Calculus

Problem 12301

Find the slope of the secant line for $g(x)=-x^{2}$ from $x=-4$ to $x=-4+h$ , where $h \neq 0$ .

See Solution

Problem 12302

Calculate the slope of the secant line for $f(x)=-x^{2}+5x$ from $x=4$ to $x=4+h$ , where $h \neq 0$ .

See Solution

Problem 12303

Finde die allgemeine Stammfunktion von $f(x)=x^{15}$ . Bestimme $F(x)=\square$ .

See Solution

Problem 12304

Calculate the average rate of change of $g(x)=5 x^{2}-12 x$ from $x=0$ to $x=h$ (where $h \neq 0$ ).

See Solution

Problem 12305

Find the derivative $\frac{d y}{d x}$ for the function $y=2 x^{4} \sqrt{6-2 x^{2}}$ .

See Solution

Problem 12306

Finde die Stammfunktion von $f(x)=x^{10}$ . Bestimme $F(x)=\square$ .

See Solution

Problem 12307

Zeichnen Sie den Graphen einer Funktion, bei der $f'(a) = 0$ , $f''(a) = 0$ und $f' \geq 0$ im Intervall gilt.

See Solution

Problem 12308

Bestimme die Funktion $f$ , sodass die Stammfunktion $F(x)=\frac{1}{9} x^{9}+2$ gilt.
$f(x)=\square$

See Solution

Problem 12309

Calculate the average rate of change of $f(x)=x^{2}-3x-20$ from $x=6$ to $x=6+h$ , where $h \neq 0$ .

See Solution

Problem 12310

Bestimme die allgemeine Stammfunktion von $f(x)=5 x^{8}$ . Welche der folgenden ist korrekt? $F(x)=40 x^{9}+c$ , $F(x)=\frac{5}{9} x^{9}+c$ , $F(x)=\frac{5}{8} x^{7}+c$ , $F(x)=40 x^{7}+c$ .

See Solution

Problem 12311

Find the derivative of $y$ where $y=\frac{\ln (20 x)}{20 x}$ .

See Solution

Problem 12312

Finde die allgemeine Stammfunktion von $f(x)=9 x^{3}$ . Wähle aus: $F(x)=27 x^{2}+c$ , $F(x)=\frac{9}{4} x^{4}+c$ , $F(x)=27 x^{4}+c$ , $F(x)=3 x^{2}+c$ .

See Solution

Problem 12313

Find $f(a)$ , $f(a+h)$ , and the difference quotient $\frac{f(a+h)-f(a)}{h}$ for $f(x) = \frac{5}{x+9}$ .

See Solution

Problem 12314

Find the slope of the secant line for $k(x)=2x^2+9x-14$ from $x=-4$ to $x=-4+h$ , with $h \neq 0$ .

See Solution

Problem 12315

Find the derivative of $y$ where $y=\ln (2 \ln x)$ .

See Solution

Problem 12316

Evaluate the expression: $\int_{0}^{\frac{\pi}{2}} 3 \cos (t) - \int_{\frac{\pi}{2}}^{\frac{3 \pi}{2}} 3 \cos (t) + \int_{\frac{3 \pi}{2}}^{\frac{5 \pi}{2}} 3 \cos (t) - \int_{\frac{5 \pi}{2}}^{3 \pi} 3 \cos (t)$ .

See Solution

Problem 12317

Bestimme die Funktion $f(x)$ , wenn $F(x)=7x+c$ und $F'(x)=f(x)$ . Was ist $f(x)=\square \cdot x^{9}$ ?

See Solution

Problem 12318

Find $\frac{dy}{dt}$ for $y=\ln(2t e^{2t})$ .

See Solution

Problem 12319

Find the derivative of $y$ with respect to $\theta$ for $y=\log _{5} 6 \theta$ .

See Solution

Problem 12320

Evaluate the integral $\int t^{5} e^{-t^{3}} d t$ .

See Solution

Problem 12321

Find the derivative of $y$ with respect to $x$ for $y=\log _{2} e^{7 x}$ .

See Solution

Problem 12322

Berechne die 1. und 2. Ableitung von $f(x)=2 x^{1.5}-\frac{2}{x}+3 \sqrt{x}$ .

See Solution

Problem 12323

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the function $y=(x^{3}-1)^{5}$ .

See Solution

Problem 12324

Bestimme den Parameter $a$ für die Funktion $f(x)=x^{3}+6 x^{2}-3 a x+1$ , sodass ein Sattelpunkt existiert.

See Solution

Problem 12325

Find the average rate of change of $f(x)=x^{2}-x-8$ on the interval $-1 \leq x \leq 5$ .

See Solution

Problem 12326

Find the average rate of change of $g(x)=x^{2}-10 x+18$ from $x=1$ to $x=11$ .

See Solution

Problem 12327

Find the average rate of change of $g(x)=-x^{2}-9 x+24$ from $x=-9$ to $x=-2$ .

See Solution

Problem 12328

Calculate the integral from -1 to 1 of the function $e^{x} - e^{-x}$ .

See Solution

Problem 12329

Find the slope of the curve $7 y^{8}+9 x^{5}=3 y+13 x$ at the point $(1,1)$ .

See Solution

Problem 12330

Find the volume increase rate of a cone with height $\frac{1}{2} \mathrm{~cm}$ and radius $6 \mathrm{~cm}$ , both growing at $\frac{1}{2}$ cm/s.

See Solution

Problem 12331

Find the net change and average rate of change for $f(t)=4t^{2}$ at $t=5$ and $t=5+h$ .

See Solution

Problem 12332

Determine the absolute maximum and minimum of $f(x)=1+\cos ^{2} x$ on the interval $\left[\frac{\pi}{4}, \pi\right]$ .

See Solution

Problem 12333

Find the absolute maximum and minimum of $f(x)=x-2 \tan^{-1} x$ on the interval $[0,4]$ .

See Solution

Problem 12334

Find the slope $m$ and equation $y=m x+b$ of the tangent line to $f(x)=\frac{9}{x}$ at $(-2,-\frac{9}{2})$ .

See Solution

Problem 12335

Find the slope $m$ of the tangent line to $f(x)=3x^2-5x+2$ at $(2,4)$ and the equation $y=mx+b$ with $b=$ .

See Solution

Problem 12336

Find the drug concentration $C(12)$ using $C(t)=0.07(1-e^{-0.2 t})$ . Round to three decimal places.

See Solution

Problem 12337

Given the function $f(x)=\sqrt{x+5}$ , find: a) $f(x+h)$ , b) $f(x+h)-f(x)$ , c) $\frac{f(x+h)-f(x)}{h}$ , d) $f^{\prime}(x)$ .

See Solution

Problem 12338

Latanya invested \ $1,800 at$ 3.5\% $continuous interest. Ethan invested \$1,800 at$ 3.75\%$ monthly. How much will Latanya have when Ethan's money doubles?

See Solution

Problem 12339

Given $f(x)=\sqrt{x+5}$ , find: a) $f(x+h)$ , b) $f(x+h)-f(x)$ , c) $\frac{f(x+h)-f(x)}{h}$ , d) $f^{\prime}(x)$ .

See Solution

Problem 12340

Latanya invested \ $1,800 at$ 3.5\% $continuous interest. Ethan invested \$1,800 at$ 3.75\%$ monthly. When will Latanya's amount equal Ethan's doubled value?

See Solution

Problem 12341

Apply the Mean Value Theorem to $f(x)=x^{3}-3 x+2$ over the interval $[-2,2]$ .

See Solution

Problem 12342

Find the derivative $f^{\prime}(x)$ of $f(x)=2 x^{3}+1$ using the limit definition, then find the tangent line at $x=-1$ .

See Solution

Problem 12343

Find $q^{\prime}(x)$ for $q(x)=\log_{8}(6x^{3}+3x+1)$ .

See Solution

Problem 12344

Find the derivative $f^{\prime}(x)$ of the function $f(x)=(3 x^{5}+5)^{3}$ .

See Solution

Problem 12345

Determine the nature of the critical point at $x=1$ for the function with $f^{\prime \prime}(x)=2x-1$ .

See Solution

Problem 12346

Find the derivative $y^{\prime}$ of the equation $y=7 \sqrt[9]{(x^{2}+1)^{2}}$ .

See Solution

Problem 12347

If $f(x)$ has a local minimum at $x=c$ with $f'(x)<0$ for $x<c$ and $f'(x)>0$ for $x>c$ , then $x=c$ is a critical point.

See Solution

Problem 12348

Determine where the function $f(x)=\frac{x^{4}}{4}-3 x^{3}-1$ is concave down (select all applicable intervals).

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Problem 12349

Find the derivative $y^{\prime}$ of the equation $y=7 \sqrt[5]{(x^{2}+1)^{4}}$ .

See Solution

Problem 12350

Calculate the sum of the series: $\sum_{n=0}^{\infty} \frac{3^{n+1}}{(-2)^{n}}$ .

See Solution

Problem 12351

Find the limit: $\lim _{n \rightarrow \infty} \frac{(n+1)(n-2)(n+3000)}{2 n^{3}+1}$ .

See Solution

Problem 12352

Find the limit: $\lim _{n \rightarrow \infty} \frac{1}{\sqrt{n}(\sqrt{n+1}-\sqrt{n})}$

See Solution

Problem 12353

Find the limit: $\lim _{x \rightarrow 0}(1+2 x)^{1 / x}$ .

See Solution

Problem 12354

Find the sum of the series: $\sum_{n=1}^{\infty} \frac{6 \cdot 2^{2 n-1}}{3^{n}}$ .

See Solution

Problem 12355

Find the derivative $y^{\prime}$ of the equation $y=5 \sqrt[7]{(x^{2}+1)^{2}}$ .

See Solution

Problem 12356

Find the derivative $y^{\prime}$ of the equation $y=7 \sqrt[4]{(x^{2}+1)^{3}}$ .

See Solution

Problem 12357

Estimate critical points of $g(x)$ for $0 < x < 10$ using $g^{\prime}(x)$ values. Classify max/min points.

See Solution

Problem 12358

Find the derivative of $y=\frac{3}{4}(x^{2}-1)^{2/3}$ .

See Solution

Problem 12359

A cannon fires a projectile with horizontal velocity $866 \, \text{m/s}$ and vertical velocity $500 \, \text{m/s}$ . Find the max height. Options: A. $2.50 \times 10^{3} \, \text{m}$ B. $1.28 \times 10^{4} \, \text{m}$ C. $1.54 \times 10^{4} \, \text{m}$ D. $4.42 \times 10^{4} \, \text{m}$ .

See Solution

Problem 12360

Find the derivative of $f(x) = x(x^{2}-1)^{-1/3}$ .

See Solution

Problem 12361

Find the limit: $\lim _{n \rightarrow \infty}\left(\frac{1+2+\cdots+n}{n+2}-\frac{n}{2}\right)$ .

See Solution

Problem 12362

Find $x$ values for which the series $\sum_{n=1}^{\infty}(-5)^{n} x^{n}$ converges and its sum.

See Solution

Problem 12363

Estimate the critical points of $f(x)$ and $g(x)$ using the given tables. Classify them as maxima, minima, or none.

See Solution

Problem 12364

Find the derivative $y^{\prime}$ of the equation $y=7 \sqrt[6]{(x^{2}+1)^{5}}$ .

See Solution

Problem 12365

Find the derivative $y^{\prime}$ of the function $y=x^{2} \sqrt{4 x-1}$ . Answer: $\mathbf{y}^{\prime}=\square$

See Solution

Problem 12366

Find the limit: $\lim _{n \rightarrow \infty} \frac{1}{\sqrt{n}(\sqrt{n+1}-\sqrt{n})}$

See Solution

Problem 12367

Given $f$ is differentiable, $f(-1)=2$ , and $f^{\prime}(x) \geq 1$ , what can we conclude about $f(8)$ ?

See Solution

Problem 12368

Find the derivative $y^{\prime}$ for the function $y=\left(\frac{x+8}{x+6}\right)^{6}$ .

See Solution

Problem 12369

If $f$ is differentiable, $f(-1)=2$ , and $f^{\prime}(x) \geq 1$ , what can we conclude about $f(8)$ ?

See Solution

Problem 12370

Estimate the critical points of $g(x)$ for $0 < x < 10$ using $g'(x)$ values. Identify maxima, minima, and non-extrema.

See Solution

Problem 12371

Estimate the critical points of $g(x)$ for $0<x<10$ using $g'(x)$ values and classify them as max/min or none.

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Problem 12372

Given $f(-1)=2$ and $f'(x) \geq 1$ , use the Mean Value Theorem to determine bounds for $f(8)$ .

See Solution

Problem 12373

Given $f$ differentiable, $f(-4)=2$ , and $f^{\prime}(x) \geq 1$ , use the Mean Value Theorem to analyze $f(8)$ .

See Solution

Problem 12374

Find the tangent line equation for $f(x)=\sqrt{x^{2}+33}$ at $x=4$ . $\mathbf{y}=\square$ (Use $x$ as the variable.)

See Solution

Problem 12375

Determine the values of $p$ for which the series $\sum_{n=1}^{\infty} \frac{\ln n}{n^{p}}$ converges.

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Problem 12376

Find the dimensions of a square-based rectangular solid with max volume and surface area of 150 in². Height = $x$ .

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Problem 12377

Find the dimensions of the largest rectangle inscribed in a semicircle of radius $r$ .

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Problem 12378

A cone's radius and height increase at $\frac{1}{2}$ cm/s. Find the volume's rate of change when height is $\frac{1}{2}$ cm and radius is $6$ cm.
Options: (a) $\frac{\pi}{2}$ (b) $4 \pi$ (c) $7 \pi$ (d) $14 \pi$ (e) $21 \pi$

See Solution

Problem 12379

Differentiate the function: $y=(8 x^{2}-7)(4 x^{2}-7 x+9)$ . Find $\frac{d y}{d x}$ .

See Solution

Problem 12380

Find the dimensions of a page with $17$ sq in of print and $1$ inch margins that minimizes paper usage using calculus.

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Problem 12381

Check if the series $\sum_{n=1}^{\infty} \frac{1}{n \sqrt{n}}$ converges or diverges.

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Problem 12382

Find the linear approximation $L(x)$ of the function $f$ at $x=3$ given $f(3)=2$ and $f'(3)=-13$ . Choose from options.

See Solution

Problem 12383

Find the derivative of $s(t)=(3t^{5}-t)(t^{3}-2t+1)$ .

See Solution

Problem 12384

Evaluate the series: $\sum_{n=1}^{\infty} \frac{1}{(3 n-1)^{4}}$

See Solution

Problem 12385

Find the rate of change of facts remembered at $t=1$ hour and $t=10$ hours for $f(t)=\frac{87 t}{99 t-87}$ .

See Solution

Problem 12386

Find $f^{\prime \prime}(2)$ from the Taylor polynomial $T_{4}(x)=1+4(x-2)-5(x-2)^{2}+18(x-2)^{3}-(x-2)^{4}$ . Choices: (a) -10, (b) -1, (c) -5, (d) 18, (e) 4.

See Solution

Problem 12387

Determine where the function $f$ is concave down given its second derivative $f^{\prime \prime}(x)=\frac{30\left(x^{2}+3\right)}{(x-3)^{3}(x+3)^{3}}$ .

See Solution

Problem 12388

A school has 160 m of fencing for a rectangular playground.
(a) Find the area function $A(x)$ in terms of side length $x$ : $A(x)=\square$
(b) What side length $x$ maximizes the area? Side length $x: \square$ m
(c) What is the maximum area? Maximum area: \square m

See Solution

Problem 12389

Approximate $\sqrt{24.6}$ using differentials and compare with a calculator. Round to four decimal places.

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Problem 12390

Find the third derivative of the function $y=-\frac{1}{3} x^{5}+\frac{1}{4} x^{-2}-\frac{2}{3}$ .

See Solution

Problem 12391

Find $\frac{d y}{d x}$ using implicit differentiation for the equation: $7 x y + y^{2} = 2 x + y$ .

See Solution

Problem 12392

Find the derivative of $f(x)=\frac{1}{3} x^{3}-\frac{1}{2} x^{2}-6 x-1$ .

See Solution

Problem 12393

Find the differential $d y$ for the function $y=\frac{x+1}{4 x-1}$ . Use " $d x$ " for $d x$ .

See Solution

Problem 12394

Find the absolute extrema and their $x$ values for $f(x)=\frac{1}{3} x^{3}-\frac{1}{2} x^{2}-6 x-1$ on $[-3,4]$ .

See Solution

Problem 12395

Find the third derivative $\frac{d^{3} y}{d x^{3}}$ for the function $y=\frac{1}{4} x+\frac{1}{6}+\frac{1}{12} x^{-2}$ .

See Solution

Problem 12396

Find the derivative of the function $f(x)=-\frac{3 \sqrt[4]{x^{5}}}{2}$ in radical form, avoiding negative exponents.

See Solution

Problem 12397

Find the differential $d y$ for the function $y=-8 x-(\sin (x))^{2}$ . What is $d y$ ?

See Solution

Problem 12398

Find the differential $d y$ for the function $y = x \sqrt{5 - x^{2}}$ . What is $d y$ ?

See Solution

Problem 12399

Find the slope of the curve $6 y^{9}+5 x^{9}=7 y+4 x$ at the point $(1,1)$ .

See Solution

Problem 12400

Find the derivative of the function $y=\frac{3}{\sqrt[3]{x^{2}}}$ , expressed in radical form and simplified.

See Solution

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Calculus

Problem 12301

Find the slope of the secant line for g(x)=−x2g(x)=-x^{2}g(x)=−x2 from x=−4x=-4x=−4 to x=−4+hx=-4+hx=−4+h, where h≠0h \neq 0h=0.

Problem 12302

Calculate the slope of the secant line for f(x)=−x2+5xf(x)=-x^{2}+5xf(x)=−x2+5x from x=4x=4x=4 to x=4+hx=4+hx=4+h, where h≠0h \neq 0h=0.

Problem 12303

Finde die allgemeine Stammfunktion von f(x)=x15f(x)=x^{15}f(x)=x15. Bestimme F(x)=□F(x)=\squareF(x)=□.

Problem 12304

Calculate the average rate of change of g(x)=5x2−12xg(x)=5 x^{2}-12 xg(x)=5x2−12x from x=0x=0x=0 to x=hx=hx=h (where h≠0h \neq 0h=0).

Problem 12305

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=2x46−2x2y=2 x^{4} \sqrt{6-2 x^{2}}y=2x46−2x2​.

Problem 12306

Finde die Stammfunktion von f(x)=x10f(x)=x^{10}f(x)=x10. Bestimme F(x)=□F(x)=\squareF(x)=□.

Problem 12307

Zeichnen Sie den Graphen einer Funktion, bei der f′(a)=0f'(a) = 0f′(a)=0, f′′(a)=0f''(a) = 0f′′(a)=0 und f′≥0f' \geq 0f′≥0 im Intervall gilt.

Problem 12308

Bestimme die Funktion fff, sodass die Stammfunktion F(x)=19x9+2F(x)=\frac{1}{9} x^{9}+2F(x)=91​x9+2 gilt. f(x)=□ f(x)=\square f(x)=□

Problem 12309

Calculate the average rate of change of f(x)=x2−3x−20f(x)=x^{2}-3x-20f(x)=x2−3x−20 from x=6x=6x=6 to x=6+hx=6+hx=6+h, where h≠0h \neq 0h=0.

Problem 12310

Bestimme die allgemeine Stammfunktion von f(x)=5x8f(x)=5 x^{8}f(x)=5x8. Welche der folgenden ist korrekt? F(x)=40x9+cF(x)=40 x^{9}+cF(x)=40x9+c, F(x)=59x9+cF(x)=\frac{5}{9} x^{9}+cF(x)=95​x9+c, F(x)=58x7+cF(x)=\frac{5}{8} x^{7}+cF(x)=85​x7+c, F(x)=40x7+cF(x)=40 x^{7}+cF(x)=40x7+c.

Problem 12311

Find the derivative of yyy where y=ln⁡(20x)20xy=\frac{\ln (20 x)}{20 x}y=20xln(20x)​.

Problem 12312

Finde die allgemeine Stammfunktion von f(x)=9x3f(x)=9 x^{3}f(x)=9x3. Wähle aus: F(x)=27x2+cF(x)=27 x^{2}+cF(x)=27x2+c, F(x)=94x4+cF(x)=\frac{9}{4} x^{4}+cF(x)=49​x4+c, F(x)=27x4+cF(x)=27 x^{4}+cF(x)=27x4+c, F(x)=3x2+cF(x)=3 x^{2}+cF(x)=3x2+c.

Problem 12313

Find f(a)f(a)f(a), f(a+h)f(a+h)f(a+h), and the difference quotient f(a+h)−f(a)h\frac{f(a+h)-f(a)}{h}hf(a+h)−f(a)​ for f(x)=5x+9f(x) = \frac{5}{x+9}f(x)=x+95​.

Problem 12314

Find the slope of the secant line for k(x)=2x2+9x−14k(x)=2x^2+9x-14k(x)=2x2+9x−14 from x=−4x=-4x=−4 to x=−4+hx=-4+hx=−4+h, with h≠0h \neq 0h=0.

Problem 12315

Find the derivative of yyy where y=ln⁡(2ln⁡x)y=\ln (2 \ln x)y=ln(2lnx).

Problem 12316

Problem 12317

Bestimme die Funktion f(x)f(x)f(x), wenn F(x)=7x+cF(x)=7x+cF(x)=7x+c und F′(x)=f(x)F'(x)=f(x)F′(x)=f(x). Was ist f(x)=□⋅x9f(x)=\square \cdot x^{9}f(x)=□⋅x9?

Problem 12318

Find dydt\frac{dy}{dt}dtdy​ for y=ln⁡(2te2t)y=\ln(2t e^{2t})y=ln(2te2t).

Problem 12319

Find the derivative of yyy with respect to θ\thetaθ for y=log⁡56θy=\log _{5} 6 \thetay=log5​6θ.

Problem 12320

Evaluate the integral ∫t5e−t3dt\int t^{5} e^{-t^{3}} d t∫t5e−t3dt.

Problem 12321

Find the derivative of yyy with respect to xxx for y=log⁡2e7xy=\log _{2} e^{7 x}y=log2​e7x.

Problem 12322

Berechne die 1. und 2. Ableitung von f(x)=2x1.5−2x+3xf(x)=2 x^{1.5}-\frac{2}{x}+3 \sqrt{x}f(x)=2x1.5−x2​+3x​.

Problem 12323

Find the second derivative d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ for the function y=(x3−1)5y=(x^{3}-1)^{5}y=(x3−1)5.

Problem 12324

Bestimme den Parameter aaa für die Funktion f(x)=x3+6x2−3ax+1f(x)=x^{3}+6 x^{2}-3 a x+1f(x)=x3+6x2−3ax+1, sodass ein Sattelpunkt existiert.

Problem 12325

Find the average rate of change of f(x)=x2−x−8f(x)=x^{2}-x-8f(x)=x2−x−8 on the interval −1≤x≤5-1 \leq x \leq 5−1≤x≤5.

Problem 12326

Find the average rate of change of g(x)=x2−10x+18g(x)=x^{2}-10 x+18g(x)=x2−10x+18 from x=1x=1x=1 to x=11x=11x=11.

Problem 12327

Find the average rate of change of g(x)=−x2−9x+24g(x)=-x^{2}-9 x+24g(x)=−x2−9x+24 from x=−9x=-9x=−9 to x=−2x=-2x=−2.

Problem 12328

Calculate the integral from -1 to 1 of the function ex−e−xe^{x} - e^{-x}ex−e−x.

Problem 12329

Find the slope of the curve 7y8+9x5=3y+13x7 y^{8}+9 x^{5}=3 y+13 x7y8+9x5=3y+13x at the point (1,1)(1,1)(1,1).

Problem 12330

Find the volume increase rate of a cone with height 12 cm\frac{1}{2} \mathrm{~cm}21​ cm and radius 6 cm6 \mathrm{~cm}6 cm, both growing at 12\frac{1}{2}21​ cm/s.

Problem 12331

Find the net change and average rate of change for f(t)=4t2f(t)=4t^{2}f(t)=4t2 at t=5t=5t=5 and t=5+ht=5+ht=5+h.

Problem 12332

Determine the absolute maximum and minimum of f(x)=1+cos⁡2xf(x)=1+\cos ^{2} xf(x)=1+cos2x on the interval [π4,π]\left[\frac{\pi}{4}, \pi\right][4π​,π].

Problem 12333

Find the absolute maximum and minimum of f(x)=x−2tan⁡−1xf(x)=x-2 \tan^{-1} xf(x)=x−2tan−1x on the interval [0,4][0,4][0,4].

Problem 12334

Find the slope mmm and equation y=mx+by=m x+by=mx+b of the tangent line to f(x)=9xf(x)=\frac{9}{x}f(x)=x9​ at (−2,−92)(-2,-\frac{9}{2})(−2,−29​).

Problem 12335

Find the slope mmm of the tangent line to f(x)=3x2−5x+2f(x)=3x^2-5x+2f(x)=3x2−5x+2 at (2,4)(2,4)(2,4) and the equation y=mx+by=mx+by=mx+b with b=b=b=.

Problem 12336

Find the drug concentration C(12)C(12)C(12) using C(t)=0.07(1−e−0.2t)C(t)=0.07(1-e^{-0.2 t})C(t)=0.07(1−e−0.2t). Round to three decimal places.

Problem 12337

Given the function f(x)=x+5f(x)=\sqrt{x+5}f(x)=x+5​, find: a) f(x+h)f(x+h)f(x+h), b) f(x+h)−f(x)f(x+h)-f(x)f(x+h)−f(x), c) f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​, d) f′(x)f^{\prime}(x)f′(x).

Problem 12338

Latanya invested \1,800at1,800 at 1,800at3.5\%continuousinterest.Ethaninvested$1,800at continuous interest. Ethan invested \$1,800 at continuousinterest.Ethaninvested$1,800at3.75\%$ monthly. How much will Latanya have when Ethan's money doubles?

Problem 12339

Given f(x)=x+5f(x)=\sqrt{x+5}f(x)=x+5​, find: a) f(x+h)f(x+h)f(x+h), b) f(x+h)−f(x)f(x+h)-f(x)f(x+h)−f(x), c) f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​, d) f′(x)f^{\prime}(x)f′(x).

Problem 12340

Latanya invested \1,800at1,800 at 1,800at3.5\%continuousinterest.Ethaninvested$1,800at continuous interest. Ethan invested \$1,800 at continuousinterest.Ethaninvested$1,800at3.75\%$ monthly. When will Latanya's amount equal Ethan's doubled value?

Find the slope of the secant line for $g(x)=-x^{2}$ from $x=-4$ to $x=-4+h$ , where $h \neq 0$ .

Calculate the slope of the secant line for $f(x)=-x^{2}+5x$ from $x=4$ to $x=4+h$ , where $h \neq 0$ .

Finde die allgemeine Stammfunktion von $f(x)=x^{15}$ . Bestimme $F(x)=\square$ .

Calculate the average rate of change of $g(x)=5 x^{2}-12 x$ from $x=0$ to $x=h$ (where $h \neq 0$ ).

Find the derivative $\frac{d y}{d x}$ for the function $y=2 x^{4} \sqrt{6-2 x^{2}}$ .

Finde die Stammfunktion von $f(x)=x^{10}$ . Bestimme $F(x)=\square$ .

Zeichnen Sie den Graphen einer Funktion, bei der $f'(a) = 0$ , $f''(a) = 0$ und $f' \geq 0$ im Intervall gilt.

Bestimme die Funktion $f$ , sodass die Stammfunktion $F(x)=\frac{1}{9} x^{9}+2$ gilt.
$f(x)=\square$

Calculate the average rate of change of $f(x)=x^{2}-3x-20$ from $x=6$ to $x=6+h$ , where $h \neq 0$ .

Bestimme die allgemeine Stammfunktion von $f(x)=5 x^{8}$ . Welche der folgenden ist korrekt? $F(x)=40 x^{9}+c$ , $F(x)=\frac{5}{9} x^{9}+c$ , $F(x)=\frac{5}{8} x^{7}+c$ , $F(x)=40 x^{7}+c$ .

Find the derivative of $y$ where $y=\frac{\ln (20 x)}{20 x}$ .

Finde die allgemeine Stammfunktion von $f(x)=9 x^{3}$ . Wähle aus: $F(x)=27 x^{2}+c$ , $F(x)=\frac{9}{4} x^{4}+c$ , $F(x)=27 x^{4}+c$ , $F(x)=3 x^{2}+c$ .

Find $f(a)$ , $f(a+h)$ , and the difference quotient $\frac{f(a+h)-f(a)}{h}$ for $f(x) = \frac{5}{x+9}$ .

Find the slope of the secant line for $k(x)=2x^2+9x-14$ from $x=-4$ to $x=-4+h$ , with $h \neq 0$ .

Find the derivative of $y$ where $y=\ln (2 \ln x)$ .

Bestimme die Funktion $f(x)$ , wenn $F(x)=7x+c$ und $F'(x)=f(x)$ . Was ist $f(x)=\square \cdot x^{9}$ ?

Find $\frac{dy}{dt}$ for $y=\ln(2t e^{2t})$ .

Find the derivative of $y$ with respect to $\theta$ for $y=\log _{5} 6 \theta$ .

Evaluate the integral $\int t^{5} e^{-t^{3}} d t$ .

Find the derivative of $y$ with respect to $x$ for $y=\log _{2} e^{7 x}$ .

Berechne die 1. und 2. Ableitung von $f(x)=2 x^{1.5}-\frac{2}{x}+3 \sqrt{x}$ .

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the function $y=(x^{3}-1)^{5}$ .

Bestimme den Parameter $a$ für die Funktion $f(x)=x^{3}+6 x^{2}-3 a x+1$ , sodass ein Sattelpunkt existiert.

Find the average rate of change of $f(x)=x^{2}-x-8$ on the interval $-1 \leq x \leq 5$ .

Find the average rate of change of $g(x)=x^{2}-10 x+18$ from $x=1$ to $x=11$ .

Find the average rate of change of $g(x)=-x^{2}-9 x+24$ from $x=-9$ to $x=-2$ .

Calculate the integral from -1 to 1 of the function $e^{x} - e^{-x}$ .

Find the slope of the curve $7 y^{8}+9 x^{5}=3 y+13 x$ at the point $(1,1)$ .

Find the volume increase rate of a cone with height $\frac{1}{2} \mathrm{~cm}$ and radius $6 \mathrm{~cm}$ , both growing at $\frac{1}{2}$ cm/s.

Find the net change and average rate of change for $f(t)=4t^{2}$ at $t=5$ and $t=5+h$ .

Determine the absolute maximum and minimum of $f(x)=1+\cos ^{2} x$ on the interval $\left[\frac{\pi}{4}, \pi\right]$ .

Find the absolute maximum and minimum of $f(x)=x-2 \tan^{-1} x$ on the interval $[0,4]$ .

Find the slope $m$ and equation $y=m x+b$ of the tangent line to $f(x)=\frac{9}{x}$ at $(-2,-\frac{9}{2})$ .

Find the slope $m$ of the tangent line to $f(x)=3x^2-5x+2$ at $(2,4)$ and the equation $y=mx+b$ with $b=$ .

Find the drug concentration $C(12)$ using $C(t)=0.07(1-e^{-0.2 t})$ . Round to three decimal places.

Given the function $f(x)=\sqrt{x+5}$ , find: a) $f(x+h)$ , b) $f(x+h)-f(x)$ , c) $\frac{f(x+h)-f(x)}{h}$ , d) $f^{\prime}(x)$ .

Latanya invested \ $1,800 at$ 3.5\% $continuous interest. Ethan invested \$1,800 at$ 3.75\%$ monthly. How much will Latanya have when Ethan's money doubles?

Given $f(x)=\sqrt{x+5}$ , find: a) $f(x+h)$ , b) $f(x+h)-f(x)$ , c) $\frac{f(x+h)-f(x)}{h}$ , d) $f^{\prime}(x)$ .

Latanya invested \ $1,800 at$ 3.5\% $continuous interest. Ethan invested \$1,800 at$ 3.75\%$ monthly. When will Latanya's amount equal Ethan's doubled value?

Apply the Mean Value Theorem to $f(x)=x^{3}-3 x+2$ over the interval $[-2,2]$ .

Find the derivative $f^{\prime}(x)$ of $f(x)=2 x^{3}+1$ using the limit definition, then find the tangent line at $x=-1$ .

Find $q^{\prime}(x)$ for $q(x)=\log_{8}(6x^{3}+3x+1)$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=(3 x^{5}+5)^{3}$ .

Determine the nature of the critical point at $x=1$ for the function with $f^{\prime \prime}(x)=2x-1$ .

Find the derivative $y^{\prime}$ of the equation $y=7 \sqrt[9]{(x^{2}+1)^{2}}$ .

If $f(x)$ has a local minimum at $x=c$ with $f'(x)<0$ for $x<c$ and $f'(x)>0$ for $x>c$ , then $x=c$ is a critical point.

Determine where the function $f(x)=\frac{x^{4}}{4}-3 x^{3}-1$ is concave down (select all applicable intervals).

Find the derivative $y^{\prime}$ of the equation $y=7 \sqrt[5]{(x^{2}+1)^{4}}$ .

Calculate the sum of the series: $\sum_{n=0}^{\infty} \frac{3^{n+1}}{(-2)^{n}}$ .

Find the limit: $\lim _{n \rightarrow \infty} \frac{(n+1)(n-2)(n+3000)}{2 n^{3}+1}$ .

Find the limit: $\lim _{n \rightarrow \infty} \frac{1}{\sqrt{n}(\sqrt{n+1}-\sqrt{n})}$

Find the limit: $\lim _{x \rightarrow 0}(1+2 x)^{1 / x}$ .

Find the sum of the series: $\sum_{n=1}^{\infty} \frac{6 \cdot 2^{2 n-1}}{3^{n}}$ .

Find the derivative $y^{\prime}$ of the equation $y=5 \sqrt[7]{(x^{2}+1)^{2}}$ .

Find the derivative $y^{\prime}$ of the equation $y=7 \sqrt[4]{(x^{2}+1)^{3}}$ .

Estimate critical points of $g(x)$ for $0 < x < 10$ using $g^{\prime}(x)$ values. Classify max/min points.

Find the derivative of $y=\frac{3}{4}(x^{2}-1)^{2/3}$ .

Find the derivative of $f(x) = x(x^{2}-1)^{-1/3}$ .

Find the limit: $\lim _{n \rightarrow \infty}\left(\frac{1+2+\cdots+n}{n+2}-\frac{n}{2}\right)$ .

Find $x$ values for which the series $\sum_{n=1}^{\infty}(-5)^{n} x^{n}$ converges and its sum.

Estimate the critical points of $f(x)$ and $g(x)$ using the given tables. Classify them as maxima, minima, or none.

Find the derivative $y^{\prime}$ of the equation $y=7 \sqrt[6]{(x^{2}+1)^{5}}$ .

Find the derivative $y^{\prime}$ of the function $y=x^{2} \sqrt{4 x-1}$ . Answer: $\mathbf{y}^{\prime}=\square$

Find the limit: $\lim _{n \rightarrow \infty} \frac{1}{\sqrt{n}(\sqrt{n+1}-\sqrt{n})}$

Given $f$ is differentiable, $f(-1)=2$ , and $f^{\prime}(x) \geq 1$ , what can we conclude about $f(8)$ ?

Find the derivative $y^{\prime}$ for the function $y=\left(\frac{x+8}{x+6}\right)^{6}$ .

If $f$ is differentiable, $f(-1)=2$ , and $f^{\prime}(x) \geq 1$ , what can we conclude about $f(8)$ ?

Estimate the critical points of $g(x)$ for $0 < x < 10$ using $g'(x)$ values. Identify maxima, minima, and non-extrema.

Estimate the critical points of $g(x)$ for $0<x<10$ using $g'(x)$ values and classify them as max/min or none.

Given $f(-1)=2$ and $f'(x) \geq 1$ , use the Mean Value Theorem to determine bounds for $f(8)$ .

Given $f$ differentiable, $f(-4)=2$ , and $f^{\prime}(x) \geq 1$ , use the Mean Value Theorem to analyze $f(8)$ .

Find the tangent line equation for $f(x)=\sqrt{x^{2}+33}$ at $x=4$ . $\mathbf{y}=\square$ (Use $x$ as the variable.)