^{2}+14.7 t+0.5$ .

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Problem 28205

Find $S(16)$ and $S^{\prime}(16)$ for $S(t)=\sqrt{t}+7$ . Estimate $S(17)$ and $S(18)$ , rounding to 1 decimal place.

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Problem 28206

Test if the function $f(x)=\left\{\begin{array}{c}6 x-5 \text { when } x \geq 1 \\ 5 x-6 \text { when } x<1\end{array}\right.$ is differentiable at $x=1$ .

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Problem 28207

Is the function $f(x)=\left\{\begin{array}{l}3 x-2 \text { if } x \geq 1 \\ 2 x-3 \text { if } x<1\end{array}\right.$ differentiable at $x=1$ ?

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Problem 28208

Find the time $t$ (from 8 to 12) when the processing rate $P(t)=t^{3}-30 t^{2}+298 t-976$ is highest. Justify your answer.

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Problem 28209

Find the absolute maximum of $f(x)=x^{3}-3 x^{2}+13$ on the interval $[-2,4]$ using calculus.

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Problem 28210

Is the function $f(x)=\left\{\begin{array}{cc}(x-1) \sin \left(\frac{1}{x-1}\right) & x \neq 1 \\ -1 & x=1\end{array}\right.$ continuous on $\mathrm{R}$ ?

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Problem 28211

Is the function $f(x)=\begin{cases} (x-1) \sin \left(\frac{1}{x-1}\right) & x \neq 1 \\ -1 & x=1 \end{cases}$ continuous on R?

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Problem 28212

Find the derivatives of these functions: 1) $y=2 x^{4}+6 x+\ln(x^{2}+5 x)$ 2) $y=(x^{5}+x) \cos x$ 3) $y=\tan^{2}(x+1)$ 4) $y=\frac{x^{2}-2 x}{x^{2}+1}$ 5) $y=\sqrt{7+\ln x}$ .

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Problem 28213

Find where the tangent line to $y=f(x)=x^{3} \ln (x)$ is horizontal and if it intersects the graph at $(1,0)$ .

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Problem 28214

Find $\frac{d y}{d x}$ using implicit differentiation for the curve $x^{3}-3 x^{2} y+2 x y^{2}=12$ .

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Problem 28215

A soccer ball's height is given by $h(t)=-4.9 t^{2}+14.7 t+0.5$ . Verify if it peaks at $t=1.5$ s using rates of change. i) Estimate IRoC at $t=1.5$ s. ii) Find RoC before $t=1.5$ s. iii) Find RoC after $t=1.5$ s.

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Problem 28216

Find the derivatives of these functions: 1) $y=2 x^{4}+6 x+\ln(x^{2}+5 x)$ 2) $y=(x^{5}+x) \cos x$ 3) $y=\tan^{2}(x+1)$ 4) $y=\frac{x^{2}-2 x}{x^{2}+1}$ 5) $y=\sqrt{7+\ln x}$ .

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Problem 28217

Is the function $f(x)=|x-2|$ differentiable at the point $x=2$ ?

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Problem 28218

Find the difference quotient for $f(x)=x$ at the point $(3,3)$ . Choose from: a) $\frac{3-h}{h}$ , b) 1, c) 3, d) $\mathrm{h}$ .

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Problem 28219

Find the difference quotient for $f(x)=x^{2}$ at the point $(5,25)$ . Choose from the options provided.

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Problem 28220

Estimate the derivative of $f(x)=\sqrt{3x}$ at the point $x=7$ .

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Problem 28221

Evaluate these integrals: 1) $\int(x^{5}+4 x^{3}-\frac{x}{3}+2) dx$ 2) $\int(\sin(x)-\sec(x)\tan(x)) dx$ 3) $\int(8x-12)(4x^{2}-12x)^{4} dx$ 4) $\int \frac{\sin(x)}{\sqrt{1-\cos(x)}} dx$ 5) $\int 2x \sec^{2}(x^{2}+1) dx$ 6) $\int \frac{e^{x}}{e^{x}+1} dx$

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Problem 28222

Find the difference quotient for $f(x)=\sqrt{x}$ at $(16,4)$ . Choose from: a) $0 \frac{16+h-16}{h}$ , b) $\bigcirc \frac{\sqrt{4+h}-\sqrt{16}}{h}$ , c) $0 \frac{\sqrt{4+h}-16}{h}$ , d) $0 \frac{\sqrt{16+h}-4}{h}$ .

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Problem 28223

Is the function $f(x)=|x-2|$ differentiable at the point $x=2$ ?

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Problem 28224

Find the derivative $f^{\prime}(x)$ of $f(x)=4x^{2}+6x+3$ using the limit definition.

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Problem 28225

Find the average rate of change of $y=2 \cos \left(x-\frac{\pi}{3}\right)+1$ for these intervals: a) $\frac{\pi}{6} \leq x \leq \frac{\pi}{2}$ , b) $\frac{\pi}{2} \leq x \leq \frac{5 \pi}{4}$ .

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Problem 28226

Which statement is true? $P: \mathbf{f}(\mathbf{x})=\sqrt{4-x^{2}}$ is continuous on $[-2,2]$ . $Q: \mathbf{g}(\mathbf{x})=\sqrt{9-x^{2}}$ is continuous on $[-2,2]$ . Options: a) Only P b) Only Q c) Both P and Q d) Neither P nor Q

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Problem 28227

Evaluate the integral from 2 to 7 of the function $8 x e^{x^{2}+4 y}$ with respect to $x$ .

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Problem 28228

Evaluate the double integral: $\int_{2}^{3} \int_{\sqrt{x}}^{x} \frac{y}{x} \, dy \, dx$

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Problem 28229

Which expression represents the derivative at $x=c$ if it exists? a) $\lim _{h \rightarrow 0} \frac{f(c-h)-f(c)}{h}$ b) $\lim _{h \rightarrow 0} \frac{f(c+h)-f(c-h)}{2 h}$ c) $\lim _{h \rightarrow 0} \frac{f(c+h)+f(c)}{h}$ d) $\lim _{h \rightarrow 0} \frac{f(c)-f(c-h)}{-c}$

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Problem 28230

Evaluate the integral: $\int_{1}^{3}\left(x^{4}+x y-y^{2}\right) d x$

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Problem 28231

Find the length of the curve $f(x)=\frac{x^{3}}{12}+\frac{1}{x}$ for $1 \leq x \leq y$ .

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Problem 28232

1. Find the slope of the tangent line for $f(x)=\frac{-5 x}{2 x+3}$ at $x=2$ and $f(x)=\frac{2 x^{2}-6 x}{3 x+5}$ at $x=-2$ . Identify where no tangent exists.
2. For the cost function $C(x)=\frac{x^{2}-4 x+20}{x}$ , calculate the average cost for 3000 shirts and estimate the rate of change at that level.
3. Calculate the average rate of change for $y=2 \cos \left(x-\frac{\pi}{3}\right)+1$ in the intervals: a) $\frac{\pi}{6} \leq x \leq \frac{\pi}{2}$ b) $\frac{\pi}{2} \leq x \leq \frac{5 \pi}{4}$

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Problem 28233

Find $\frac{d y}{d x}$ for $x=2 \sin \left(\frac{y}{3}\right)$ , and express your answer in terms of $x$ .

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Problem 28234

Evaluate the integral $\iint_{R} 6 y d A$ for the region $R$ between $y=6-x^{2}$ and the lines $y=x$ , $y=-x$ .

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Problem 28235

Find the slope of the secant line between points $(c, f(c))$ and $(c+2 h, f(c+2 h))$ :
a) $0 \frac{f(c+2 h)+f(c)}{2 h}$ b) $0 \frac{f(c+h)+f(c)}{c-h}$ c) $0 \frac{f(c)-f(h)}{c-h}$ d) $0 \frac{f(c+2 h)-f(c)}{2 h}$

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Problem 28236

Evaluate the double integral $\iint_{\mathbb{R}} \sqrt[3]{3 x+6 y} d \mathbf{A}$ over the region $1 \leq x \leq 6$ , $0 \leq y \leq 2$ .

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Problem 28237

Assignment: Teach the Teacher
Explain how to find the maximum instantaneous rate of change for $P(t)=0.5 \sin \left(\frac{\pi}{6} t\right)+4$ using centered intervals. Compare accuracy of estimates for intervals: i) 1 h, ii) 0.5 h, iii) 0.25 h.

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Problem 28238

Identify two points on the graph of $f(x)=4\cos(x-\frac{\pi}{4})+2$ where the instantaneous rate of change is: a) zero, b) negative, c) positive.

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Problem 28239

Evaluate the integral: $\int_{-2}^{3}(3 x^{4} y^{3}+5 x y-6) \, dx$

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Problem 28240

Evaluate the integral: $\int_{-2}^{3}(3 x^{4} y^{3}+5 x y-6) dx$

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Problem 28241

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

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Problem 28242

Evaluate the double integral: $\int_{3}^{4} \int_{1}^{2} 12 x^{3} y^{2} \, dx \, dy$

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Problem 28243

3. Given $P(t)=-1.5 t^{2}+36 t+6$ , find the average population change from 2000 to 2024 and analyze its meaning. Also, find when the instantaneous rate is 0.
4. For the graph of $y=f(x)$ , identify intervals where the average rate of change is positive, negative, or zero.

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Problem 28244

Berechnen Sie die Untersumme $U_{10}$ für das Integral $\int_{0}^{2} 0,5 x \, dx$ mit 10 gleichen Intervallen.

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Problem 28245

Ein Gartenpool wird mit $v(t)=0,1 t^{2}-\frac{1}{t^{3}+1}+40$ gefüllt. Bestimmen Sie die Füllhöhe und das Volumen nach 10 Minuten.

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Problem 28246

Determine where the average rate of change of $y=f(x)$ is positive, negative, or zero based on its graph.

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Problem 28247

1. Find the average rate of change of $h(t)=18t-0.8t^{2}$ from $t=10$ to $t=15$ for the golf ball's height.
2. A car worth \$23500 depreciates to \$8750 in 8 years. Calculate the average annual percent decrease in value.

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Problem 28248

Evaluate the double integral $\iint_{R} 4 x^{3} y^{2} \, dA$ where $R=\{(x, y) | 0 \leq x \leq 3, 0 \leq y \leq 2\}$ .

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Problem 28249

Population function: $P(t)=-1.5 t^{2}+36 t+6$ . Find average change from 2000 to 2024, and check changes for 2000-2012 & 2012-2024. When is the rate 0?

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Problem 28250

A rock sample of $256 \mathrm{~g}$ tungsten-187 decays to $0.25 \mathrm{~g}$ . With a half-life of 1 day, how long does this take?

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Problem 28251

Find the average rate of change of area with respect to radius for a circle from $0 \mathrm{~cm}$ to $100 \mathrm{~cm}$ and at $120 \mathrm{~cm}$ .

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Problem 28252

Find where the slope of the tangent line equals the slope of the secant line through $A(2,-4)$ and $B(3,0)$ for $f(x)=x^{3}-3x^{2}$ .
Also, for a diver on a $10 \mathrm{~m}$ platform, use $h(t)=10+2t-4.9t^{2}$ to find when they hit the water and their height change rate.

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Problem 28253

A diver on a $10 \mathrm{~m}$ platform dives with height $h(t)=10+2t-4.9t^{2}$ . Find when they enter the water and the height change rate.

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Problem 28254

Find the limit: $\lim _{x \rightarrow 1} \frac{\frac{1}{\sqrt{x}}-1}{x-1}$ .

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Problem 28255

Find the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{5}-\sqrt{x+5}}{x}$ .

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Problem 28256

Find the derivative of the function $y=\frac{3x-2}{2-5x}$ .

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Problem 28257

A ship's height is modeled by $h(t)=\sin \frac{\pi}{5} t$ . Find the average rate of change over 5s and the instantaneous rate at $t=6s$ .

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Problem 28258

A pilot ejects from 10,000 ft, with altitude $h(t)=-16 t^{2}+90 t+10000$ . Find time $t$ for max altitude and relate it to slope.

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Problem 28259

A ship's height is modeled by $h(t) = \sin \frac{\pi}{5} t$ . Find the average rate of change over 5s and the instantaneous rate at $t=6s$ .

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Problem 28260

Solve the differential equation $(4 x+y+3) dy=(4 x+y+1)^{2} dx$ . What is the solution? (a), (b), (c), or (d)?

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Problem 28261

A ship's height $h(t) = \sin \frac{\pi}{5} t$ in meters varies with time $t$ in seconds. Find the average rate of change over 5s and the instantaneous rate at $t=6s$ . Mark Value: 4

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Problem 28262

A radioactive substance decays exponentially. Given $P(t)=100(1.2)^{-t}$ , find the half-life and rate of decay after the first half-life.

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Problem 28263

Find values of $a$ and $b$ if $(1,2)$ is a relative extremum of $P(x)=2ax^2-3bx+3$ .

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Problem 28264

The town's population is 12000 and decreases at $1.8\%$ per year. Find the population model and rates of change in 10 years and at half population.

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Problem 28265

Untersuchen Sie die Flächeninhalte der Funktionen: I. $f(x)=\frac{1}{x^{3}}$ , II. $f(x)=\frac{1}{x^{2}}$ , III. $f(x)=\frac{1}{\sqrt{x}}$ für $x \geq 1$ .

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Problem 28266

The town's population is 12000 and decreases at $1.8\%$ per year. Find the population model and rates of change in 10 years and at half population.

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Problem 28267

Find the average wind speed change from mile 10 to mile 100 using $S(d)=93 \log d+65$ . Also, estimate speed changes at miles 10 and 100.

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Problem 28268

The town's population is 12,000 and decreases by $1.8\%$ per year.
a) Model the population with an equation. b) Find the rate of change in 10 years. c) Find the rate when the population is 6,000.
Mark Value: 3

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Problem 28269

The town's population is 12000 and decreases at $1.8\%$ per year.
a) Model the population with an equation. b) Find the rate of change in 10 years. c) Find the rate when the population is 6000.

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Problem 28270

Find the average wind speed change from mile 10 to 100 using $S(d)=93 \log d+65$ . Also, estimate speed change at miles 10 and 100.

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Problem 28271

Demand for snack cakes is $p(x)=\frac{15}{2 x^{2}+11 x+5}$ . Find revenue function and marginal revenue at $x=0.75$ and $x=2$ .

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Problem 28272

Calculate the average wind speed change from mile 10 to mile 100 using $S(d)=93 \log d+65$ . Also, find the speed change at miles 10 and 100.

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Problem 28273

Bestimme die Ableitungen der Funktionen: a) $f(x)=5 e^{7 x^{2}+4 x}$ b) $g(x)=(2 x^{4}-7 x)(x^{3}+x)$ c) $h(x)=3(2 x^{4}+7)^{13}$

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Problem 28274

Find the limit: $\lim _{x \rightarrow 0} \frac{\frac{4}{x+2}-2}{x}$

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Problem 28275

Bestimme die Terme der 1. Ableitung von $f(x)=\sqrt{8 x}$ .

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Problem 28276

Untersuchen Sie die Funktion $f(x) = 3 \cdot e^{0.3 x^{3} - 1.2 x} - 3$ auf Nullstellen, Extrempunkte, Wendestellen und die Tangente im Ursprung.

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Problem 28277

Untersuchen Sie die Funktion $f(x)=3 \cdot e^{0.3 x^{3}-1.2 x}-3$ auf Nullstellen, Extrempunkte, Wendepunkte und Tangente im Ursprung.

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Problem 28278

Bestimme die Stammfunktion von $g(x) = \frac{1}{2} x^{3} + \frac{1}{2} x^{2} - x$ .

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Problem 28279

Berechnen Sie den Flächeninhalt der Funktion $f(x)=-\frac{1}{2} x^{2}+2$ zwischen den Nullstellen und $x=-2$ , $x=2$ .

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Problem 28280

Calculate the area between $y=4 \sin (x)$ and $y=3 \cos (x)$ from $x=0$ to $x=\frac{3 \pi}{4}$ .

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Problem 28281

Find the tangent slope of $f(x)=\frac{3x+1}{2x+7}$ at $x=-2$ , rounded to one decimal place.

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Problem 28282

Find the limit: $\lim _{x \rightarrow 2} \frac{3 x^{2}-5 x-2}{4 x^{2}-5 x-6}$ .

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Problem 28283

Find the average rate of change of $f(x)=4(x+2)^{2}+1$ on the interval $-1<x<3$ .

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Problem 28284

Find the marginal cost function $M C(x)$ for $C(x)=200+\frac{7}{x}+\frac{x^{2}}{7}$ . Then, calculate $M C(12)$ and the cost of the 13th processor.

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Problem 28285

Find the slope of the tangent to $f(x)=\frac{3 x+1}{2 x+7}$ at $x=-2$ , rounded to one decimal place.

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Problem 28286

Find the velocity $v(t)$ and acceleration $a(t)$ of the particle with position $s(t)=\frac{t}{1+t^{2}}$ for $t \geq 0$ . Also, determine when it is slowing down or speeding up.

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Problem 28287

Given an initial deposit of \ $100, the balance after$ t $years is$ f(t)=100(1.08)^{t} $. Find the units of rate of change of$ f(t) $and average rates over$ [0,0.5] $and$ [0,1]$. Round answers to two decimals.

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Problem 28288

Find the limit: $\lim _{x \rightarrow 7} \frac{\sqrt[5]{3-5 x}}{(x-5)^{3}}$

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Problem 28289

Find the value of $c$ in $[1,4]$ satisfying the Mean Value Theorem for $f(x)=5 \cos ^{2}\left(\frac{x}{2}\right)+\ln (x+1)-3$ .

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Problem 28290

Calculate the limit: $\lim _{x \rightarrow -3} \frac{x+3}{3-\sqrt{x+12}}$

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Problem 28291

Given $f(0)=-4$ and $f(10)=11$ , which must be true for some $c$ in $(0,10)$ ? (A) $f^{\prime}(c)=0$ (B) $f^{\prime}(c)=\frac{11+(-4)}{10-0}$ (C) $f^{\prime}(c)=\frac{11-(-4)}{10-0}$ (D) $f^{\prime}(c)=1.5$

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Problem 28292

On which closed interval does the function $f(x)=\frac{x+4}{(x-1)(x+3)}$ have an absolute max and min? Options: (A) $[-5,5]$ , (B) $[-3,1]$ , (C) $[-2,0]$ , (D) $[0,5]$

See Solution

Problem 28293

Two resistors $a$ and $b$ are in parallel with $R=\frac{15}{8}$ ohms and $b=a+2$ . Find $a$ and $b$ . Also, for $M=3t^4-2t^3+5$ , find the average change from day 1 to 3 and instantaneous change after 2 days.

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Problem 28294

Find the limits: w. $\lim _{x \rightarrow-3} \frac{x+3}{3-\sqrt{x+12}}$ x. $\lim _{x \rightarrow 4}\left(\frac{8}{x^{2}-16}-\frac{1}{x-4}\right)$ y. $\lim _{x \rightarrow \pi} \frac{1+\sin x}{\cos x}$
2. $\lim _{x \rightarrow \frac{3 \pi}{2}} \frac{\sin x}{\cos x}$

See Solution

Problem 28295

Find the average rate of change of $M=3t^{4}-2t^{3}+5$ from day 1 to day 3 and estimate the rate after 2 days.

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Problem 28296

Two resistors $a$ and $b$ are in parallel with $R=15/8$ ohms. If $b=a+2$ , find $a$ and $b$ . Also, find the average and instantaneous rate of change of $M=3t^4-2t^3+5$ from day 1 to 3 and at day 2.

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Problem 28297

Evaluate these limits: a. $\lim _{x \rightarrow 3} \sqrt[3]{\frac{x-4}{6 x^{2}+2}}$ b. $\lim _{x \rightarrow 3} \frac{\frac{1}{x^{2}}-\frac{1}{9}}{x-3}$ c. $\lim _{h \rightarrow 0} \frac{(-3+h)^{2}-9}{h}$ d. $\lim _{h \rightarrow 0} \frac{(2+h)^{3}-8}{h}$

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Problem 28298

Find the derivative $\frac{d y}{d x}$ at the point $(0,0)$ for the curve defined by $\tan (x y)=x+y$ .

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Problem 28299

Show that $\lim _{x \rightarrow 0} x^{3}\left(\cos \frac{1}{x^{2}}\right)=0$ using the Squeeze Theorem.

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Problem 28300

Find the limits: e. $\lim _{x \rightarrow 0} \frac{x^{3}}{2 x^{3}+3 x^{4}}$ f. $\lim _{x \rightarrow-1} \frac{2 x+3}{3 x+2}$ g. $\lim _{x \rightarrow 3} \frac{x+3}{3-\sqrt{x+12}}$ h. $\lim _{x \rightarrow 8} \frac{x+8}{\sqrt[3]{x}+2}$ i. $\lim _{x \rightarrow 0} \frac{(3+x)^{3}-3^{3}}{(3+x)^{2}-3^{2}}$ j. $\lim _{x \rightarrow 9} \frac{x-9}{3 \sqrt{x}}$ k. $\lim _{x \rightarrow 9} \frac{x^{2}-81}{3-\sqrt{x}}$ l. $\lim _{x \rightarrow 16} \frac{2 \sqrt{x}+x^{3 / 2}}{\sqrt[4]{x}+5}$

See Solution

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Calculus

Problem 28201

Find two positive numbers that add up to 20 and have the maximum product.

Problem 28202

Find the derivative f′(x)f^{\prime}(x)f′(x) of f(x)=25x+5f(x)=\frac{2}{5 x+5}f(x)=5x+52​ using the limit definition.

Problem 28203

Find when the instantaneous rate of change (IRoc) of N=200(2)0.2tN=200(2)^{0.2 t}N=200(2)0.2t is least, and estimate the IRoc at that time.

Problem 28204

Prove if the soccer ball reaches its highest point at t=1.5t=1.5t=1.5 s using the height function h(t)=−4.9t2+14.7t+0.5h(t)=-4.9 t^{2}+14.7 t+0.5h(t)=−4.9t2+14.7t+0.5.

Problem 28205

Find S(16)S(16)S(16) and S′(16)S^{\prime}(16)S′(16) for S(t)=t+7S(t)=\sqrt{t}+7S(t)=t​+7. Estimate S(17)S(17)S(17) and S(18)S(18)S(18), rounding to 1 decimal place.

Problem 28206

Test if the function f(x)={6x−5 when x≥15x−6 when x<1f(x)=\left\{\begin{array}{c}6 x-5 \text { when } x \geq 1 \\ 5 x-6 \text { when } x<1\end{array}\right.f(x)={6x−5 when x≥15x−6 when x<1​ is differentiable at x=1x=1x=1.

Problem 28207

Is the function f(x)={3x−2 if x≥12x−3 if x<1f(x)=\left\{\begin{array}{l}3 x-2 \text { if } x \geq 1 \\ 2 x-3 \text { if } x<1\end{array}\right.f(x)={3x−2 if x≥12x−3 if x<1​ differentiable at x=1x=1x=1?

Problem 28208

Find the time ttt (from 8 to 12) when the processing rate P(t)=t3−30t2+298t−976P(t)=t^{3}-30 t^{2}+298 t-976P(t)=t3−30t2+298t−976 is highest. Justify your answer.

Problem 28209

Find the absolute maximum of f(x)=x3−3x2+13f(x)=x^{3}-3 x^{2}+13f(x)=x3−3x2+13 on the interval [−2,4][-2,4][−2,4] using calculus.

Problem 28210

Is the function f(x)={(x−1)sin⁡(1x−1)x≠1−1x=1f(x)=\left\{\begin{array}{cc}(x-1) \sin \left(\frac{1}{x-1}\right) & x \neq 1 \\ -1 & x=1\end{array}\right.f(x)={(x−1)sin(x−11​)−1​x=1x=1​ continuous on R\mathrm{R}R?

Problem 28211

Is the function f(x)={(x−1)sin⁡(1x−1)x≠1−1x=1f(x)=\begin{cases} (x-1) \sin \left(\frac{1}{x-1}\right) & x \neq 1 \\ -1 & x=1 \end{cases}f(x)={(x−1)sin(x−11​)−1​x=1x=1​ continuous on R?

Problem 28212

Problem 28213

Find where the tangent line to y=f(x)=x3ln⁡(x)y=f(x)=x^{3} \ln (x)y=f(x)=x3ln(x) is horizontal and if it intersects the graph at (1,0)(1,0)(1,0).

Problem 28214

Find dydx\frac{d y}{d x}dxdy​ using implicit differentiation for the curve x3−3x2y+2xy2=12x^{3}-3 x^{2} y+2 x y^{2}=12x3−3x2y+2xy2=12.

Problem 28215

A soccer ball's height is given by h(t)=−4.9t2+14.7t+0.5h(t)=-4.9 t^{2}+14.7 t+0.5h(t)=−4.9t2+14.7t+0.5. Verify if it peaks at t=1.5t=1.5t=1.5 s using rates of change. i) Estimate IRoC at t=1.5t=1.5t=1.5 s. ii) Find RoC before t=1.5t=1.5t=1.5 s. iii) Find RoC after t=1.5t=1.5t=1.5 s.

Problem 28216

Problem 28217

Is the function f(x)=∣x−2∣f(x)=|x-2|f(x)=∣x−2∣ differentiable at the point x=2x=2x=2?

Problem 28218

Find the difference quotient for f(x)=xf(x)=xf(x)=x at the point (3,3)(3,3)(3,3). Choose from: a) 3−hh\frac{3-h}{h}h3−h​, b) 1, c) 3, d) h\mathrm{h}h.

Problem 28219

Find the difference quotient for f(x)=x2f(x)=x^{2}f(x)=x2 at the point (5,25)(5,25)(5,25). Choose from the options provided.

Problem 28220

Estimate the derivative of f(x)=3xf(x)=\sqrt{3x}f(x)=3x​ at the point x=7x=7x=7.

Problem 28221

Problem 28222

Problem 28223

Is the function f(x)=∣x−2∣f(x)=|x-2|f(x)=∣x−2∣ differentiable at the point x=2x=2x=2?

Problem 28224

Find the derivative f′(x)f^{\prime}(x)f′(x) of f(x)=4x2+6x+3f(x)=4x^{2}+6x+3f(x)=4x2+6x+3 using the limit definition.

Problem 28225

Find the average rate of change of y=2cos⁡(x−π3)+1y=2 \cos \left(x-\frac{\pi}{3}\right)+1y=2cos(x−3π​)+1 for these intervals: a) π6≤x≤π2\frac{\pi}{6} \leq x \leq \frac{\pi}{2}6π​≤x≤2π​, b) π2≤x≤5π4\frac{\pi}{2} \leq x \leq \frac{5 \pi}{4}2π​≤x≤45π​.

Problem 28226

Problem 28227

Evaluate the integral from 2 to 7 of the function 8xex2+4y8 x e^{x^{2}+4 y}8xex2+4y with respect to xxx.

Problem 28228

Evaluate the double integral: ∫23∫xxyx dy dx\int_{2}^{3} \int_{\sqrt{x}}^{x} \frac{y}{x} \, dy \, dx∫23​∫x​x​xy​dydx

Problem 28229

Problem 28230

Evaluate the integral: ∫13(x4+xy−y2)dx\int_{1}^{3}\left(x^{4}+x y-y^{2}\right) d x∫13​(x4+xy−y2)dx

Problem 28231

Find the length of the curve f(x)=x312+1xf(x)=\frac{x^{3}}{12}+\frac{1}{x}f(x)=12x3​+x1​ for 1≤x≤y1 \leq x \leq y1≤x≤y.

Problem 28232

Problem 28233

Find dydx\frac{d y}{d x}dxdy​ for x=2sin⁡(y3)x=2 \sin \left(\frac{y}{3}\right)x=2sin(3y​), and express your answer in terms of xxx.

Problem 28234

Evaluate the integral ∬R6ydA\iint_{R} 6 y d A∬R​6ydA for the region RRR between y=6−x2y=6-x^{2}y=6−x2 and the lines y=xy=xy=x, y=−xy=-xy=−x.

Problem 28235

Problem 28236

Evaluate the double integral ∬R3x+6y3dA\iint_{\mathbb{R}} \sqrt[3]{3 x+6 y} d \mathbf{A}∬R​33x+6y​dA over the region 1≤x≤61 \leq x \leq 61≤x≤6, 0≤y≤20 \leq y \leq 20≤y≤2.

Problem 28237

Assignment: Teach the TeacherExplain how to find the maximum instantaneous rate of change for P(t)=0.5sin⁡(π6t)+4P(t)=0.5 \sin \left(\frac{\pi}{6} t\right)+4P(t)=0.5sin(6π​t)+4 using centered intervals. Compare accuracy of estimates for intervals: i) 1 h, ii) 0.5 h, iii) 0.25 h.

Problem 28238

Identify two points on the graph of f(x)=4cos⁡(x−π4)+2f(x)=4\cos(x-\frac{\pi}{4})+2f(x)=4cos(x−4π​)+2 where the instantaneous rate of change is: a) zero, b) negative, c) positive.

Problem 28239

Evaluate the integral: ∫−23(3x4y3+5xy−6) dx\int_{-2}^{3}(3 x^{4} y^{3}+5 x y-6) \, dx∫−23​(3x4y3+5xy−6)dx

Problem 28240

Evaluate the integral: ∫−23(3x4y3+5xy−6)dx\int_{-2}^{3}(3 x^{4} y^{3}+5 x y-6) dx∫−23​(3x4y3+5xy−6)dx

Problem 28241

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=(x2+3x+1)ex2f(x)=(x^{2}+3x+1)e^{x^{2}}f(x)=(x2+3x+1)ex2.

Problem 28242

Evaluate the double integral: ∫34∫1212x3y2 dx dy\int_{3}^{4} \int_{1}^{2} 12 x^{3} y^{2} \, dx \, dy∫34​∫12​12x3y2dxdy

Problem 28243

Problem 28244

Berechnen Sie die Untersumme U10U_{10}U10​ für das Integral ∫020,5x dx\int_{0}^{2} 0,5 x \, dx∫02​0,5xdx mit 10 gleichen Intervallen.

Find the derivative $f^{\prime}(x)$ of $f(x)=\frac{2}{5 x+5}$ using the limit definition.

Find when the instantaneous rate of change (IRoc) of $N=200(2)^{0.2 t}$ is least, and estimate the IRoc at that time.

Prove if the soccer ball reaches its highest point at $t=1.5$ s using the height function $h(t)=-4.9 t^{2}+14.7 t+0.5$ .

Find $S(16)$ and $S^{\prime}(16)$ for $S(t)=\sqrt{t}+7$ . Estimate $S(17)$ and $S(18)$ , rounding to 1 decimal place.

Test if the function $f(x)=\left\{\begin{array}{c}6 x-5 \text { when } x \geq 1 \\ 5 x-6 \text { when } x<1\end{array}\right.$ is differentiable at $x=1$ .

Is the function $f(x)=\left\{\begin{array}{l}3 x-2 \text { if } x \geq 1 \\ 2 x-3 \text { if } x<1\end{array}\right.$ differentiable at $x=1$ ?

Find the time $t$ (from 8 to 12) when the processing rate $P(t)=t^{3}-30 t^{2}+298 t-976$ is highest. Justify your answer.

Find the absolute maximum of $f(x)=x^{3}-3 x^{2}+13$ on the interval $[-2,4]$ using calculus.

Is the function $f(x)=\left\{\begin{array}{cc}(x-1) \sin \left(\frac{1}{x-1}\right) & x \neq 1 \\ -1 & x=1\end{array}\right.$ continuous on $\mathrm{R}$ ?

Is the function $f(x)=\begin{cases} (x-1) \sin \left(\frac{1}{x-1}\right) & x \neq 1 \\ -1 & x=1 \end{cases}$ continuous on R?

Find where the tangent line to $y=f(x)=x^{3} \ln (x)$ is horizontal and if it intersects the graph at $(1,0)$ .

Find $\frac{d y}{d x}$ using implicit differentiation for the curve $x^{3}-3 x^{2} y+2 x y^{2}=12$ .

A soccer ball's height is given by $h(t)=-4.9 t^{2}+14.7 t+0.5$ . Verify if it peaks at $t=1.5$ s using rates of change. i) Estimate IRoC at $t=1.5$ s. ii) Find RoC before $t=1.5$ s. iii) Find RoC after $t=1.5$ s.

Is the function $f(x)=|x-2|$ differentiable at the point $x=2$ ?

Find the difference quotient for $f(x)=x$ at the point $(3,3)$ . Choose from: a) $\frac{3-h}{h}$ , b) 1, c) 3, d) $\mathrm{h}$ .

Find the difference quotient for $f(x)=x^{2}$ at the point $(5,25)$ . Choose from the options provided.

Estimate the derivative of $f(x)=\sqrt{3x}$ at the point $x=7$ .

Is the function $f(x)=|x-2|$ differentiable at the point $x=2$ ?

Find the derivative $f^{\prime}(x)$ of $f(x)=4x^{2}+6x+3$ using the limit definition.

Find the average rate of change of $y=2 \cos \left(x-\frac{\pi}{3}\right)+1$ for these intervals: a) $\frac{\pi}{6} \leq x \leq \frac{\pi}{2}$ , b) $\frac{\pi}{2} \leq x \leq \frac{5 \pi}{4}$ .

Evaluate the integral from 2 to 7 of the function $8 x e^{x^{2}+4 y}$ with respect to $x$ .

Evaluate the double integral: $\int_{2}^{3} \int_{\sqrt{x}}^{x} \frac{y}{x} \, dy \, dx$

Evaluate the integral: $\int_{1}^{3}\left(x^{4}+x y-y^{2}\right) d x$

Find the length of the curve $f(x)=\frac{x^{3}}{12}+\frac{1}{x}$ for $1 \leq x \leq y$ .

Find $\frac{d y}{d x}$ for $x=2 \sin \left(\frac{y}{3}\right)$ , and express your answer in terms of $x$ .

Evaluate the integral $\iint_{R} 6 y d A$ for the region $R$ between $y=6-x^{2}$ and the lines $y=x$ , $y=-x$ .

Evaluate the double integral $\iint_{\mathbb{R}} \sqrt[3]{3 x+6 y} d \mathbf{A}$ over the region $1 \leq x \leq 6$ , $0 \leq y \leq 2$ .

Assignment: Teach the Teacher
Explain how to find the maximum instantaneous rate of change for $P(t)=0.5 \sin \left(\frac{\pi}{6} t\right)+4$ using centered intervals. Compare accuracy of estimates for intervals: i) 1 h, ii) 0.5 h, iii) 0.25 h.

Identify two points on the graph of $f(x)=4\cos(x-\frac{\pi}{4})+2$ where the instantaneous rate of change is: a) zero, b) negative, c) positive.

Evaluate the integral: $\int_{-2}^{3}(3 x^{4} y^{3}+5 x y-6) \, dx$

Evaluate the integral: $\int_{-2}^{3}(3 x^{4} y^{3}+5 x y-6) dx$

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

Evaluate the double integral: $\int_{3}^{4} \int_{1}^{2} 12 x^{3} y^{2} \, dx \, dy$

Berechnen Sie die Untersumme $U_{10}$ für das Integral $\int_{0}^{2} 0,5 x \, dx$ mit 10 gleichen Intervallen.

Ein Gartenpool wird mit $v(t)=0,1 t^{2}-\frac{1}{t^{3}+1}+40$ gefüllt. Bestimmen Sie die Füllhöhe und das Volumen nach 10 Minuten.

Determine where the average rate of change of $y=f(x)$ is positive, negative, or zero based on its graph.

1. Find the average rate of change of $h(t)=18t-0.8t^{2}$ from $t=10$ to $t=15$ for the golf ball's height.
2. A car worth \$23500 depreciates to \$8750 in 8 years. Calculate the average annual percent decrease in value.

Evaluate the double integral $\iint_{R} 4 x^{3} y^{2} \, dA$ where $R=\{(x, y) | 0 \leq x \leq 3, 0 \leq y \leq 2\}$ .

Population function: $P(t)=-1.5 t^{2}+36 t+6$ . Find average change from 2000 to 2024, and check changes for 2000-2012 & 2012-2024. When is the rate 0?

A rock sample of $256 \mathrm{~g}$ tungsten-187 decays to $0.25 \mathrm{~g}$ . With a half-life of 1 day, how long does this take?

Find the average rate of change of area with respect to radius for a circle from $0 \mathrm{~cm}$ to $100 \mathrm{~cm}$ and at $120 \mathrm{~cm}$ .

A diver on a $10 \mathrm{~m}$ platform dives with height $h(t)=10+2t-4.9t^{2}$ . Find when they enter the water and the height change rate.

Find the limit: $\lim _{x \rightarrow 1} \frac{\frac{1}{\sqrt{x}}-1}{x-1}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{5}-\sqrt{x+5}}{x}$ .

Find the derivative of the function $y=\frac{3x-2}{2-5x}$ .

A ship's height is modeled by $h(t)=\sin \frac{\pi}{5} t$ . Find the average rate of change over 5s and the instantaneous rate at $t=6s$ .

A pilot ejects from 10,000 ft, with altitude $h(t)=-16 t^{2}+90 t+10000$ . Find time $t$ for max altitude and relate it to slope.

A ship's height is modeled by $h(t) = \sin \frac{\pi}{5} t$ . Find the average rate of change over 5s and the instantaneous rate at $t=6s$ .

Solve the differential equation $(4 x+y+3) dy=(4 x+y+1)^{2} dx$ . What is the solution? (a), (b), (c), or (d)?

A ship's height $h(t) = \sin \frac{\pi}{5} t$ in meters varies with time $t$ in seconds. Find the average rate of change over 5s and the instantaneous rate at $t=6s$ . Mark Value: 4

A radioactive substance decays exponentially. Given $P(t)=100(1.2)^{-t}$ , find the half-life and rate of decay after the first half-life.

Find values of $a$ and $b$ if $(1,2)$ is a relative extremum of $P(x)=2ax^2-3bx+3$ .

The town's population is 12000 and decreases at $1.8\%$ per year. Find the population model and rates of change in 10 years and at half population.

Untersuchen Sie die Flächeninhalte der Funktionen: I. $f(x)=\frac{1}{x^{3}}$ , II. $f(x)=\frac{1}{x^{2}}$ , III. $f(x)=\frac{1}{\sqrt{x}}$ für $x \geq 1$ .

The town's population is 12000 and decreases at $1.8\%$ per year. Find the population model and rates of change in 10 years and at half population.

Find the average wind speed change from mile 10 to mile 100 using $S(d)=93 \log d+65$ . Also, estimate speed changes at miles 10 and 100.

The town's population is 12,000 and decreases by $1.8\%$ per year.
a) Model the population with an equation. b) Find the rate of change in 10 years. c) Find the rate when the population is 6,000.
Mark Value: 3

The town's population is 12000 and decreases at $1.8\%$ per year.
a) Model the population with an equation. b) Find the rate of change in 10 years. c) Find the rate when the population is 6000.

Find the average wind speed change from mile 10 to 100 using $S(d)=93 \log d+65$ . Also, estimate speed change at miles 10 and 100.

Demand for snack cakes is $p(x)=\frac{15}{2 x^{2}+11 x+5}$ . Find revenue function and marginal revenue at $x=0.75$ and $x=2$ .

Calculate the average wind speed change from mile 10 to mile 100 using $S(d)=93 \log d+65$ . Also, find the speed change at miles 10 and 100.

Bestimme die Ableitungen der Funktionen: a) $f(x)=5 e^{7 x^{2}+4 x}$ b) $g(x)=(2 x^{4}-7 x)(x^{3}+x)$ c) $h(x)=3(2 x^{4}+7)^{13}$

Find the limit: $\lim _{x \rightarrow 0} \frac{\frac{4}{x+2}-2}{x}$