Calculus

Problem 15401

Find the derivative of $f(x)=e^{\left[\frac{x^{3}}{3}\right]}$ , so $f'(x)=\square$ .

See Solution

Problem 15402

Find the derivative of $f(x)=\ln \left(x^{2}+1\right)^{5}$ . What is $f^{\prime}(x)=?$

See Solution

Problem 15403

Find the derivative of the function $h(x)=\cos \left(\frac{6 x^{3}}{5+4 x^{2}}\right)$ , denoted as $h^{\prime}(x)$ .

See Solution

Problem 15404

Bestätigen Sie die Nullstellen, Extrem- und Wendepunkte der Funktionen: a) $f(x) = x^{3}-3 x^{2}-x+3$ b) $f(x) = \frac{1}{2} x^{4}-3 x^{2}+4$ Geben Sie Monotonie- und Krümmungsintervalle an.

See Solution

Problem 15405

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

See Solution

Problem 15406

Find the derivative of the function $y=\left(\frac{4-3 x}{4 x^{2}-3}\right)^{5}$ with respect to $x$ .

See Solution

Problem 15407

Find the derivative of $f(x)=\frac{e^{x}}{x^{5}}$ . What is $f^{\prime}(x)=\square$ ?

See Solution

Problem 15408

Check if the Mean Value Theorem applies to $f(x)=|x-5|$ on $[-5,11]$ and find the guaranteed point(s).

See Solution

Problem 15409

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

See Solution

Problem 15410

Find the derivative of $f(x)=x^{6} \ln (x)$ .

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Problem 15411

Find the absolute max and min of $p(x)=x^{2}-1$ on the interval $[-5,3]$ .

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Problem 15412

Gegeben ist die Funktion $f(x) = -\frac{1}{6} x^{3}+2 x$ . Analysiere Nullstellen, Symmetrie, Extrem- und Wendepunkte.

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Problem 15413

Differentiate $y=\frac{x \sqrt{x^{5}+9}}{(x-5)^{2 / 3}}$ using logarithmic differentiation. Show your work.

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Problem 15414

Find the derivative of $y=\sin ^{3}(x)$ , i.e., calculate $\frac{d y}{d x}$ .

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Problem 15415

Differentiate $y=\frac{x \sqrt{x^{3}+9}}{(x-5)^{2 / 3}}$ using logarithmic differentiation. Show your work.

See Solution

Problem 15416

Find the derivative of $y=\ln(-9x^{3}-5x^{2})$ .

See Solution

Problem 15417

Find the derivative of the function $f(x)=-\csc(4x)$ .

See Solution

Problem 15418

Find the derivatives for the function $f(x)=\ln(e^{x}-5x)$ . Calculate $f^{\prime}(x)$ and $f^{\prime}(0)$ .

See Solution

Problem 15419

Find the derivative of the function $y=\log _{6}(7 x^{5})$ .

See Solution

Problem 15420

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

See Solution

Problem 15421

Find the derivative of the function $y=-3 \sec (4 x)$ , which is $\frac{d y}{d x}$ .

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Problem 15422

To minimize an objective function with one local maximum on a closed interval, where should the student look?
A. Find where the first derivative is 0. B. Find where the second derivative is 0. C. Find where the function equals 0. D. Check the endpoints of the interval.

See Solution

Problem 15423

Find the derivative of the function $y=5^{-9 x^{6}+8 x^{5}}$ .

See Solution

Problem 15424

Find the derivative of $y=e^{-8 x^{5}-6 x^{4}}$ .

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Problem 15425

For the function $f(x)=x^{3} \ln (x)-x^{3}$ , find: (a) $f^{\prime}(x)$ and (b) $f^{\prime}(e)$ (round to five decimals).

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Problem 15426

Find $h^{\prime}(4)$ if $h(x)=f(g(x))$ using the values from the chart.

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Problem 15427

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

See Solution

Problem 15428

Find all $x$ -values where the function $f(x)=\int_{-1}^{x} \frac{1}{x}(t^{5}-10 t^{4}+25 t^{3}) dt$ has a point of inflection.

See Solution

Problem 15429

Find the population $N(9)$ in 2031 given $N(0)=5000$ and $\frac{d N}{d t}=400+600\sqrt{t}$ for $0 \leq t \leq 9$ .

See Solution

Problem 15430

a. With $400 \mathrm{~m}$ of fencing for three sides, find dimensions that maximize the area of a pen against a barn. b. For four adjacent pens of area $100 \mathrm{~m}^{2}$ each, determine dimensions that minimize fencing used.

See Solution

Problem 15431

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

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Problem 15432

A satellite orbits the Earth 40 times a year. Find its angular velocity in radians per week, rounded to 2 decimal places.

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Problem 15433

Find the time in hours for a bacteria population to double with a growth rate of $2.5\%$ per hour. Round to the nearest hundredth.

See Solution

Problem 15434

Find the half-life of a substance with a decay rate of $8.9\%$ per day using the continuous exponential decay model.

See Solution

Problem 15435

How many years will \$ 2100 invested at 4.75% interest, compounded continuously, take to double? Round to the nearest hundredth.

See Solution

Problem 15436

Encuentra la tasa de crecimiento horario de una población de bacterias que pasa de 1200 a 1607 en 6 horas, en %.

See Solution

Problem 15437

Find the tangent line equation for $g(x)=f(3x+3)$ at $x=1$ , given $f(1)=4, f'(1)=5$ .

See Solution

Problem 15438

Find $h^{\prime}(5)$ for $h(x)=f(g(x))$ using the given values in the chart.

See Solution

Problem 15439

Find the value of the integral $\int_{0}^{5} \frac{g^{\prime}(x)}{1+(g(x))^{2}} d x$ given $g(0)=0$ and $g(5)=1$ .

See Solution

Problem 15440

Find the half-life of a substance decaying at $5.4\%$ per day using the continuous exponential decay model.

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Problem 15441

Find the derivative $\frac{d y}{d x}$ for the equation: $3 x^{3} y^{2}-3 x^{2} y=6$ .

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Problem 15442

Calculate the future value of a \ $7,000 investment at 9\% interest compounded continuously for 6 years. \$\square$

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Problem 15443

Find the hourly growth rate of a bacteria population that grows from 1600 to 1881 in 4 hours. Express as a percentage.

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Problem 15444

Find $\frac{d y}{d x}$ in terms of $x$ and $y$ for the equation $-x^{3}-2 y^{3}+4 y^{2}+5 y=0$ .

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Problem 15445

Find the tangent line equation for $f(x)=-\frac{1}{6} e^{-\frac{x}{3}}+1$ at slope $\frac{1}{6}$ .

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Problem 15446

Find the limit as $x$ approaches 0 of $\frac{\int_{x}^{0} \sqrt{t^{2}+9} \, dt}{x}$ .

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Problem 15447

Find $\frac{d y}{d x}$ in terms of $x$ and $y$ for the equation $-x^{2} y+y=x^{3}+2$ .

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Problem 15448

Find $F'(2)$ for $F(x)=\int_{5x}^{x^2} f(t) dt$ using $f(2)=2$ , $f(4)=-1$ , $f(6)=6$ , $f(8)=-1$ .

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Problem 15449

Bestimmen Sie die Wendepunkte der Funktion $f(x)$ mit der ersten Ableitung $f^{\prime}(x)=e^{-b x} \cdot(1-b x-b^{2})$ .

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Problem 15450

Find the tangent lines to the curve $4 x^{3}=-5 x y+2$ at $x=2$ .

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Problem 15451

Find the tangent line equations for the curve $4xy=5y+2x^4$ at the point where $x=1$ .

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Problem 15452

Find the zeros, extrema, and inflection points of $f_{+}(x)=\frac{1}{x^{2}} e^{\frac{2 t}{x}}$ .

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Problem 15453

Calculez les dérivées des fonctions suivantes : 1. $f(x)=\mathrm{e}^{-\sqrt{x}}+\ln(x^{3}+1)$ 2. $f(x)=\log_{2}(1-x^{2})-\log_{2}(1+x)$ 3. $f(x)=\tan\left(\frac{1}{x+1}\right)$ 4. $f(x)=\arccos x+\arcsin(2x)$ 5. $f(x)=\pi^{x^{3}+2x-7}$ .

See Solution

Problem 15454

Find the tangent lines to the curve given by $-x y=4-2 x$ at the point where $x=4$ .

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Problem 15455

Find the value of $f^{\prime}\left(\frac{2}{5}\right)$ if $f(x)=\sin^{-1}(x)$ .

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Problem 15456

Find the value of $g^{\prime}(6)$ if $g(x)=f^{-1}(x)$ and $f(2)=6$ , $f^{\prime}(2)=-10$ .

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Problem 15457

Find $g^{\prime}(-4)$ for $g(x)=f^{-1}(x)$ where $f(x)=-x^{3}+3x^{2}-6x$ and $(1,-4)$ is on $f$ .

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Problem 15458

Find the third derivative $f^{\prime \prime \prime}(x)$ of the function $f(x)=\frac{1}{6}-2 x-\frac{4}{3} x^{-1}+\frac{3}{4} x^{4}$ .

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Problem 15459

Find the Maclaurin series for $f(x)=\frac{11}{1+x}$ .

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Problem 15460

Find the Taylor series for $f(x)=\frac{7}{x^{2}}$ at $x=1$ .

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Problem 15461

Find the limit using I'Hopital's Rule: $\lim _{x \rightarrow 0^{+}} \frac{\int_{0}^{x} \sqrt{t} \cos t \, dt}{x^{2}}=\square$

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Problem 15462

Find the second derivative of the function $y=-\frac{5}{2} x^{3}+\frac{1}{2} x^{2}-\frac{1}{3} x^{6}+\frac{1}{24} x$ .

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Problem 15463

Find the Maclaurin series for these functions: 1. $f(x)=\frac{11}{1+x}$ 2. $f(x)=6 \cos (-x)$

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Problem 15464

Find the third derivative of the function $y=\frac{5}{2} x^{3}+\frac{1}{6} x^{4}-\frac{1}{3} x^{-2}$ .

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Problem 15465

Find the third derivative $f^{\prime \prime \prime}(x)$ of the function $f(x)=\frac{1}{12} x^{-3}-\frac{1}{4} x^{4}-\frac{1}{6} x+2 x^{3}$ .

See Solution

Problem 15466

Calculate the sum of the series: $\sum_{j=1}^{\infty} \frac{5}{100^{j}}$ .

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Problem 15467

Find the third derivative $\frac{d^{3} y}{d x^{3}}$ for the function $y=\frac{1}{6} x^{4}+\frac{1}{6}-\frac{1}{3} x^{5}+\frac{1}{12} x^{6}$ .

See Solution

Problem 15468

A. Find dimensions for a pen with $400 \mathrm{~m}$ of fencing to maximize area. B. Determine dimensions for four pens of area $100 \mathrm{~m}^{2}$ each to minimize fencing used.

See Solution

Problem 15469

Find the limit: $\lim _{n \rightarrow \infty} \frac{n}{n^{2}+1}$ .

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Problem 15470

Find the derivative of $g_{t}(x)=\sqrt{t \cdot x \cdot e^{-x}+0.1 \cdot t}$ .

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Problem 15471

Is the series $\sum_{n=1}^{\infty}(-1)^{n}\left(1-\frac{7}{n}\right)^{n^{2}}$ convergent or divergent?

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Problem 15472

Caffeine's half-life is 1.5 hours. If tea has $40 \text{ mg}$ , how long $(t)$ until only $1 \text{ mg}$ remains?

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Problem 15473

Find $F(x)$ , the antiderivative of $f(x)=\sqrt{x}-x^{4}$ . Simplify $F(x)$ .

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Problem 15474

Find the slope of the tangent line to $f(x)=\frac{x^{2}}{4}$ at $x=3$ . Reflect it to find $g(x)$ and $g^{\prime}\left(\_\right)=$ .

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Problem 15475

Evaluate the limit: $\lim _{n \rightarrow \infty} \frac{1-n^{3}}{1+2 n^{3}}$

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Problem 15476

Construct the arc length integral for the parametric equations $x=\cos t$ and $y=\cos 3t$ for $0 \leq t \leq \pi$ .

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Problem 15477

Find the differential of the function $y=\tan (\sqrt{5 t})$ .

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Problem 15478

How does linear approximation help estimate $f$ near a point where $f$ and $f'$ are easy to evaluate? Select the correct option.

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Problem 15479

Calculate the integral: $\int \left( \frac{4}{3 x^{2}}+\frac{7}{2 x} \right) d x$

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Problem 15480

Determine the convergence of these series: a. $\sum_{n=1}^{\infty} \frac{n^{8}}{(-8)^{n}}$ b. $\sum_{n=1}^{\infty}(-1)^{n} \frac{n^{2}(n+3) !}{n ! 8^{3 n}}$

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Problem 15481

Estimate $f(5.85)$ using linear approximation with $f(6)=2$ and $f^{\prime}(6)=4$ .

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Problem 15482

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

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Problem 15483

Find the tangent line equation for the curve $y=\frac{1}{\sqrt[5]{5 x^{3}-2 x^{2}}}$ at $x=2$ .

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Problem 15484

Find the line equation for the linear approximation of $f(x)=\frac{x}{x+1}$ at $a=1$ and estimate $f(1.3)$ . Compute percent error: $100 \cdot \frac{\text { |approximation - exact } \mid}{\mid \text { exact } \mid}$ .

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Problem 15485

Find and simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=-5 x^{2}-3$ . Show your work.

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Problem 15486

Estimate $e^{0.01}$ using a linear approximation for a small error. Find $e^{0.01} \approx \square$ .

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Problem 15487

Estimate the gradient $|d P / d r|$ using Jeans mass, number density, and compare with $|d P / d r|=\mathrm{GM} \rho / \mathrm{r}^{2}$ .

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Problem 15488

Find dy for $y=\sin(15x^2)$ . What is $dy$ in terms of $dx$ ?

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Problem 15489

A bank offers 5% interest compounded continuously. Find when a deposit will (a) quadruple (b) increase by 60%.

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Problem 15490

Find $f^{\prime}(1)$ if $f(x)=g(2x+1)$ and the slope of $g(x)$ at $x=-1$ is -2.

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Problem 15491

Find $f^{\prime}(1)$ given $72 + 8 f(x) + 7 x^{2} (f(x))^{3} = 0$ and $f(1) = -2$ .

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Problem 15492

Approximate the volume change of a sphere as its radius goes from $r=20 \mathrm{ft}$ to $r=20.1 \mathrm{ft}$ . Use $V(r)=\frac{4}{3} \pi r^{3}$ . Find $\Delta V \approx \square \mathrm{ft}^{3}$ (round to the nearest hundredth).

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Problem 15493

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

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Problem 15494

Find the average velocity of the particle defined by $s(t)=\frac{t^{3}}{3}-4 t^{2}+3$ on $[0,3]$ seconds.

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Problem 15495

Show that the derivative operator $\frac{d}{d t}$ is a linear mapping from $\mathcal{P}_{3}$ to $\mathcal{P}_{2}$ .

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Problem 15496

Find the limit as $n$ approaches infinity for the expression $\frac{6 n+9}{3 n-2}$ .

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Problem 15497

Find the derivative $f^{\prime}(\pi)$ for the function $f(x)=2 \cos x+e^{x}$ .

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Problem 15498

Find the initial velocity and the velocity after 5 seconds for $v(t)=52(1-e^{-0.16 t})$ . Round to the nearest whole number.

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Problem 15499

Find the derivative of $h(x)=4 \sqrt{x} \ln x$ .

See Solution

Problem 15500

A circle's circumference $C$ is increasing at $2 \mathrm{~cm/s}$ . Find the area's increase rate when $r=6 \mathrm{~cm}$ .

See Solution

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Calculus

Problem 15401

Find the derivative of f(x)=e[x33]f(x)=e^{\left[\frac{x^{3}}{3}\right]}f(x)=e[3x3​], so f′(x)=□f'(x)=\squaref′(x)=□.

Problem 15402

Find the derivative of f(x)=ln⁡(x2+1)5f(x)=\ln \left(x^{2}+1\right)^{5}f(x)=ln(x2+1)5. What is f′(x)=?f^{\prime}(x)=?f′(x)=?

Problem 15403

Find the derivative of the function h(x)=cos⁡(6x35+4x2)h(x)=\cos \left(\frac{6 x^{3}}{5+4 x^{2}}\right)h(x)=cos(5+4x26x3​), denoted as h′(x)h^{\prime}(x)h′(x).

Problem 15404

Bestätigen Sie die Nullstellen, Extrem- und Wendepunkte der Funktionen: a) f(x)=x3−3x2−x+3f(x) = x^{3}-3 x^{2}-x+3f(x)=x3−3x2−x+3 b) f(x)=12x4−3x2+4f(x) = \frac{1}{2} x^{4}-3 x^{2}+4f(x)=21​x4−3x2+4 Geben Sie Monotonie- und Krümmungsintervalle an.

Problem 15405

Find the derivative of f(x)=ln⁡(x)f(x)=\ln (\sqrt{x})f(x)=ln(x​). What is f′(x)f^{\prime}(x)f′(x)?

Problem 15406

Find the derivative of the function y=(4−3x4x2−3)5y=\left(\frac{4-3 x}{4 x^{2}-3}\right)^{5}y=(4x2−34−3x​)5 with respect to xxx.

Problem 15407

Find the derivative of f(x)=exx5f(x)=\frac{e^{x}}{x^{5}}f(x)=x5ex​. What is f′(x)=□f^{\prime}(x)=\squaref′(x)=□?

Problem 15408

Check if the Mean Value Theorem applies to f(x)=∣x−5∣f(x)=|x-5|f(x)=∣x−5∣ on [−5,11][-5,11][−5,11] and find the guaranteed point(s).

Problem 15409

Find the derivative of f(x)=ln⁡(x2)f(x)=\ln(x^{2})f(x)=ln(x2). What is f′′(x)f^{\prime \prime}(x)f′′(x)?

Problem 15410

Find the derivative of f(x)=x6ln⁡(x)f(x)=x^{6} \ln (x)f(x)=x6ln(x).

Problem 15411

Find the absolute max and min of p(x)=x2−1p(x)=x^{2}-1p(x)=x2−1 on the interval [−5,3][-5,3][−5,3].

Problem 15412

Gegeben ist die Funktion f(x)=−16x3+2xf(x) = -\frac{1}{6} x^{3}+2 xf(x)=−61​x3+2x. Analysiere Nullstellen, Symmetrie, Extrem- und Wendepunkte.

Problem 15413

Differentiate y=xx5+9(x−5)2/3y=\frac{x \sqrt{x^{5}+9}}{(x-5)^{2 / 3}}y=(x−5)2/3xx5+9​​ using logarithmic differentiation. Show your work.

Problem 15414

Find the derivative of y=sin⁡3(x)y=\sin ^{3}(x)y=sin3(x), i.e., calculate dydx\frac{d y}{d x}dxdy​.

Problem 15415

Differentiate y=xx3+9(x−5)2/3y=\frac{x \sqrt{x^{3}+9}}{(x-5)^{2 / 3}}y=(x−5)2/3xx3+9​​ using logarithmic differentiation. Show your work.

Problem 15416

Find the derivative of y=ln⁡(−9x3−5x2)y=\ln(-9x^{3}-5x^{2})y=ln(−9x3−5x2).

Problem 15417

Find the derivative of the function f(x)=−csc⁡(4x)f(x)=-\csc(4x)f(x)=−csc(4x).

Problem 15418

Find the derivatives for the function f(x)=ln⁡(ex−5x)f(x)=\ln(e^{x}-5x)f(x)=ln(ex−5x). Calculate f′(x)f^{\prime}(x)f′(x) and f′(0)f^{\prime}(0)f′(0).

Problem 15419

Find the derivative of the function y=log⁡6(7x5)y=\log _{6}(7 x^{5})y=log6​(7x5).

Problem 15420

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

Problem 15421

Find the derivative of the function y=−3sec⁡(4x)y=-3 \sec (4 x)y=−3sec(4x), which is dydx\frac{d y}{d x}dxdy​.

Problem 15422

To minimize an objective function with one local maximum on a closed interval, where should the student look?A. Find where the first derivative is 0. B. Find where the second derivative is 0. C. Find where the function equals 0. D. Check the endpoints of the interval.

Problem 15423

Find the derivative of the function y=5−9x6+8x5y=5^{-9 x^{6}+8 x^{5}}y=5−9x6+8x5.

Problem 15424

Find the derivative of y=e−8x5−6x4y=e^{-8 x^{5}-6 x^{4}}y=e−8x5−6x4.

Problem 15425

For the function f(x)=x3ln⁡(x)−x3f(x)=x^{3} \ln (x)-x^{3}f(x)=x3ln(x)−x3, find: (a) f′(x)f^{\prime}(x)f′(x) and (b) f′(e)f^{\prime}(e)f′(e) (round to five decimals).

Problem 15426

Find h′(4)h^{\prime}(4)h′(4) if h(x)=f(g(x))h(x)=f(g(x))h(x)=f(g(x)) using the values from the chart.

Problem 15427

Find the limit: lim⁡n→∞(2n3+n2+12n3+n2−1)3n+2\lim _{n \rightarrow \infty}\left(\frac{2 n^{3}+n^{2}+1}{2 n^{3}+n^{2}-1}\right)^{3 n+2}limn→∞​(2n3+n2−12n3+n2+1​)3n+2.

Problem 15428

Find all xxx-values where the function f(x)=∫−1x1x(t5−10t4+25t3)dtf(x)=\int_{-1}^{x} \frac{1}{x}(t^{5}-10 t^{4}+25 t^{3}) dtf(x)=∫−1x​x1​(t5−10t4+25t3)dt has a point of inflection.

Problem 15429

Find the population N(9)N(9)N(9) in 2031 given N(0)=5000N(0)=5000N(0)=5000 and dNdt=400+600t\frac{d N}{d t}=400+600\sqrt{t}dtdN​=400+600t​ for 0≤t≤90 \leq t \leq 90≤t≤9.

Problem 15430

a. With 400 m400 \mathrm{~m}400 m of fencing for three sides, find dimensions that maximize the area of a pen against a barn. b. For four adjacent pens of area 100 m2100 \mathrm{~m}^{2}100 m2 each, determine dimensions that minimize fencing used.

Problem 15431

Find the limit: lim⁡n→∞(2u3+u2+12u3+u2−1)3u+2\lim _{n \rightarrow \infty}\left(\frac{2 u^{3}+u^{2}+1}{2 u^{3}+u^{2}-1}\right)^{3 u+2}limn→∞​(2u3+u2−12u3+u2+1​)3u+2.

Problem 15432

A satellite orbits the Earth 40 times a year. Find its angular velocity in radians per week, rounded to 2 decimal places.

Problem 15433

Find the time in hours for a bacteria population to double with a growth rate of 2.5%2.5\%2.5% per hour. Round to the nearest hundredth.

Problem 15434

Find the half-life of a substance with a decay rate of 8.9%8.9\%8.9% per day using the continuous exponential decay model.

Problem 15435

How many years will \$ 2100 invested at 4.75% interest, compounded continuously, take to double? Round to the nearest hundredth.

Problem 15436

Encuentra la tasa de crecimiento horario de una población de bacterias que pasa de 1200 a 1607 en 6 horas, en %.

Problem 15437

Find the tangent line equation for g(x)=f(3x+3)g(x)=f(3x+3)g(x)=f(3x+3) at x=1x=1x=1, given f(1)=4,f′(1)=5f(1)=4, f'(1)=5f(1)=4,f′(1)=5.

Problem 15438

Find h′(5)h^{\prime}(5)h′(5) for h(x)=f(g(x))h(x)=f(g(x))h(x)=f(g(x)) using the given values in the chart.

Problem 15439

Find the value of the integral ∫05g′(x)1+(g(x))2dx\int_{0}^{5} \frac{g^{\prime}(x)}{1+(g(x))^{2}} d x∫05​1+(g(x))2g′(x)​dx given g(0)=0g(0)=0g(0)=0 and g(5)=1g(5)=1g(5)=1.

Problem 15440

Find the derivative of $f(x)=e^{\left[\frac{x^{3}}{3}\right]}$ , so $f'(x)=\square$ .

Find the derivative of $f(x)=\ln \left(x^{2}+1\right)^{5}$ . What is $f^{\prime}(x)=?$

Find the derivative of the function $h(x)=\cos \left(\frac{6 x^{3}}{5+4 x^{2}}\right)$ , denoted as $h^{\prime}(x)$ .

Bestätigen Sie die Nullstellen, Extrem- und Wendepunkte der Funktionen: a) $f(x) = x^{3}-3 x^{2}-x+3$ b) $f(x) = \frac{1}{2} x^{4}-3 x^{2}+4$ Geben Sie Monotonie- und Krümmungsintervalle an.

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

Find the derivative of the function $y=\left(\frac{4-3 x}{4 x^{2}-3}\right)^{5}$ with respect to $x$ .

Find the derivative of $f(x)=\frac{e^{x}}{x^{5}}$ . What is $f^{\prime}(x)=\square$ ?

Check if the Mean Value Theorem applies to $f(x)=|x-5|$ on $[-5,11]$ and find the guaranteed point(s).

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

Find the derivative of $f(x)=x^{6} \ln (x)$ .

Find the absolute max and min of $p(x)=x^{2}-1$ on the interval $[-5,3]$ .

Gegeben ist die Funktion $f(x) = -\frac{1}{6} x^{3}+2 x$ . Analysiere Nullstellen, Symmetrie, Extrem- und Wendepunkte.

Differentiate $y=\frac{x \sqrt{x^{5}+9}}{(x-5)^{2 / 3}}$ using logarithmic differentiation. Show your work.

Find the derivative of $y=\sin ^{3}(x)$ , i.e., calculate $\frac{d y}{d x}$ .

Differentiate $y=\frac{x \sqrt{x^{3}+9}}{(x-5)^{2 / 3}}$ using logarithmic differentiation. Show your work.

Find the derivative of $y=\ln(-9x^{3}-5x^{2})$ .

Find the derivative of the function $f(x)=-\csc(4x)$ .

Find the derivatives for the function $f(x)=\ln(e^{x}-5x)$ . Calculate $f^{\prime}(x)$ and $f^{\prime}(0)$ .

Find the derivative of the function $y=\log _{6}(7 x^{5})$ .

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

Find the derivative of the function $y=-3 \sec (4 x)$ , which is $\frac{d y}{d x}$ .

To minimize an objective function with one local maximum on a closed interval, where should the student look?
A. Find where the first derivative is 0. B. Find where the second derivative is 0. C. Find where the function equals 0. D. Check the endpoints of the interval.

Find the derivative of the function $y=5^{-9 x^{6}+8 x^{5}}$ .

Find the derivative of $y=e^{-8 x^{5}-6 x^{4}}$ .

For the function $f(x)=x^{3} \ln (x)-x^{3}$ , find: (a) $f^{\prime}(x)$ and (b) $f^{\prime}(e)$ (round to five decimals).

Find $h^{\prime}(4)$ if $h(x)=f(g(x))$ using the values from the chart.

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

Find all $x$ -values where the function $f(x)=\int_{-1}^{x} \frac{1}{x}(t^{5}-10 t^{4}+25 t^{3}) dt$ has a point of inflection.

Find the population $N(9)$ in 2031 given $N(0)=5000$ and $\frac{d N}{d t}=400+600\sqrt{t}$ for $0 \leq t \leq 9$ .

a. With $400 \mathrm{~m}$ of fencing for three sides, find dimensions that maximize the area of a pen against a barn. b. For four adjacent pens of area $100 \mathrm{~m}^{2}$ each, determine dimensions that minimize fencing used.

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

Find the time in hours for a bacteria population to double with a growth rate of $2.5\%$ per hour. Round to the nearest hundredth.

Find the half-life of a substance with a decay rate of $8.9\%$ per day using the continuous exponential decay model.

Find the tangent line equation for $g(x)=f(3x+3)$ at $x=1$ , given $f(1)=4, f'(1)=5$ .

Find $h^{\prime}(5)$ for $h(x)=f(g(x))$ using the given values in the chart.

Find the value of the integral $\int_{0}^{5} \frac{g^{\prime}(x)}{1+(g(x))^{2}} d x$ given $g(0)=0$ and $g(5)=1$ .

Find the half-life of a substance decaying at $5.4\%$ per day using the continuous exponential decay model.

Find the derivative $\frac{d y}{d x}$ for the equation: $3 x^{3} y^{2}-3 x^{2} y=6$ .

Calculate the future value of a \ $7,000 investment at 9\% interest compounded continuously for 6 years. \$\square$

Find $\frac{d y}{d x}$ in terms of $x$ and $y$ for the equation $-x^{3}-2 y^{3}+4 y^{2}+5 y=0$ .

Find the tangent line equation for $f(x)=-\frac{1}{6} e^{-\frac{x}{3}}+1$ at slope $\frac{1}{6}$ .

Find the limit as $x$ approaches 0 of $\frac{\int_{x}^{0} \sqrt{t^{2}+9} \, dt}{x}$ .

Find $\frac{d y}{d x}$ in terms of $x$ and $y$ for the equation $-x^{2} y+y=x^{3}+2$ .

Find $F'(2)$ for $F(x)=\int_{5x}^{x^2} f(t) dt$ using $f(2)=2$ , $f(4)=-1$ , $f(6)=6$ , $f(8)=-1$ .

Bestimmen Sie die Wendepunkte der Funktion $f(x)$ mit der ersten Ableitung $f^{\prime}(x)=e^{-b x} \cdot(1-b x-b^{2})$ .

Find the tangent lines to the curve $4 x^{3}=-5 x y+2$ at $x=2$ .

Find the tangent line equations for the curve $4xy=5y+2x^4$ at the point where $x=1$ .

Find the zeros, extrema, and inflection points of $f_{+}(x)=\frac{1}{x^{2}} e^{\frac{2 t}{x}}$ .

Find the tangent lines to the curve given by $-x y=4-2 x$ at the point where $x=4$ .

Find the value of $f^{\prime}\left(\frac{2}{5}\right)$ if $f(x)=\sin^{-1}(x)$ .

Find the value of $g^{\prime}(6)$ if $g(x)=f^{-1}(x)$ and $f(2)=6$ , $f^{\prime}(2)=-10$ .

Find $g^{\prime}(-4)$ for $g(x)=f^{-1}(x)$ where $f(x)=-x^{3}+3x^{2}-6x$ and $(1,-4)$ is on $f$ .

Find the third derivative $f^{\prime \prime \prime}(x)$ of the function $f(x)=\frac{1}{6}-2 x-\frac{4}{3} x^{-1}+\frac{3}{4} x^{4}$ .

Find the Maclaurin series for $f(x)=\frac{11}{1+x}$ .

Find the Taylor series for $f(x)=\frac{7}{x^{2}}$ at $x=1$ .

Find the limit using I'Hopital's Rule: $\lim _{x \rightarrow 0^{+}} \frac{\int_{0}^{x} \sqrt{t} \cos t \, dt}{x^{2}}=\square$

Find the second derivative of the function $y=-\frac{5}{2} x^{3}+\frac{1}{2} x^{2}-\frac{1}{3} x^{6}+\frac{1}{24} x$ .

Find the Maclaurin series for these functions: 1. $f(x)=\frac{11}{1+x}$ 2. $f(x)=6 \cos (-x)$

Find the third derivative of the function $y=\frac{5}{2} x^{3}+\frac{1}{6} x^{4}-\frac{1}{3} x^{-2}$ .

Find the third derivative $f^{\prime \prime \prime}(x)$ of the function $f(x)=\frac{1}{12} x^{-3}-\frac{1}{4} x^{4}-\frac{1}{6} x+2 x^{3}$ .

Calculate the sum of the series: $\sum_{j=1}^{\infty} \frac{5}{100^{j}}$ .

Find the third derivative $\frac{d^{3} y}{d x^{3}}$ for the function $y=\frac{1}{6} x^{4}+\frac{1}{6}-\frac{1}{3} x^{5}+\frac{1}{12} x^{6}$ .

A. Find dimensions for a pen with $400 \mathrm{~m}$ of fencing to maximize area. B. Determine dimensions for four pens of area $100 \mathrm{~m}^{2}$ each to minimize fencing used.

Find the limit: $\lim _{n \rightarrow \infty} \frac{n}{n^{2}+1}$ .

Find the derivative of $g_{t}(x)=\sqrt{t \cdot x \cdot e^{-x}+0.1 \cdot t}$ .

Is the series $\sum_{n=1}^{\infty}(-1)^{n}\left(1-\frac{7}{n}\right)^{n^{2}}$ convergent or divergent?

Caffeine's half-life is 1.5 hours. If tea has $40 \text{ mg}$ , how long $(t)$ until only $1 \text{ mg}$ remains?

Find $F(x)$ , the antiderivative of $f(x)=\sqrt{x}-x^{4}$ . Simplify $F(x)$ .

Find the slope of the tangent line to $f(x)=\frac{x^{2}}{4}$ at $x=3$ . Reflect it to find $g(x)$ and $g^{\prime}\left(\_\right)=$ .