Calculus

Problem 6301

Find the limit as $t$ approaches 0 for the expression $\frac{2^{t}-1}{t}$ .

See Solution

Problem 6302

Find the derivative of $y=\sin(\tan(8x))$ . What is $y'$ ?

See Solution

Problem 6303

Find the rate of change of the curve $f(x)=-x^{3}$ from $x=0$ to $x=2$ .

See Solution

Problem 6304

Find the integral of $5 \tan^{5} x \sec^{3} x \, dx$ .

See Solution

Problem 6305

Find the integral: $\int 7 \tan^{4} x \sec^{4} x \, dx$

See Solution

Problem 6306

Calculate the integral: $\int \frac{x^{2}-3}{x^{2}-4} d x$

See Solution

Problem 6307

Calculate the integral: $\int \frac{2 x+3}{6 x^{2}+5 x+1} d x$

See Solution

Problem 6308

Find the limit as $x$ approaches 4 for the expression $\frac{x^{2}+8 x}{x^{2}-8 x+16}$ .

See Solution

Problem 6309

Find the derivative of the function $f(x)=\frac{3 x}{1+x}$ using the limit definition of the derivative.

See Solution

Problem 6310

Find the rate of change of food spending percentage on May 1, given $y=35 x^{-0.25}$ and $x=7+0.2t$ .

See Solution

Problem 6311

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

See Solution

Problem 6312

Find the derivative using the limit process for $f(x)=3x+1$ . What is $f'(x)$ ?

See Solution

Problem 6313

Find the derivative of the function $f(x)=(2x^2-4)^{-1}$ . What is $f'(x)$ ?

See Solution

Problem 6314

Find the derivative of the function $r(x)=(0.1 x^{2}-4.2 x+7.5)^{1.5}$ . What is $r'(x)$ ?

See Solution

Problem 6315

Find the derivative of the constant function $f(x)=2$ using the limit process. What is $f^{\prime}(x)=?$

See Solution

Problem 6316

Find the derivative of the function $h(x)=\frac{1}{(x^{2}+x+1)^{2}}$ . What is $h^{\prime}(x)$ ?

See Solution

Problem 6317

Find the derivative of $h(x)=3-\frac{3}{4} x$ using the limit process. What is $h^{\prime}(x)=?$

See Solution

Problem 6318

Find the derivative using the limit process for $f(x)=\frac{6}{x-2}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 6319

Find the derivative of the function $h(x)=(3.1 x-3)^{2}-\frac{1}{(3.1 x-3)^{2}}$ .

See Solution

Problem 6320

Find the slope of the tangent line for $g(x)=\frac{4}{9} x+9$ at the point (-9,5).

See Solution

Problem 6321

Find the derivative using the limit process for $f(x)=2 x^{3}+4 x^{2}$ . What is $f^{\prime}(x)=?$

See Solution

Problem 6322

A projectile is launched at 160 ft/s from a 30-ft stage. What is its maximum height?

See Solution

Problem 6323

Find the derivative of the function $f(x)=\sqrt{9x+x^{2}}$ .

See Solution

Problem 6324

Find the slope of the tangent line to $g(x)=19-x^{2}$ at the point $(1,18)$ .

See Solution

Problem 6325

Find the derivative using the limit process for $f(x)=\sqrt{x+8}$ . What is $f^{\prime}(x)=?$

See Solution

Problem 6326

Find the derivative of the function $f(x)=\frac{1}{(x+9)^{2}}$ .

See Solution

Problem 6327

Find $f(5)$ and $f(5.1)$ to estimate $f^{\prime}(5)$ , rounding to one decimal place. Given $f(x)=x(8-x)$ .

See Solution

Problem 6328

Find the $x$ -values where the function $y=x^{2/5}$ is differentiable.

See Solution

Problem 6329

Find the left and right derivatives of the function $f(x)=|x-9|$ at $x=9$ . Enter NONE if a derivative doesn't exist.

See Solution

Problem 6330

Find the derivative of $h(x)=|x+4|$ at $x=-4$ . If it doesn't exist, write UNDEFINED. $h^{\prime}(-4)=$

See Solution

Problem 6331

Find the derivative of the function $f(x)=\sqrt{9x+x^{2}}$ , denoted as $f^{\prime}(x)$ .

See Solution

Problem 6332

Find the derivative of $f(x)=x^{2}-3$ at $x=4$ . If it doesn't exist, write UNDEFINED. $f^{\prime}(4)=$

See Solution

Problem 6333

Find a function $f$ such that $f(-5)=f(4)=0$ , $f'(-0.5)=0$ , $f'<0$ for $x<-0.5$ , and $f'>0$ for $x>-0.5$ .

See Solution

Problem 6334

Find the derivative of $f(x)=\frac{6}{\sqrt{x}}$ using the limit process: $f^{\prime}(x)=\square$ .

See Solution

Problem 6335

Determine where the function $f$ is differentiable given:
$f(x)=\left\{\begin{array}{ll} x^{2}-8 & , x \leq 0 \\ 8-x^{2} & , x>0 \end{array}\right.$

See Solution

Problem 6336

Is it true or false that if a function is differentiable at a point, it must be continuous there? Provide examples if false.

See Solution

Problem 6337

Find the tangent line equation for the function $f(x)=x^{2}+2$ at the point (1,3). $y=$

See Solution

Problem 6338

Find the tangent line equation for the function $f(x)=x+\frac{5}{x}$ at the point $(-5,-6)$ .

See Solution

Problem 6339

Find the tangent line equation for the function $f(x)=\sqrt{x+2}$ at the point $(2,2)$ . $y=$

See Solution

Problem 6340

Identify where the function increases, decreases, or is constant based on its graph starting at (-2,-5).

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Problem 6341

Determine concavity, inflection points, and critical points for $f(x)=-(4x+4\sin(x))$ , $0 \leq x \leq 2\pi$ .

See Solution

Problem 6342

Find the tangent line to $f(x)=x^2$ parallel to $6x-y+9=0$ . Steps: 1) Find $f'(x)$ , 2) Find slope $m$ , 3) Solve $f'(x)=m$ , 4) Get $y$ from $f(x)$ , 5) Use point-slope to find line equation.

See Solution

Problem 6343

Find the tangent line to $f(x)=2x^{2}$ that is parallel to $2x-y+4=0$ .

See Solution

Problem 6344

Find $g(5)$ and $g^{\prime}(5)$ if the tangent line at $(5,3)$ passes through $(3,5)$ .

See Solution

Problem 6345

Evaluate $f(5)$ and $f(5,1)$ , then approximate $f^{\prime}(5)$ for $f(x)=x(8-x)$ . Round to one decimal place.

See Solution

Problem 6346

Calculate the derivative $y'$ of the function $y = x^2$ at the point $(3,9)$ .

See Solution

Problem 6347

Estimate the slope of the tangent line to $y=x^{1/3}$ at the point (1,1) and verify it analytically. $y'=$

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Problem 6348

Estimate the slope of the tangent line to $y=x^{-1/2}$ at $(1,1)$ and verify: $y^{\prime}(1)=$ .

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Problem 6349

Find the second derivative of $y=x^{-2}$ at $x=1$ . What is $y^{\prime \prime}(1)=?$

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Problem 6350

Find the derivative of the function $y=8$ . What is $y^{\prime}=$ ?

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Problem 6351

Find the derivative of the function $f(x)=-324$ . What is $f^{\prime}(x)=?$

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Problem 6352

Find the derivative of $f(x)=\sqrt[5]{x}$ .

See Solution

Problem 6353

Find the derivative of the function $g(x) = 18x - 11$ . What is $g'(x)$ ?

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Problem 6354

Find the derivative of the function $h(x)=7 x^{8}$ . What is $h^{\prime}(x)=?$

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Problem 6355

Find the derivative of the function $g(x)=18x-11$ . What is $g^{\prime}(x)=?$

See Solution

Problem 6356

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

See Solution

Problem 6357

Find the derivative of the function $f(t) = 4t^{2} - 8t - 9$ . What is $f^{\prime}(t)$ ?

See Solution

Problem 6358

Find the derivative of the function $y=\frac{\pi}{2} \cos (\theta)-\cos (\theta)$ . What is $y^{\prime}$ ?

See Solution

Problem 6359

Find the derivative of the function $y=\frac{4}{(3 x)^{2}}+7 \sin (x)$ . What is $y^{\prime}=?$

See Solution

Problem 6360

Find the derivative of the function $g(t)=t^{2}-\frac{8}{t^{3}}$ . What is $g^{\prime}(t)$ ?

See Solution

Problem 6361

Find the derivative of $y=\frac{7}{3 x^{7}}$ . What is $y^{\prime}=?$

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Problem 6362

Find the derivative of the function $y=4+5 \sin (x)$ . What is $y^{\prime}(x)=?$

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Problem 6363

Find the slope of $f(x)=\frac{4}{x^{2}}$ at the point (2,1) and confirm using a graphing utility. $f^{\prime}(2)=$

See Solution

Problem 6364

Complete the tasks: Rewrite $y=\frac{7}{3 x^{2}}$ , differentiate to find $y^{\prime}$ , then simplify $y^{\prime}$ .

See Solution

Problem 6365

Find the slope of $f(\theta)=7 \sin (\theta)-\theta$ at the point $(0,0)$ and confirm with a graphing utility.

See Solution

Problem 6366

Find the derivative of $f(x)=\sqrt{x}-10 \sqrt[3]{x}$ : $f^{\prime}(x)=\square$ .

See Solution

Problem 6367

Find the antiderivative of the function $\int \frac{x^{2}-3}{x^{2}-4} d x$ .

See Solution

Problem 6368

Find the derivative of $f(x)=\frac{7 x^{3}+5 x^{2}}{x}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 6369

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

See Solution

Problem 6370

Find the integral of $7 \tan^4x \sec^4x$ . What is the result?

See Solution

Problem 6371

Find the average rate of change of $f(t)=2 t^{2}-5$ over $[5,5.1]$ and compare it with the rates at $t=5$ and $t=5.1$ .

See Solution

Problem 6372

Find the derivative of the function $h(x)=7 x^{8}$ . What is $h^{\prime}(x)$ ?

See Solution

Problem 6373

Find where the graph of $y=x^{3}+9x$ has a horizontal tangent line. If none, enter NONE.

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Problem 6374

1. Find $\lim _{x \rightarrow 0} \frac{\sqrt{1+x}+\sqrt{1-x}}{x}$ .
2. Calculate $\lim _{x \rightarrow 2} \frac{x^{3}-3}{x-2}$ .
3. Determine $\lim _{x \rightarrow 2} \frac{\sqrt{3-x}-1}{2-x}$ .
4. Evaluate $\lim _{x \rightarrow 2}[2(x+3)+7]$ .
5. Compute $\lim _{x \rightarrow 0} x^{2}+3 x+7$ .
6. Find $\lim _{x \rightarrow 1}\left[(x+3)^{2}-16\right]$ .
7. Calculate $\lim _{x \rightarrow 1}\left[(x+1)^{2}+1\right]$ .
8. Determine $\lim _{x \rightarrow 0}\left[(2 x+1)^{3}-5\right]$ .
9. Evaluate $\lim _{x \rightarrow 2} \frac{x-5}{x+2}$ .

See Solution

Problem 6375

Find the derivative $y' = \frac{d}{dx} \left( \frac{4}{(3x)^{2}} + 7 \sin(x) \right)$ .

See Solution

Problem 6376

Find the antiderivative of $\int \frac{x^{2}-3}{x^{2}-4} d x$ .

See Solution

Problem 6377

Evaluate the integral: $\int 7 \tan ^{4} x \sec ^{4} x \, dx = ?$

See Solution

Problem 6378

Find $x$ -values where $f(x)=2x^3+3x^2-120x+7$ has horizontal tangents and their equations. Answers: $x=-5,4$ .

See Solution

Problem 6379

Find the antiderivative of $\int \frac{x^{2}-3}{x^{2}-4} dx$ .

See Solution

Problem 6380

Find the limit: $\lim _{x \rightarrow \infty} \frac{81-\sqrt{x}}{9+\sqrt{x}}$ . Enter $\infty$ , $-\infty$ , or DNE if it doesn't exist.

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Problem 6381

Find the limits: 12. $\lim _{x \rightarrow 2} \frac{x^{2}-x-2}{x^{2}-3 x+2}$ , 13. $\lim _{x \rightarrow 0} \frac{x^{3}+7 x}{x^{2}+2 x}$ , 14. $\lim _{x \rightarrow 0} \frac{\sqrt{3+x}+\sqrt{2}}{x}$ , 15. $\lim _{x \rightarrow 2} \frac{\sqrt{3 x-2}+x}{2-\sqrt{6-x}}$ .

See Solution

Problem 6382

Find the limit: $\lim _{u \rightarrow-\infty} \frac{(u^{2}+1)(8 u^{2}-1)}{(u^{2}+3)^{2}}$ . Enter $\infty$ , $-\infty$ , or DNE if it doesn't exist.

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Problem 6383

Berechnen Sie die Grenzwerte: a) $\lim _{n \rightarrow \infty}(\sqrt{n+100}-\sqrt{n})$ b) $\lim _{n \rightarrow \infty} \sqrt{n} \cdot(\sqrt{n+10}-\sqrt{n})$ c) $\lim _{n \rightarrow \infty}\left(\sqrt{4 n^{2}+3 n}-2 n\right)$

See Solution

Problem 6384

Sketch the graph of a function $f$ with these limits and values: $\lim_{x \to 4^+} f(x)=5$ , $\lim_{x \to 4^-} f(x)=3$ , $\lim_{x \to -1} f(x)=3$ , $f(4)=4$ , $f(-1)=2$ .

See Solution

Problem 6385

Find and simplify $\frac{d y}{d x}$ for $y=x^{3}(6-x^{2})$ .

See Solution

Problem 6386

Find and simplify $\frac{d y}{d x}$ for $y=x^{3}(6-x^{2})$ , $y=5x(2x^{2}-1)$ , and $y=\frac{2x+8}{3x-1}$ .

See Solution

Problem 6387

Find $\frac{d y}{d x}$ for $y=(4 x^{2}+x)(x-x^{2})$ . No need to expand your answer.

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Problem 6388

Find and simplify $\frac{d y}{d x}$ for $y=5 x(2 x^{2}-1)$ .

See Solution

Problem 6389

Find the derivative $\frac{d y}{d x}$ for $y=\frac{3 x^{2}-9 x+13}{2 x+4}$ without expanding.

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Problem 6390

Given values of $f(t)$ at $t = 0, 2, 4, 6, 8, 10$ are 135, 133, 129, 124, 118, 107. Determine the sign of the first and second derivatives. Estimate $f^{\prime}(2)$ and $f^{\prime}(8)$ using right difference quotients.

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Problem 6391

Find $q^{\prime}(t)$ , $p^{\prime}(t)$ , and revenue change at $t=5$ for sales $q(t)=6000t-60t^2$ and price $p(t)=3000-t^2$ .

See Solution

Problem 6392

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

See Solution

Problem 6393

Find the derivative of $f(x)=\sqrt{x}-\frac{6}{\sqrt{x}}$ and calculate $f^{\prime}(6)$ in simplest radical form.

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Problem 6394

Find the derivative $f^{\prime}(4)$ of the function $f(x)=-\frac{4}{x}+\frac{2 \sqrt{x}}{5}$ as a simplified fraction.

See Solution

Problem 6395

Find the rate of change of monthly revenue 5 months after the sound system's introduction using $R(t) = p(t) \cdot q(t)$ .

See Solution

Problem 6396

Find the antiderivative of $f'(x)=4ax^3 + 3bx^2 + 2cx + d$ .

See Solution

Problem 6397

The cost of airing $x$ commercials is $C(x)=20+5,000x+0.04x^{2}$ .
(a) Find $C^{\prime}(x)$ and estimate the cost increase at $x=4$ .
(b) Find $\bar{C}(x)$ and evaluate $\bar{C}(4)$ . What does this tell you?

See Solution

Problem 6398

Find the marginal revenue $R'(x)$ and profit $P'(x)$ for producing $x$ servings of pasta with cost $C(x)=350+0.10x+0.002x^2$ . Compute revenue and profit for 200 servings, and find when marginal profit is zero ( $x=250$ plates).

See Solution

Problem 6399

The cost of airing $x$ commercials is $C(x)=20+5000x+0.04x^2$ .
(a) Find the marginal cost $C'(x)$ and estimate for $x=4$ .
(b) Find the average cost $\bar{C}(x)$ and evaluate $\bar{C}(4)$ . What does this tell you?

See Solution

Problem 6400

Determine the signs of $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ for the given functions over the intervals provided.

See Solution

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Calculus

Problem 6301

Find the limit as ttt approaches 0 for the expression 2t−1t\frac{2^{t}-1}{t}t2t−1​.

Problem 6302

Find the derivative of y=sin⁡(tan⁡(8x))y=\sin(\tan(8x))y=sin(tan(8x)). What is y′y'y′?

Problem 6303

Find the rate of change of the curve f(x)=−x3f(x)=-x^{3}f(x)=−x3 from x=0x=0x=0 to x=2x=2x=2.

Problem 6304

Find the integral of 5tan⁡5xsec⁡3x dx5 \tan^{5} x \sec^{3} x \, dx5tan5xsec3xdx.

Problem 6305

Find the integral: ∫7tan⁡4xsec⁡4x dx\int 7 \tan^{4} x \sec^{4} x \, dx∫7tan4xsec4xdx

Problem 6306

Calculate the integral: ∫x2−3x2−4dx\int \frac{x^{2}-3}{x^{2}-4} d x∫x2−4x2−3​dx

Problem 6307

Calculate the integral: ∫2x+36x2+5x+1dx\int \frac{2 x+3}{6 x^{2}+5 x+1} d x∫6x2+5x+12x+3​dx

Problem 6308

Find the limit as xxx approaches 4 for the expression x2+8xx2−8x+16\frac{x^{2}+8 x}{x^{2}-8 x+16}x2−8x+16x2+8x​.

Problem 6309

Find the derivative of the function f(x)=3x1+xf(x)=\frac{3 x}{1+x}f(x)=1+x3x​ using the limit definition of the derivative.

Problem 6310

Find the rate of change of food spending percentage on May 1, given y=35x−0.25y=35 x^{-0.25}y=35x−0.25 and x=7+0.2tx=7+0.2tx=7+0.2t.

Problem 6311

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

Problem 6312

Find the derivative using the limit process for f(x)=3x+1f(x)=3x+1f(x)=3x+1. What is f′(x)f'(x)f′(x)?

Problem 6313

Find the derivative of the function f(x)=(2x2−4)−1f(x)=(2x^2-4)^{-1}f(x)=(2x2−4)−1. What is f′(x)f'(x)f′(x)?

Problem 6314

Find the derivative of the function r(x)=(0.1x2−4.2x+7.5)1.5r(x)=(0.1 x^{2}-4.2 x+7.5)^{1.5}r(x)=(0.1x2−4.2x+7.5)1.5. What is r′(x)r'(x)r′(x)?

Problem 6315

Find the derivative of the constant function f(x)=2f(x)=2f(x)=2 using the limit process. What is f′(x)=?f^{\prime}(x)=?f′(x)=?

Problem 6316

Find the derivative of the function h(x)=1(x2+x+1)2h(x)=\frac{1}{(x^{2}+x+1)^{2}}h(x)=(x2+x+1)21​. What is h′(x)h^{\prime}(x)h′(x)?

Problem 6317

Find the derivative of h(x)=3−34xh(x)=3-\frac{3}{4} xh(x)=3−43​x using the limit process. What is h′(x)=?h^{\prime}(x)=?h′(x)=?

Problem 6318

Find the derivative using the limit process for f(x)=6x−2f(x)=\frac{6}{x-2}f(x)=x−26​. What is f′(x)f^{\prime}(x)f′(x)?

Problem 6319

Find the derivative of the function h(x)=(3.1x−3)2−1(3.1x−3)2h(x)=(3.1 x-3)^{2}-\frac{1}{(3.1 x-3)^{2}}h(x)=(3.1x−3)2−(3.1x−3)21​.

Problem 6320

Find the slope of the tangent line for g(x)=49x+9g(x)=\frac{4}{9} x+9g(x)=94​x+9 at the point (-9,5).

Problem 6321

Find the derivative using the limit process for f(x)=2x3+4x2f(x)=2 x^{3}+4 x^{2}f(x)=2x3+4x2. What is f′(x)=?f^{\prime}(x)=?f′(x)=?

Problem 6322

A projectile is launched at 160 ft/s from a 30-ft stage. What is its maximum height?

Problem 6323

Find the derivative of the function f(x)=9x+x2f(x)=\sqrt{9x+x^{2}}f(x)=9x+x2​.

Problem 6324

Find the slope of the tangent line to g(x)=19−x2g(x)=19-x^{2}g(x)=19−x2 at the point (1,18)(1,18)(1,18).

Problem 6325

Find the derivative using the limit process for f(x)=x+8f(x)=\sqrt{x+8}f(x)=x+8​. What is f′(x)=?f^{\prime}(x)=?f′(x)=?

Problem 6326

Find the derivative of the function f(x)=1(x+9)2f(x)=\frac{1}{(x+9)^{2}}f(x)=(x+9)21​.

Problem 6327

Find f(5)f(5)f(5) and f(5.1)f(5.1)f(5.1) to estimate f′(5)f^{\prime}(5)f′(5), rounding to one decimal place. Given f(x)=x(8−x)f(x)=x(8-x)f(x)=x(8−x).

Problem 6328

Find the xxx-values where the function y=x2/5y=x^{2/5}y=x2/5 is differentiable.

Problem 6329

Find the left and right derivatives of the function f(x)=∣x−9∣f(x)=|x-9|f(x)=∣x−9∣ at x=9x=9x=9. Enter NONE if a derivative doesn't exist.

Problem 6330

Find the derivative of h(x)=∣x+4∣h(x)=|x+4|h(x)=∣x+4∣ at x=−4x=-4x=−4. If it doesn't exist, write UNDEFINED. h′(−4)=h^{\prime}(-4)=h′(−4)=

Problem 6331

Find the derivative of the function f(x)=9x+x2f(x)=\sqrt{9x+x^{2}}f(x)=9x+x2​, denoted as f′(x)f^{\prime}(x)f′(x).

Problem 6332

Find the derivative of f(x)=x2−3f(x)=x^{2}-3f(x)=x2−3 at x=4x=4x=4. If it doesn't exist, write UNDEFINED. f′(4)=f^{\prime}(4)=f′(4)=

Problem 6333

Find a function fff such that f(−5)=f(4)=0f(-5)=f(4)=0f(−5)=f(4)=0, f′(−0.5)=0f'(-0.5)=0f′(−0.5)=0, f′<0f'<0f′<0 for x<−0.5x<-0.5x<−0.5, and f′>0f'>0f′>0 for x>−0.5x>-0.5x>−0.5.

Problem 6334

Find the derivative of f(x)=6xf(x)=\frac{6}{\sqrt{x}}f(x)=x​6​ using the limit process: f′(x)=□f^{\prime}(x)=\squaref′(x)=□.

Problem 6335

Determine where the function fff is differentiable given: f(x)={x2−8,x≤08−x2,x>0 f(x)=\left\{\begin{array}{ll} x^{2}-8 & , x \leq 0 \\ 8-x^{2} & , x>0 \end{array}\right. f(x)={x2−88−x2​,x≤0,x>0​

Problem 6336

Is it true or false that if a function is differentiable at a point, it must be continuous there? Provide examples if false.

Problem 6337

Find the tangent line equation for the function f(x)=x2+2f(x)=x^{2}+2f(x)=x2+2 at the point (1,3). y=y=y=

Problem 6338

Find the tangent line equation for the function f(x)=x+5xf(x)=x+\frac{5}{x}f(x)=x+x5​ at the point (−5,−6)(-5,-6)(−5,−6).

Problem 6339

Find the tangent line equation for the function f(x)=x+2f(x)=\sqrt{x+2}f(x)=x+2​ at the point (2,2)(2,2)(2,2). y=y= y=

Problem 6340

Find the limit as $t$ approaches 0 for the expression $\frac{2^{t}-1}{t}$ .

Find the derivative of $y=\sin(\tan(8x))$ . What is $y'$ ?

Find the rate of change of the curve $f(x)=-x^{3}$ from $x=0$ to $x=2$ .

Find the integral of $5 \tan^{5} x \sec^{3} x \, dx$ .

Find the integral: $\int 7 \tan^{4} x \sec^{4} x \, dx$

Calculate the integral: $\int \frac{x^{2}-3}{x^{2}-4} d x$

Calculate the integral: $\int \frac{2 x+3}{6 x^{2}+5 x+1} d x$

Find the limit as $x$ approaches 4 for the expression $\frac{x^{2}+8 x}{x^{2}-8 x+16}$ .

Find the derivative of the function $f(x)=\frac{3 x}{1+x}$ using the limit definition of the derivative.

Find the rate of change of food spending percentage on May 1, given $y=35 x^{-0.25}$ and $x=7+0.2t$ .

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

Find the derivative using the limit process for $f(x)=3x+1$ . What is $f'(x)$ ?

Find the derivative of the function $f(x)=(2x^2-4)^{-1}$ . What is $f'(x)$ ?

Find the derivative of the function $r(x)=(0.1 x^{2}-4.2 x+7.5)^{1.5}$ . What is $r'(x)$ ?

Find the derivative of the constant function $f(x)=2$ using the limit process. What is $f^{\prime}(x)=?$

Find the derivative of the function $h(x)=\frac{1}{(x^{2}+x+1)^{2}}$ . What is $h^{\prime}(x)$ ?

Find the derivative of $h(x)=3-\frac{3}{4} x$ using the limit process. What is $h^{\prime}(x)=?$

Find the derivative using the limit process for $f(x)=\frac{6}{x-2}$ . What is $f^{\prime}(x)$ ?

Find the derivative of the function $h(x)=(3.1 x-3)^{2}-\frac{1}{(3.1 x-3)^{2}}$ .

Find the slope of the tangent line for $g(x)=\frac{4}{9} x+9$ at the point (-9,5).

Find the derivative using the limit process for $f(x)=2 x^{3}+4 x^{2}$ . What is $f^{\prime}(x)=?$

Find the derivative of the function $f(x)=\sqrt{9x+x^{2}}$ .

Find the slope of the tangent line to $g(x)=19-x^{2}$ at the point $(1,18)$ .

Find the derivative using the limit process for $f(x)=\sqrt{x+8}$ . What is $f^{\prime}(x)=?$

Find the derivative of the function $f(x)=\frac{1}{(x+9)^{2}}$ .

Find $f(5)$ and $f(5.1)$ to estimate $f^{\prime}(5)$ , rounding to one decimal place. Given $f(x)=x(8-x)$ .

Find the $x$ -values where the function $y=x^{2/5}$ is differentiable.

Find the left and right derivatives of the function $f(x)=|x-9|$ at $x=9$ . Enter NONE if a derivative doesn't exist.

Find the derivative of $h(x)=|x+4|$ at $x=-4$ . If it doesn't exist, write UNDEFINED. $h^{\prime}(-4)=$

Find the derivative of the function $f(x)=\sqrt{9x+x^{2}}$ , denoted as $f^{\prime}(x)$ .

Find the derivative of $f(x)=x^{2}-3$ at $x=4$ . If it doesn't exist, write UNDEFINED. $f^{\prime}(4)=$

Find a function $f$ such that $f(-5)=f(4)=0$ , $f'(-0.5)=0$ , $f'<0$ for $x<-0.5$ , and $f'>0$ for $x>-0.5$ .

Find the derivative of $f(x)=\frac{6}{\sqrt{x}}$ using the limit process: $f^{\prime}(x)=\square$ .

Determine where the function $f$ is differentiable given:
$f(x)=\left\{\begin{array}{ll} x^{2}-8 & , x \leq 0 \\ 8-x^{2} & , x>0 \end{array}\right.$

Find the tangent line equation for the function $f(x)=x^{2}+2$ at the point (1,3). $y=$

Find the tangent line equation for the function $f(x)=x+\frac{5}{x}$ at the point $(-5,-6)$ .

Find the tangent line equation for the function $f(x)=\sqrt{x+2}$ at the point $(2,2)$ . $y=$

Determine concavity, inflection points, and critical points for $f(x)=-(4x+4\sin(x))$ , $0 \leq x \leq 2\pi$ .

Find the tangent line to $f(x)=x^2$ parallel to $6x-y+9=0$ . Steps: 1) Find $f'(x)$ , 2) Find slope $m$ , 3) Solve $f'(x)=m$ , 4) Get $y$ from $f(x)$ , 5) Use point-slope to find line equation.

Find the tangent line to $f(x)=2x^{2}$ that is parallel to $2x-y+4=0$ .

Find $g(5)$ and $g^{\prime}(5)$ if the tangent line at $(5,3)$ passes through $(3,5)$ .

Evaluate $f(5)$ and $f(5,1)$ , then approximate $f^{\prime}(5)$ for $f(x)=x(8-x)$ . Round to one decimal place.

Calculate the derivative $y'$ of the function $y = x^2$ at the point $(3,9)$ .

Estimate the slope of the tangent line to $y=x^{1/3}$ at the point (1,1) and verify it analytically. $y'=$

Estimate the slope of the tangent line to $y=x^{-1/2}$ at $(1,1)$ and verify: $y^{\prime}(1)=$ .

Find the second derivative of $y=x^{-2}$ at $x=1$ . What is $y^{\prime \prime}(1)=?$

Find the derivative of the function $y=8$ . What is $y^{\prime}=$ ?

Find the derivative of the function $f(x)=-324$ . What is $f^{\prime}(x)=?$

Find the derivative of $f(x)=\sqrt[5]{x}$ .

Find the derivative of the function $g(x) = 18x - 11$ . What is $g'(x)$ ?

Find the derivative of the function $h(x)=7 x^{8}$ . What is $h^{\prime}(x)=?$

Find the derivative of the function $g(x)=18x-11$ . What is $g^{\prime}(x)=?$

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

Find the derivative of the function $f(t) = 4t^{2} - 8t - 9$ . What is $f^{\prime}(t)$ ?

Find the derivative of the function $y=\frac{\pi}{2} \cos (\theta)-\cos (\theta)$ . What is $y^{\prime}$ ?

Find the derivative of the function $y=\frac{4}{(3 x)^{2}}+7 \sin (x)$ . What is $y^{\prime}=?$

Find the derivative of the function $g(t)=t^{2}-\frac{8}{t^{3}}$ . What is $g^{\prime}(t)$ ?

Find the derivative of $y=\frac{7}{3 x^{7}}$ . What is $y^{\prime}=?$

Find the derivative of the function $y=4+5 \sin (x)$ . What is $y^{\prime}(x)=?$

Find the slope of $f(x)=\frac{4}{x^{2}}$ at the point (2,1) and confirm using a graphing utility. $f^{\prime}(2)=$

Complete the tasks: Rewrite $y=\frac{7}{3 x^{2}}$ , differentiate to find $y^{\prime}$ , then simplify $y^{\prime}$ .

Find the slope of $f(\theta)=7 \sin (\theta)-\theta$ at the point $(0,0)$ and confirm with a graphing utility.

Find the derivative of $f(x)=\sqrt{x}-10 \sqrt[3]{x}$ : $f^{\prime}(x)=\square$ .

Find the antiderivative of the function $\int \frac{x^{2}-3}{x^{2}-4} d x$ .

Find the derivative of $f(x)=\frac{7 x^{3}+5 x^{2}}{x}$ . What is $f^{\prime}(x)$ ?

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

Find the integral of $7 \tan^4x \sec^4x$ . What is the result?

Find the average rate of change of $f(t)=2 t^{2}-5$ over $[5,5.1]$ and compare it with the rates at $t=5$ and $t=5.1$ .

Find the derivative of the function $h(x)=7 x^{8}$ . What is $h^{\prime}(x)$ ?

Find where the graph of $y=x^{3}+9x$ has a horizontal tangent line. If none, enter NONE.