Calculus

Problem 2301

Find the limit as $x$ approaches -10 for the expression $\frac{x^{2}+20 x+100}{x+10}$ . If it doesn't exist, write "DNE".

See Solution

Problem 2302

Given $f(5)=1$ , $f^{\prime}(5)=7$ , $g(5)=-8$ , and $g^{\prime}(5)=6$ , find: (a) $(f g)^{\prime}(5)$ (b) $\left(\frac{f}{g}\right)^{\prime}(5)$ (c) $\left(\frac{g}{f}\right)^{\prime}(5)$

See Solution

Problem 2303

Find the derivative of $f(x)=5 x^{2} \cos (x)-4 x$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 2304

Find the derivative of the function $f(t)=\frac{\sqrt[3]{t}}{t-6}$ . What is $f^{\prime}(t)$ ?

See Solution

Problem 2305

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

See Solution

Problem 2306

Find the derivative of $G(x)=(-8 x^{3}+5 x^{2})^{5}$ . What is $G^{\prime}(x)$ ?

See Solution

Problem 2307

Evaluate the limit: $\lim _{x \rightarrow 16} \frac{x-16}{x^{2}-256}=$ (Enter "DNE" if it doesn't exist.)

See Solution

Problem 2308

Find the derivative $g^{\prime}(-1)$ for the function $g(x)=\frac{1}{4 x^{2}+3 x}$ .

See Solution

Problem 2309

Find the limit: $\lim _{x \rightarrow 1} \frac{\sqrt{x+1}-1}{x-1}$ . If it doesn't exist, write DNE.

See Solution

Problem 2310

Classify the asymptotes for the functions: (a) $y=\frac{\sqrt{x}+5 x}{6 \sqrt{x}-2 x}$ , (b) $y=\sqrt{16 x^{2}+2 x}-\sqrt{16 x^{2}-4 x}$ . Use limits for justification.

See Solution

Problem 2311

Find the limit $\lim _{x \rightarrow \infty} \frac{x^{2}}{2^{x}}$ by evaluating it at $x=0,1,2,\ldots,100$ and sketch a graph.

See Solution

Problem 2312

Evaluate the limit: $\lim _{x \rightarrow 11} \frac{2 x^{2}-6 x-176}{x^{2}-121}=$ (Enter "DNE" if it doesn't exist.)

See Solution

Problem 2313

Evaluate the limit: $\lim _{x \rightarrow 9} \frac{81-x^{2}}{x-9}$ using algebraic transformation and continuity. Enter "DNE" if it doesn't exist.

See Solution

Problem 2314

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{(5+2 x)^{3}-125}{x}=$ (Enter "DNE" if it doesn't exist.)

See Solution

Problem 2315

Find $\lim _{x \rightarrow 8} f(x)$ given the graph with points (0,0), (8,4), and (10,6.5).

See Solution

Problem 2316

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

See Solution

Problem 2317

Evaluate the limit as $x$ approaches -4 for $\frac{x^{2}+8x+16}{x+4}$ . If it doesn't exist, enter "DNE". $\lim _{x \rightarrow-4} \frac{x^{2}+8 x+16}{x+4}=$

See Solution

Problem 2318

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

See Solution

Problem 2319

Find $h'(0)$ for $h(x)=f(x) \times g(x)$ given $f(0)=-6$ , $f'(0)=5$ , $g(0)=2$ , $g'(0)=-2$ .

See Solution

Problem 2320

Evaluate the limit: $\lim _{x \rightarrow 9} \frac{x-9}{x^{2}-81}=$ (Enter "DNE" if it doesn't exist.)

See Solution

Problem 2321

Find the limit: $\lim _{x \rightarrow 7} \frac{2 x^{2}+8 x-154}{x^{2}-49}=$ (Enter "DNE" if it doesn't exist.)

See Solution

Problem 2322

Find the limit: $\lim _{x \rightarrow 100} \frac{\sqrt{x}-10}{x-100}=$ . If it doesn't exist, enter "DNE".

See Solution

Problem 2323

Find the derivative of $f(x) = 3x^{2}$ using the limit: $\lim_{\Delta x \to 0} \frac{3(x+\Delta x)^{2}-3 x^{2}}{\Delta x}$ .

See Solution

Problem 2324

Evaluate the limit: $\lim _{x \rightarrow 49} \frac{\sqrt{50-x}-1}{7-\sqrt{x}} \cdot 4.$

See Solution

Problem 2325

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{2+h}-10}{h}=$ (Enter "DNE" if it doesn't exist.)

See Solution

Problem 2326

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{(7+5 x)^{3}-343}{x}=$ (Enter "DNE" if it doesn't exist.)

See Solution

Problem 2327

Evaluate the limit: $\lim _{x \rightarrow 25} \frac{\sqrt{x}-5}{x-25}=$ (Enter "DNE" if it doesn't exist.)

See Solution

Problem 2328

Find the 35th derivative of the function $f(x) = x \cos x$ .

See Solution

Problem 2329

Evaluate the limit: $\lim _{x \rightarrow-4} \frac{x^{3}+64}{x^{2}+15 x+44}$ . Use $a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})$ .

See Solution

Problem 2330

Find the limit as $x$ approaches 9 for the expression $9 \cdot a + x$ . Enter "DNE" if it doesn't exist.

See Solution

Problem 2331

Find the limit as $t$ approaches 5 for $4t + 2at + 8a$ . Enter "DNE" if it doesn't exist. $\lim _{t \rightarrow 5}(4 t+2 a t+8 a) =$

See Solution

Problem 2332

Evaluate the limit: $\lim _{t \rightarrow-3} \frac{4 t+12}{144-16 t^{2}}=$ (Exact answer, use DNE if it doesn't exist.)

See Solution

Problem 2333

Evaluate the limit:
$\lim _{x \rightarrow 2}\left(\frac{1}{x-2}-\frac{12}{x^{2}+8 x-20}\right)$
Choose the justification for your answer.

See Solution

Problem 2334

Find the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{x+h+9}-\sqrt{x+9}}{15 h}$ .

See Solution

Problem 2335

Evaluate the limit: $\lim _{x \rightarrow \frac{\pi}{6}} \frac{\csc (6 x)}{\cot (6 x)}$ . Justify your answer.

See Solution

Problem 2336

Evaluate the limit: $\lim _{\theta \rightarrow \frac{\pi}{2}}(3 \sec (\theta)-3 \tan (\theta))=$

See Solution

Problem 2337

Evaluate the limit: $\lim _{x \rightarrow-4} \frac{x^{3}+64}{x^{2}+13 x+36}$ using $a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})$ .

See Solution

Problem 2338

Evaluate the limit as $x$ approaches 6 for $9 \cdot a + x$ . Enter "DNE" if it doesn't exist. $\lim _{x \rightarrow 6}(9 \cdot a+x) =$

See Solution

Problem 2339

Approximate the limit of the piecewise function $f(x)=\begin{cases} x+8 & x \leq 2 \\ 3x+2 & x>2 \end{cases}$ as $x \to 2$ .

See Solution

Problem 2340

Find the limit as $x$ approaches $4a$ : $\lim _{x \rightarrow 4 a} \frac{\frac{3}{x}-\frac{3}{a}}{x-a}=$

See Solution

Problem 2341

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{x+h+4}-\sqrt{x+4}}{18 h}$ .

See Solution

Problem 2342

Evaluate the limits of the piecewise function:
$f(x)=\left\{\begin{array}{ll} 4 x+8 & x \leq 1 \\ x^{2}-6 & x>1 \end{array}\right.$
Find:
$\lim _{x \rightarrow 1^{-}} f(x), \quad \lim _{x \rightarrow 1^{+}} f(x), \quad \lim _{x \rightarrow 1} f(x)$
Enter DNE if a limit does not exist.

See Solution

Problem 2343

Find the limit: $\lim _{x \rightarrow 1}\left(\frac{1}{x-1}-\frac{9}{x^{2}+7 x-8}\right)$ . Provide the exact answer or DNE.

See Solution

Problem 2344

Define the function $f(x)=\begin{cases} x+6 & \text{if } x<0 \\ 6 & \text{if } x>0 \end{cases}$ . Sketch its graph and find: 1. $\lim _{x \rightarrow 0^{-}} f(x)$ , 2. $\lim _{x \rightarrow 0^{+}} f(x)$ , 3. $\lim _{x \rightarrow 0} f(x)$ .

See Solution

Problem 2345

Evaluate the limit: $\lim _{x \rightarrow \frac{\pi}{6}} \frac{\csc (6 x)}{\cot (6 x)}$ . Provide the answer in exact form or DNE if it doesn't exist.

See Solution

Problem 2346

Find the derivative of $G(t)=\frac{6t}{t+7}$ using the definition and state its domain in interval notation.

See Solution

Problem 2347

Determine if the derivative $f^{\prime}(0)$ exists for the function $f(x)=x^{2} \sin \left(\frac{3}{x}\right)$ when $x \neq 0$ and $f(0)=0$ . If it exists, find its value.

See Solution

Problem 2348

Find the marginal revenue at 96 units for the revenue function $R(q)=-2 q^{2}+900 q$ . Calculate $M R(96)=\$$ per unit.

See Solution

Problem 2349

Find the limit: $\lim _{h \rightarrow 0} \frac{7 f(x+h)-7 f(x-h)}{h}$ for a differentiable function $f$ .

See Solution

Problem 2350

Find the marginal cost of producing 1000 stuffed alligator toys given $C(x)=0.004 x^{2}+6 x+7000$ .

See Solution

Problem 2351

Find the extreme values of $z=39x+10y+90$ with constraints $y \geq 0$ and $y \leq 25-x^{2}$ .

See Solution

Problem 2352

Find the extreme values of $z=39x+10y+90$ with constraints $y \geq 0$ and $y \leq 25-x^{2}$ . Complete: $f_{\min }=-1$ at $(-5, 0)$ , $f_{\max }=\square$ at $(\square, \square)$ .

See Solution

Problem 2353

Find the average velocity for $L(t)=t^{2}-t-8$ in these intervals:
1. $[5, 8]$
2. $[1, 6]$
3. $[2, 4]$ (3+ decimal places)

See Solution

Problem 2354

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

See Solution

Problem 2355

What is the expression for $f^{\prime}(a)$ for a differentiable function $f$ ? A, B, or C?

See Solution

Problem 2356

Find the expression for $f^{\prime}(a)$ if $f$ is differentiable: A. $\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$ , B. $\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}$ , C. $\lim _{x \rightarrow a} \frac{f(x+h)-f(x)}{h}$ .

See Solution

Problem 2357

Find the derivative of the function $y=\frac{x}{4-5 x^{4}}$ in simplified form.

See Solution

Problem 2358

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

See Solution

Problem 2359

Evaluate the integral: $\int_{0}^{1} 2 x^{3} \ln \left(x^{2}+1\right) d x$

See Solution

Problem 2360

A ball is dropped from 79 m. Find when it hits the ground, its velocity $v(t)$ , and acceleration $a(t)$ .

See Solution

Problem 2361

A particle's position is $s(t)=2 t^{3}-30 t^{2}+126 t$ . Find velocity at $t=0$ , rest times, position at $t=20$ , and total distance from $t=0$ to $t=20$ .

See Solution

Problem 2362

A particle's position is $s(t)=2 t^{3}-15 t^{2}+24 t$ . Find its velocity at $t=0$ , when it stops, its position at $t=10$ , and total distance from $t=0$ to $t=10$ .

See Solution

Problem 2363

A particle's position is $s(t)=2 t^{3}-24 t^{2}+42 t$ . Find its velocity at $t=0$ , when it stops, its position at $t=16$ , and total distance from $t=0$ to $t=16$ .

See Solution

Problem 2364

Compute the surface integral $\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S$ for the given vector field $\mathbf{F}$ .

See Solution

Problem 2365

Compute the surface integral $\iint_{S} \mathbf{F} \cdot \mathbf{n} d S$ for $\mathbf{F}(x, y, z)=\langle x, 1-y+e^{z}, y-e^{z}\rangle$ on $y+z=10$ inside $x^{2}+y^{2}=1$ .

See Solution

Problem 2366

Calculate the work done to compress a spring from $40 \mathrm{~cm}$ to $10 \mathrm{~cm}$ with a force of $60 \mathrm{~N}$ .

See Solution

Problem 2367

Consider $\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|^{3}}$ . Which statement is true: (i) $\int_{C} \mathbf{F} \cdot d \mathbf{r}$ path-independent, (ii) $\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=0$ for closed $S$ , (iii) $\operatorname{div}(\mathbf{F})=0$ ? A. None, B. (i) and (ii), C. (i) and (iii), D. (ii) and (iii), E. All.

See Solution

Problem 2368

Find the volume of the solid formed by rotating the area between $x=y^{3}$ and $x=y$ in the first quadrant about the $y$ -axis using the washer method.

See Solution

Problem 2369

Find the sixth derivative of $y=x^{2} \sin x$ at $x=0$ .

See Solution

Problem 2370

Calculate the integral of $x^{2}$ with respect to $x$ : $\int x^{2} d x=$

See Solution

Problem 2371

Find the integral of $x^{7}$ with respect to $x$ : $\int x^{7} d x=$

See Solution

Problem 2372

Solve for $2 x d x = \int\left(e^{x}+\operatorname{cosec}^{2} x+\sec x \cdot \tan x\right) d x$ .

See Solution

Problem 2373

Find the integral: $\int\left(e^{x}+\operatorname{cosec}^{2} x+\sec x \cdot \tan x\right) d x$

See Solution

Problem 2374

Calculate the integral $\int_{0}^{?} e^{5x+11} dx$ .

See Solution

Problem 2375

Calculate the integral: $\int 2^{\log _{4} 2} dx$ .

See Solution

Problem 2376

Calculate the integral: $\int 2^{\log _{4} x} dx$

See Solution

Problem 2377

Calculate the integral: $\int 2^{\log^{x} x} \, dx$

See Solution

Problem 2378

Find the integral of $5^{\ln x}$ with respect to $x$ : $\int 5^{\ln x} d x=$

See Solution

Problem 2379

Find the integral of the function $(x^{2}-2)^{2}$ with respect to $x$ : $\int (x^{2}-2)^{2} \, dx$ .

See Solution

Problem 2380

Calculate the integral: $\int\left(1-\frac{3}{x}\right) d x$ .

See Solution

Problem 2381

Evaluate the integral: $\int\left(\frac{1}{x^{4}}+\frac{1}{\sqrt[4]{x}}\right) d x$

See Solution

Problem 2382

Calculate the integral: $\int \frac{x^{4}+2 x^{2}-1}{\sqrt{x}} \, dx$

See Solution

Problem 2383

Evaluate the integral: $\int x \sqrt[3]{(4-x^{2})^{2}} \, dx$

See Solution

Problem 2384

Evaluate the integral: $\int \frac{1+\cos x}{x+\sin x} d x$

See Solution

Problem 2385

Find the integral of $\frac{\sec ^{2} x}{\tan x}$ with respect to $x$ .

See Solution

Problem 2386

Find $j^{\prime}(-2)$ for $j(x)=\frac{f(x)}{g(x)}$ using the quotient rule with given values for $f$ and $g$ .

See Solution

Problem 2387

Find $h^{\prime}(-5)$ for $h(x)=f(x) \cdot g(x)$ , where $f(x)=9-5 x^{2}$ , $g(-5)=-4$ , $g^{\prime}(-5)=-2$ .

See Solution

Problem 2388

Find $h^{\prime}(-3)$ for $h(x)=\frac{f(x)}{g(x)}$ , where $f(x)=x^{2}+8$ , $g(-3)=8$ , $g^{\prime}(-3)=-3$ . Answer: $h^{\prime}(-3)=$

See Solution

Problem 2389

Find $j^{\prime}(8)$ for $j(x)=f(x) \cdot g(x)$ using the product rule, given $f(8)=-9$ , $f^{\prime}(8)=8$ , $g(8)=0$ , $g^{\prime}(8)=-3$ . Answer: $j^{\prime}(8)=$

See Solution

Problem 2390

Find $k^{\prime}(0)$ for $k(x)=(3x^{2})g(x)h(x)$ using the given values for $g(x)$ and $h(x)$ .

See Solution

Problem 2391

Find $k^{\prime}(3)$ for $k(x)=\frac{x^{2}(f(x))}{1-2x}$ , given $f(3)=1$ and $f^{\prime}(3)=2$ .

See Solution

Problem 2392

Find the derivative $h^{\prime}(x)$ using the quotient rule for $h(x)=-\frac{7 x^{2}}{-10 x^{2}+4 x-3}$ .

See Solution

Problem 2393

Find $k'(3)$ for $k(x)=\frac{x^{2}(f(x))}{1-2x}$ , given $f(3)=1$ and $f'(3)=2$ .

See Solution

Problem 2394

Find $h^{\prime}(3)$ for $h(x)=\frac{3}{x^{2}}+\frac{f(x)}{g(x)}$ using given values at $x=3$ .

See Solution

Problem 2395

Find $h^{\prime}(3)$ for $h(x)=\frac{3}{x^{2}}+\frac{f(x)}{g(x)}$ using values: $f(3)=-4$ , $g(3)=-4$ , $f^{\prime}(3)=1$ , $g^{\prime}(3)=3$ .

See Solution

Problem 2396

Find $h^{\prime}(8)$ for $h(x)=f(x) \cdot g(x)$ where $f(x)=-4x^{2}-5$ , $g(8)=2$ , $g^{\prime}(8)=8$ .

See Solution

Problem 2397

Find $k^{\prime}(-1)$ for $k(x)=-2 x(h(x)+2 x^{2} g(x))$ using given values for $h(x)$ and $g(x)$ .

See Solution

Problem 2398

Find $k^{\prime}(-1)$ for $k(x)=-2 x(h(x)+2 x^{2} g(x))$ using given values of $h(x)$ and $g(x)$ .

See Solution

Problem 2399

Find $k'(2)$ for $k(x)=5h(x)+\frac{3x^2}{g(x)}$ using given values of $h$ , $g$ , $h'$ , and $g'$ .

See Solution

Problem 2400

Find $k^{\prime}(2)$ for $k(x)=(-x^{2}) g(x) h(x)$ using given values for $g(x)$ and $h(x)$ .

See Solution

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Calculus

Problem 2301

Find the limit as xxx approaches -10 for the expression x2+20x+100x+10\frac{x^{2}+20 x+100}{x+10}x+10x2+20x+100​. If it doesn't exist, write "DNE".

Problem 2302

Problem 2303

Find the derivative of f(x)=5x2cos⁡(x)−4xf(x)=5 x^{2} \cos (x)-4 xf(x)=5x2cos(x)−4x. What is f′(x)f^{\prime}(x)f′(x)?

Problem 2304

Find the derivative of the function f(t)=t3t−6f(t)=\frac{\sqrt[3]{t}}{t-6}f(t)=t−63t​​. What is f′(t)f^{\prime}(t)f′(t)?

Problem 2305

Find the derivative of the function f(x)=−2x+43x−4f(x)=-\frac{2 x+4}{3 x-4}f(x)=−3x−42x+4​.

Problem 2306

Find the derivative of G(x)=(−8x3+5x2)5G(x)=(-8 x^{3}+5 x^{2})^{5}G(x)=(−8x3+5x2)5. What is G′(x)G^{\prime}(x)G′(x)?

Problem 2307

Evaluate the limit: lim⁡x→16x−16x2−256=\lim _{x \rightarrow 16} \frac{x-16}{x^{2}-256}=x→16lim​x2−256x−16​= (Enter "DNE" if it doesn't exist.)

Problem 2308

Find the derivative g′(−1)g^{\prime}(-1)g′(−1) for the function g(x)=14x2+3xg(x)=\frac{1}{4 x^{2}+3 x}g(x)=4x2+3x1​.

Problem 2309

Find the limit: lim⁡x→1x+1−1x−1\lim _{x \rightarrow 1} \frac{\sqrt{x+1}-1}{x-1}limx→1​x−1x+1​−1​. If it doesn't exist, write DNE.

Problem 2310

Classify the asymptotes for the functions: (a) y=x+5x6x−2xy=\frac{\sqrt{x}+5 x}{6 \sqrt{x}-2 x}y=6x​−2xx​+5x​, (b) y=16x2+2x−16x2−4xy=\sqrt{16 x^{2}+2 x}-\sqrt{16 x^{2}-4 x}y=16x2+2x​−16x2−4x​. Use limits for justification.

Problem 2311

Find the limit lim⁡x→∞x22x\lim _{x \rightarrow \infty} \frac{x^{2}}{2^{x}}limx→∞​2xx2​ by evaluating it at x=0,1,2,…,100x=0,1,2,\ldots,100x=0,1,2,…,100 and sketch a graph.

Problem 2312

Evaluate the limit: lim⁡x→112x2−6x−176x2−121=\lim _{x \rightarrow 11} \frac{2 x^{2}-6 x-176}{x^{2}-121}=x→11lim​x2−1212x2−6x−176​= (Enter "DNE" if it doesn't exist.)

Problem 2313

Evaluate the limit: lim⁡x→981−x2x−9\lim _{x \rightarrow 9} \frac{81-x^{2}}{x-9}x→9lim​x−981−x2​ using algebraic transformation and continuity. Enter "DNE" if it doesn't exist.

Problem 2314

Evaluate the limit: lim⁡x→0(5+2x)3−125x=\lim _{x \rightarrow 0} \frac{(5+2 x)^{3}-125}{x}=x→0lim​x(5+2x)3−125​= (Enter "DNE" if it doesn't exist.)

Problem 2315

Find lim⁡x→8f(x)\lim _{x \rightarrow 8} f(x)limx→8​f(x) given the graph with points (0,0), (8,4), and (10,6.5).

Problem 2316

Find f′(1)f^{\prime}(1)f′(1) if f(x)=−3h(x)−4xh(x)f(x)=-3h(x)-\frac{4x}{h(x)}f(x)=−3h(x)−h(x)4x​, with h(1)=4h(1)=4h(1)=4 and h′(1)=−3h^{\prime}(1)=-3h′(1)=−3.

Problem 2317

Evaluate the limit as x x x approaches -4 for x2+8x+16x+4 \frac{x^{2}+8x+16}{x+4} x+4x2+8x+16​. If it doesn't exist, enter "DNE". lim⁡x→−4x2+8x+16x+4=\lim _{x \rightarrow-4} \frac{x^{2}+8 x+16}{x+4}=x→−4lim​x+4x2+8x+16​=

Problem 2318

Find f′(1)f^{\prime}(1)f′(1) given f(x)=−3⋅h(x)−4xh(x)f(x)=-3 \cdot h(x)-\frac{4 x}{h(x)}f(x)=−3⋅h(x)−h(x)4x​, h(1)=4h(1)=4h(1)=4, h′(1)=−3h^{\prime}(1)=-3h′(1)=−3.

Problem 2319

Find h′(0)h'(0)h′(0) for h(x)=f(x)×g(x)h(x)=f(x) \times g(x)h(x)=f(x)×g(x) given f(0)=−6f(0)=-6f(0)=−6, f′(0)=5f'(0)=5f′(0)=5, g(0)=2g(0)=2g(0)=2, g′(0)=−2g'(0)=-2g′(0)=−2.

Problem 2320

Evaluate the limit: lim⁡x→9x−9x2−81=\lim _{x \rightarrow 9} \frac{x-9}{x^{2}-81}=x→9lim​x2−81x−9​= (Enter "DNE" if it doesn't exist.)

Problem 2321

Find the limit: lim⁡x→72x2+8x−154x2−49=\lim _{x \rightarrow 7} \frac{2 x^{2}+8 x-154}{x^{2}-49}=x→7lim​x2−492x2+8x−154​= (Enter "DNE" if it doesn't exist.)

Problem 2322

Find the limit: lim⁡x→100x−10x−100=\lim _{x \rightarrow 100} \frac{\sqrt{x}-10}{x-100}=x→100lim​x−100x​−10​=. If it doesn't exist, enter "DNE".

Problem 2323

Find the derivative of f(x)=3x2f(x) = 3x^{2}f(x)=3x2 using the limit: lim⁡Δx→03(x+Δx)2−3x2Δx\lim_{\Delta x \to 0} \frac{3(x+\Delta x)^{2}-3 x^{2}}{\Delta x}limΔx→0​Δx3(x+Δx)2−3x2​.

Problem 2324

Evaluate the limit: lim⁡x→4950−x−17−x⋅4.\lim _{x \rightarrow 49} \frac{\sqrt{50-x}-1}{7-\sqrt{x}} \cdot 4.x→49lim​7−x​50−x​−1​⋅4.

Problem 2325

Evaluate the limit: lim⁡h→02+h−10h=\lim _{h \rightarrow 0} \frac{\sqrt{2+h}-10}{h}=h→0lim​h2+h​−10​= (Enter "DNE" if it doesn't exist.)

Problem 2326

Evaluate the limit: lim⁡x→0(7+5x)3−343x=\lim _{x \rightarrow 0} \frac{(7+5 x)^{3}-343}{x}=x→0lim​x(7+5x)3−343​= (Enter "DNE" if it doesn't exist.)

Problem 2327

Evaluate the limit: lim⁡x→25x−5x−25=\lim _{x \rightarrow 25} \frac{\sqrt{x}-5}{x-25}=x→25lim​x−25x​−5​= (Enter "DNE" if it doesn't exist.)

Problem 2328

Find the 35th derivative of the function f(x)=xcos⁡xf(x) = x \cos xf(x)=xcosx.

Problem 2329

Evaluate the limit: lim⁡x→−4x3+64x2+15x+44\lim _{x \rightarrow-4} \frac{x^{3}+64}{x^{2}+15 x+44}limx→−4​x2+15x+44x3+64​. Use a3+b3=(a+b)(a2−ab+b2)a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})a3+b3=(a+b)(a2−ab+b2).

Problem 2330

Find the limit as xxx approaches 9 for the expression 9⋅a+x9 \cdot a + x9⋅a+x. Enter "DNE" if it doesn't exist.

Problem 2331

Find the limit as ttt approaches 5 for 4t+2at+8a4t + 2at + 8a4t+2at+8a. Enter "DNE" if it doesn't exist. lim⁡t→5(4t+2at+8a)=\lim _{t \rightarrow 5}(4 t+2 a t+8 a) =t→5lim​(4t+2at+8a)=

Problem 2332

Evaluate the limit: lim⁡t→−34t+12144−16t2=\lim _{t \rightarrow-3} \frac{4 t+12}{144-16 t^{2}}=t→−3lim​144−16t24t+12​= (Exact answer, use DNE if it doesn't exist.)

Problem 2333

Evaluate the limit: lim⁡x→2(1x−2−12x2+8x−20) \lim _{x \rightarrow 2}\left(\frac{1}{x-2}-\frac{12}{x^{2}+8 x-20}\right) x→2lim​(x−21​−x2+8x−2012​) Choose the justification for your answer.

Problem 2334

Find the limit: lim⁡h→0x+h+9−x+915h\lim _{h \rightarrow 0} \frac{\sqrt{x+h+9}-\sqrt{x+9}}{15 h}h→0lim​15hx+h+9​−x+9​​.

Problem 2335

Evaluate the limit: lim⁡x→π6csc⁡(6x)cot⁡(6x)\lim _{x \rightarrow \frac{\pi}{6}} \frac{\csc (6 x)}{\cot (6 x)}x→6π​lim​cot(6x)csc(6x)​. Justify your answer.

Problem 2336

Evaluate the limit: lim⁡θ→π2(3sec⁡(θ)−3tan⁡(θ))=\lim _{\theta \rightarrow \frac{\pi}{2}}(3 \sec (\theta)-3 \tan (\theta))=θ→2π​lim​(3sec(θ)−3tan(θ))=

Problem 2337

Evaluate the limit: lim⁡x→−4x3+64x2+13x+36\lim _{x \rightarrow-4} \frac{x^{3}+64}{x^{2}+13 x+36}x→−4lim​x2+13x+36x3+64​ using a3+b3=(a+b)(a2−ab+b2)a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})a3+b3=(a+b)(a2−ab+b2).

Problem 2338

Evaluate the limit as xxx approaches 6 for 9⋅a+x9 \cdot a + x9⋅a+x. Enter "DNE" if it doesn't exist. lim⁡x→6(9⋅a+x)=\lim _{x \rightarrow 6}(9 \cdot a+x) =x→6lim​(9⋅a+x)=

Problem 2339

Approximate the limit of the piecewise function f(x)={x+8x≤23x+2x>2f(x)=\begin{cases} x+8 & x \leq 2 \\ 3x+2 & x>2 \end{cases}f(x)={x+83x+2​x≤2x>2​ as x→2x \to 2x→2.

Problem 2340

Find the limit as xxx approaches 4a4a4a: lim⁡x→4a3x−3ax−a=\lim _{x \rightarrow 4 a} \frac{\frac{3}{x}-\frac{3}{a}}{x-a}=x→4alim​x−ax3​−a3​​=

Find the limit as $x$ approaches -10 for the expression $\frac{x^{2}+20 x+100}{x+10}$ . If it doesn't exist, write "DNE".

Find the derivative of $f(x)=5 x^{2} \cos (x)-4 x$ . What is $f^{\prime}(x)$ ?

Find the derivative of the function $f(t)=\frac{\sqrt[3]{t}}{t-6}$ . What is $f^{\prime}(t)$ ?

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

Find the derivative of $G(x)=(-8 x^{3}+5 x^{2})^{5}$ . What is $G^{\prime}(x)$ ?

Evaluate the limit: $\lim _{x \rightarrow 16} \frac{x-16}{x^{2}-256}=$ (Enter "DNE" if it doesn't exist.)

Find the derivative $g^{\prime}(-1)$ for the function $g(x)=\frac{1}{4 x^{2}+3 x}$ .

Find the limit: $\lim _{x \rightarrow 1} \frac{\sqrt{x+1}-1}{x-1}$ . If it doesn't exist, write DNE.

Classify the asymptotes for the functions: (a) $y=\frac{\sqrt{x}+5 x}{6 \sqrt{x}-2 x}$ , (b) $y=\sqrt{16 x^{2}+2 x}-\sqrt{16 x^{2}-4 x}$ . Use limits for justification.

Find the limit $\lim _{x \rightarrow \infty} \frac{x^{2}}{2^{x}}$ by evaluating it at $x=0,1,2,\ldots,100$ and sketch a graph.

Evaluate the limit: $\lim _{x \rightarrow 11} \frac{2 x^{2}-6 x-176}{x^{2}-121}=$ (Enter "DNE" if it doesn't exist.)

Evaluate the limit: $\lim _{x \rightarrow 9} \frac{81-x^{2}}{x-9}$ using algebraic transformation and continuity. Enter "DNE" if it doesn't exist.

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{(5+2 x)^{3}-125}{x}=$ (Enter "DNE" if it doesn't exist.)

Find $\lim _{x \rightarrow 8} f(x)$ given the graph with points (0,0), (8,4), and (10,6.5).

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

Evaluate the limit as $x$ approaches -4 for $\frac{x^{2}+8x+16}{x+4}$ . If it doesn't exist, enter "DNE". $\lim _{x \rightarrow-4} \frac{x^{2}+8 x+16}{x+4}=$

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

Find $h'(0)$ for $h(x)=f(x) \times g(x)$ given $f(0)=-6$ , $f'(0)=5$ , $g(0)=2$ , $g'(0)=-2$ .

Evaluate the limit: $\lim _{x \rightarrow 9} \frac{x-9}{x^{2}-81}=$ (Enter "DNE" if it doesn't exist.)

Find the limit: $\lim _{x \rightarrow 7} \frac{2 x^{2}+8 x-154}{x^{2}-49}=$ (Enter "DNE" if it doesn't exist.)

Find the limit: $\lim _{x \rightarrow 100} \frac{\sqrt{x}-10}{x-100}=$ . If it doesn't exist, enter "DNE".

Find the derivative of $f(x) = 3x^{2}$ using the limit: $\lim_{\Delta x \to 0} \frac{3(x+\Delta x)^{2}-3 x^{2}}{\Delta x}$ .

Evaluate the limit: $\lim _{x \rightarrow 49} \frac{\sqrt{50-x}-1}{7-\sqrt{x}} \cdot 4.$

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{2+h}-10}{h}=$ (Enter "DNE" if it doesn't exist.)

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{(7+5 x)^{3}-343}{x}=$ (Enter "DNE" if it doesn't exist.)

Evaluate the limit: $\lim _{x \rightarrow 25} \frac{\sqrt{x}-5}{x-25}=$ (Enter "DNE" if it doesn't exist.)

Find the 35th derivative of the function $f(x) = x \cos x$ .

Evaluate the limit: $\lim _{x \rightarrow-4} \frac{x^{3}+64}{x^{2}+15 x+44}$ . Use $a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})$ .

Find the limit as $x$ approaches 9 for the expression $9 \cdot a + x$ . Enter "DNE" if it doesn't exist.

Find the limit as $t$ approaches 5 for $4t + 2at + 8a$ . Enter "DNE" if it doesn't exist. $\lim _{t \rightarrow 5}(4 t+2 a t+8 a) =$

Evaluate the limit: $\lim _{t \rightarrow-3} \frac{4 t+12}{144-16 t^{2}}=$ (Exact answer, use DNE if it doesn't exist.)

Evaluate the limit:
$\lim _{x \rightarrow 2}\left(\frac{1}{x-2}-\frac{12}{x^{2}+8 x-20}\right)$
Choose the justification for your answer.

Find the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{x+h+9}-\sqrt{x+9}}{15 h}$ .

Evaluate the limit: $\lim _{x \rightarrow \frac{\pi}{6}} \frac{\csc (6 x)}{\cot (6 x)}$ . Justify your answer.

Evaluate the limit: $\lim _{\theta \rightarrow \frac{\pi}{2}}(3 \sec (\theta)-3 \tan (\theta))=$

Evaluate the limit: $\lim _{x \rightarrow-4} \frac{x^{3}+64}{x^{2}+13 x+36}$ using $a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})$ .

Evaluate the limit as $x$ approaches 6 for $9 \cdot a + x$ . Enter "DNE" if it doesn't exist. $\lim _{x \rightarrow 6}(9 \cdot a+x) =$

Approximate the limit of the piecewise function $f(x)=\begin{cases} x+8 & x \leq 2 \\ 3x+2 & x>2 \end{cases}$ as $x \to 2$ .

Find the limit as $x$ approaches $4a$ : $\lim _{x \rightarrow 4 a} \frac{\frac{3}{x}-\frac{3}{a}}{x-a}=$

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{x+h+4}-\sqrt{x+4}}{18 h}$ .

Find the limit: $\lim _{x \rightarrow 1}\left(\frac{1}{x-1}-\frac{9}{x^{2}+7 x-8}\right)$ . Provide the exact answer or DNE.

Evaluate the limit: $\lim _{x \rightarrow \frac{\pi}{6}} \frac{\csc (6 x)}{\cot (6 x)}$ . Provide the answer in exact form or DNE if it doesn't exist.

Find the derivative of $G(t)=\frac{6t}{t+7}$ using the definition and state its domain in interval notation.

Determine if the derivative $f^{\prime}(0)$ exists for the function $f(x)=x^{2} \sin \left(\frac{3}{x}\right)$ when $x \neq 0$ and $f(0)=0$ . If it exists, find its value.

Find the marginal revenue at 96 units for the revenue function $R(q)=-2 q^{2}+900 q$ . Calculate $M R(96)=\$$ per unit.

Find the limit: $\lim _{h \rightarrow 0} \frac{7 f(x+h)-7 f(x-h)}{h}$ for a differentiable function $f$ .

Find the marginal cost of producing 1000 stuffed alligator toys given $C(x)=0.004 x^{2}+6 x+7000$ .

Find the extreme values of $z=39x+10y+90$ with constraints $y \geq 0$ and $y \leq 25-x^{2}$ .

Find the extreme values of $z=39x+10y+90$ with constraints $y \geq 0$ and $y \leq 25-x^{2}$ . Complete: $f_{\min }=-1$ at $(-5, 0)$ , $f_{\max }=\square$ at $(\square, \square)$ .

Find the average velocity for $L(t)=t^{2}-t-8$ in these intervals:
1. $[5, 8]$
2. $[1, 6]$
3. $[2, 4]$ (3+ decimal places)

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

What is the expression for $f^{\prime}(a)$ for a differentiable function $f$ ? A, B, or C?

Find the derivative of the function $y=\frac{x}{4-5 x^{4}}$ in simplified form.

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

Evaluate the integral: $\int_{0}^{1} 2 x^{3} \ln \left(x^{2}+1\right) d x$

A ball is dropped from 79 m. Find when it hits the ground, its velocity $v(t)$ , and acceleration $a(t)$ .

A particle's position is $s(t)=2 t^{3}-30 t^{2}+126 t$ . Find velocity at $t=0$ , rest times, position at $t=20$ , and total distance from $t=0$ to $t=20$ .

A particle's position is $s(t)=2 t^{3}-15 t^{2}+24 t$ . Find its velocity at $t=0$ , when it stops, its position at $t=10$ , and total distance from $t=0$ to $t=10$ .

A particle's position is $s(t)=2 t^{3}-24 t^{2}+42 t$ . Find its velocity at $t=0$ , when it stops, its position at $t=16$ , and total distance from $t=0$ to $t=16$ .

Compute the surface integral $\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S$ for the given vector field $\mathbf{F}$ .

Compute the surface integral $\iint_{S} \mathbf{F} \cdot \mathbf{n} d S$ for $\mathbf{F}(x, y, z)=\langle x, 1-y+e^{z}, y-e^{z}\rangle$ on $y+z=10$ inside $x^{2}+y^{2}=1$ .

Calculate the work done to compress a spring from $40 \mathrm{~cm}$ to $10 \mathrm{~cm}$ with a force of $60 \mathrm{~N}$ .

Find the volume of the solid formed by rotating the area between $x=y^{3}$ and $x=y$ in the first quadrant about the $y$ -axis using the washer method.

Find the sixth derivative of $y=x^{2} \sin x$ at $x=0$ .

Calculate the integral of $x^{2}$ with respect to $x$ : $\int x^{2} d x=$

Find the integral of $x^{7}$ with respect to $x$ : $\int x^{7} d x=$

Solve for $2 x d x = \int\left(e^{x}+\operatorname{cosec}^{2} x+\sec x \cdot \tan x\right) d x$ .

Find the integral: $\int\left(e^{x}+\operatorname{cosec}^{2} x+\sec x \cdot \tan x\right) d x$

Calculate the integral $\int_{0}^{?} e^{5x+11} dx$ .