Calculus

Problem 7401

Find the derivative of the function $f(x)=\cos(5x-1)\sin(5x+3)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 7402

Find the derivative $\frac{d y}{d x}$ for $y=\frac{e^{x^{4}}}{\sqrt{3-x^{4}}$.

See Solution

Problem 7403

Find the derivative of $-8 \sin^{2}(-4 x^{5})$ using the chain rule twice.

See Solution

Problem 7404

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

See Solution

Problem 7405

Find the derivative of the function $f(x)=\cos (-3 x-6) \sin (3 x-4)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 7406

Calculate the difference quotient for $f(x) = 4x^2 + 3x$ using $\frac{f(x + h) - f(x)}{h}$ .

See Solution

Problem 7407

Find if there exists a value of $0 < x < 5$ such that the average rate of change of $f$ as $x$ approaches 0 equals 0.

See Solution

Problem 7408

Find the derivative of the implicit function given by $(x^{2}+y^{2})^{2}=y^{2}x^{2}-xy$ .

See Solution

Problem 7409

Una población $P$ crece según $dP/dt=0.5P-0.00125P^2$ . Halla $P(16)$ para $P(0)=40, 500, 100$ y compara los resultados.

See Solution

Problem 7410

Given $f(3)=5$ , $f^{\prime}(3)=2$ , find $D^{\prime}(3)$ for $D(x)=x^{2} f(x)$ . Use derivative rules.

See Solution

Problem 7411

Find the derivative $C^{\prime}(2)$ for $C(x)=\frac{f(x)}{g(x)}$ where $f(x) = x$ and $g(x) = -\frac{1}{2}x +2$ .

See Solution

Problem 7412

Select all true statements about derivatives:
1. If $f(x)=\frac{1}{x^{5}}$ , then $f^{\prime}(x)=\frac{1}{5 x^{4}}$ .
2. If $f(x)=x(x^{3}+5)$ , then $f^{\prime}(x)=3 x^{2}$ .
3. If $f(x)=\frac{x^{2}}{\sqrt{x}}$ , then $f^{\prime}(x)=\frac{2 x \sqrt{x}-\frac{1}{2} x^{\frac{3}{2}}}{x}$ .
4. If $f(x)=\sqrt[3]{x}$ , then $f^{\prime}(x)=\frac{1}{3} x^{-2/3}$ .
5. If $f(x)=5^{2}$ , then $f^{\prime}(x)=0$ .

See Solution

Problem 7413

If $f(5)=4$ , $f'(5)=-2$ , $g(5)=3$ , $g'(5)=7$ , then is $A'(5)=-14$ for $A(x)=f(x)g(x)$ ? True or False?

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Problem 7414

Find $b > 1$ such that the average rate of change of $f(x) = e^{2x}$ over $[1, b]$ equals 20. What equation to use?

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Problem 7415

Find when the displacement $s=t^{3}-3t^{2}$ is 0 for $t>0$ and the distance traveled in the first 4 seconds.

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Problem 7416

Find the global minimum and maximum of $f(x, y)=4 x^{3}+4 x^{2} y+5 y^{2}$ for $x, y \geq 0$ and $x+y \leq 1$ .

See Solution

Problem 7417

Find the global minimum and maximum of $f(x, y)=4 x^{3}-8 y$ for $0 \leq x, y \leq 1$ . $f_{\min }=$ $f_{\max }=$

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Problem 7418

Find max and min of $f(x, y, z)=y z+x y$ with constraints $y^{2}+z^{2}=4$ and $x y=5$ . Max value is Min value is.

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Problem 7419

Find the derivative of $y=(6-x^{9})^{7}$ with respect to $x$ : $\frac{d y}{d x}=$

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Problem 7420

Find the limit: $\lim _{x \rightarrow 0} x\left(13+\frac{6}{x}\right)$ .

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Problem 7421

Find the derivative of the function $F(x)=(4 x+7)^{3}(x^{2}-6 x+7)^{4}$ . What is $F^{\prime}(x)$ ?

See Solution

Problem 7422

Evaluate: a $\lim _{x \rightarrow 0} 5$

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Problem 7423

Find the limit: $\lim _{x \rightarrow 3}(x+4)$ .

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Problem 7424

Gegeben ist die Funktion $f_{a}(x)=(x^{2}+a) \cdot e^{0,5-x}$ .
a) Bestimmen Sie die Nullstellen von $f_{a}$ in Abhängigkeit von $a$ und das Verhalten für $x \rightarrow \infty$ und $x \rightarrow -\infty$ .
b) Geben Sie den $y$ -Achsen-Schnittpunkt von $G_{a}$ an und identifizieren Sie die Graphen für ganzzahlige $a$ .
c) Berechnen Sie die Fläche $A$ zwischen $G_{2}$ , $G_{0}$ , der $y$ -Achse und der Linie $x=3$ .
d) Zeigen Sie, dass $G_{a}$ für $a>1$ keine Extrempunkte hat.
e) Beweisen Sie $f_{2}^{\prime \prime}(x)=(x-2)^{2} e^{0,5-x}$ und erläutern Sie die Auswirkungen auf $G_{2}$ .
f) Bestimmen Sie die Tangentengleichung $t$ an $G_{2}$ in $x=2$ und zeigen Sie, dass der Funktionswert von $t$ bei $x=1$ weniger als $2\%$ vom Wert von $f_{2}$ abweicht.
g) Finden Sie die zwei Stellen, an denen die Vase einen maximalen Radius von ca. 1,07 dm hat.
h) Interpretieren Sie die Funktion $b(t)=\pi \cdot \int_{3-t}^{3}\left(f_{0,65}(x)\right)^{2} dx$ im Kontext.

See Solution

Problem 7425

Set up a definite integral for the volume of the solid formed by rotating $y=\sqrt{x-1}$ , $y=0$ , $x=5$ about the $x$ -axis.

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Problem 7426

Find the first and second derivatives of $f(x)=2x\left(\frac{2}{x^{6}}-x\right)$ .

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Problem 7427

Finde die Tangentengleichung $t_{1}$ an $f(x)=\frac{1}{3} x^{2}$ bei $x_{0}=-2$ und die orthogonale Tangente.

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Problem 7428

Calculate the first and second derivatives of $g(x)=12(\sqrt{x})^{3}+5 \sqrt[4]{x}$ .

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Problem 7429

Find the first and second derivatives of $h(x)=2 \ln \left(x^{5 / 2}\right)$ .

See Solution

Problem 7430

Find critical points and absolute max/min of $f(x)=3 x^{\frac{1}{2}}-x^{\frac{5}{2}}$ on $[0,9]$ . Critical point(s): $\mathrm{x}=$

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Problem 7431

Find critical points and absolute max/min of $f(x)=3 x^{\frac{1}{2}}-x^{\frac{5}{2}}$ on $[0,9]$ . Critical point(s): $x=$

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Problem 7432

Find critical points and absolute max/min of $f(x)=2 x^{\frac{1}{2}}-x^{\frac{5}{2}}$ on $[0,25]$ . Critical points: $x=$

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Problem 7433

Differentiate $h(x)=e^{-a x} \cos (b x)$ using the product rule. Find $h^{\prime}(x)=$

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Problem 7434

Find the tangent line of $y=e^{4 x}$ parallel to $y=7 x+17$ . Give the equation in the form $y=m x+b$ .

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Problem 7435

Find the critical points of $f(x)=\frac{12 x}{x^{2}+36}$ and its derivative $f'(x)$ .

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Problem 7436

Find critical points of $f(x)=6 \sqrt{x}-7 x$ on the interval $[0,4]$ .

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Problem 7437

Find the critical points of the function $f(x)=\frac{x}{x^{2}+25}$ .

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Problem 7438

Find the critical points of the function $f(x)=x^{2} \sqrt{x+22}$ .

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Problem 7439

Find the derivative of $y = e^{\sqrt{6x + 4}}$ . What is $y'$ ?

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Problem 7440

Find the derivative of $h(x) = \frac{7 x^{3} e^{x}}{x^{4}-x^{2}}$ using the quotient rule. What is $h^{\prime}(x)$ ?

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Problem 7441

Set up a definite integral to find the area between the curves $y=12-x^{2}$ and $y=x^{2}-6$ .

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Problem 7442

Find the absolute maximum of $f(x)=x^{3}-6x^{2}$ on $[-2,6]$ . What is the max value and its location? A or B?

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Problem 7443

Find the absolute extreme values of $f(x)=(x-3)^{\frac{4}{3}}$ on the interval $[-5,5]$ . What are they?

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Problem 7444

Find the absolute maximum of $f(x)=x^{3}-57 x^{2}$ on $[-19,57]$ . Where is it located? A or B?

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Problem 7445

Find the storm's speed at 3:00 PM, given $s(t)=25+25t-6t^{2}$ for $t$ hours after noon.

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Problem 7446

Find the absolute extrema of $f(x)=(x-3)^{\frac{4}{3}}$ on the interval $[-5,5]$ .

See Solution

Problem 7447

Find the first and second derivatives of $k(x)=b^{\log_b(x)}$ , where $b>1$ .

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Problem 7448

Find the stone's velocity after it rises 8 feet, given height $h(t)=5t-\frac{1}{2}t^{2}$ .

See Solution

Problem 7449

Find the critical point(s) of the function $f(x)=-4 \sqrt{x}+3 x$ on the interval $[0,4]$ .

See Solution

Problem 7450

Find the absolute max and min of $f(x)=x^{3}-6x^{2}$ on $[-2,6]$ . What are their values and locations?

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Problem 7451

Find $\frac{d P}{d h}$ using the Chain Rule for $T=15.04-0.00649 h$ and $P=101.29\left(\frac{T+273.1}{288.08}\right)^{5.256}$ .

See Solution

Problem 7452

Find the absolute extreme values of $f(x)=5 \cos ^{2} x$ on the interval $[0, \pi]$ .

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Problem 7453

Find the derivative of the composite function $f(g(x))$ at $x = 6$ using the given values. Provide a whole number answer.

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Problem 7454

Determine how fast the area of a square changes as side length $x$ changes when $x=3 \mathrm{~cm}$ .

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Problem 7455

Find functions $f(x)$ and $g(x)$ such that $y=(2x+3\sin(x))^{5}=f(g(x))$ , then compute the derivative using the Chain Rule.

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Problem 7456

Find the derivative of $y=\cos^{9}(\theta+5)$ using the General Power Rule or Chain Rule.

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Problem 7457

Find the absolute extremes of $f(x)=5 \cos ^{2} x$ on $[0, \pi]$ . Choose A, B, C, or D and fill in values.

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Problem 7458

Find the absolute extreme values of $f(x)=\cos 5x$ on $[-\frac{\pi}{6}, \frac{\pi}{3}]$ . What are they?

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Problem 7459

Find the absolute extreme values of $f(x)=-4 \csc x$ on the interval $\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right]$ .

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Problem 7460

A stone is launched from a cliff $384 \mathrm{ft}$ high at $32 \mathrm{ft/s}$ . When does it reach max height? Find $s'$ . $s' =$

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Problem 7461

Find the absolute extreme values of $f(x)=\cos 5x$ on the interval $\left[-\frac{\pi}{6}, \frac{\pi}{3}\right]$ .

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Problem 7462

A stone is launched from a cliff 128 ft high at 32 ft/s. Find when it reaches max height and the derivative $s'$ .

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Problem 7463

Find the absolute extreme values of $f(x)=-4 \csc x$ on the interval $\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right]$ .

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Problem 7464

A scientist has 176 grams of goo. After 180 minutes, it decays to 11 grams. How much remains after 20 minutes?

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Problem 7465

A stone is launched from a cliff $192 \mathrm{ft}$ high at $64 \mathrm{ft/s}$ . When does it reach max height? Find $s'$ . $s' =$

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Problem 7466

A stone is launched from a cliff at $192 \mathrm{ft}$ with a speed of $64 \mathrm{ft/s}$ . Find when it reaches max height using $s=-16 t^{2}+64 t+192$ .

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Problem 7467

Find the derivative of $15 \sin(x^{7}) \cos(x^{7})$ .

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Problem 7468

Find the slope $m$ of the tangent line to the curve $y=\frac{x+3}{x-6}$ at the point $(5,-8)$ and its equation.

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Problem 7469

Find the elevator's descent speed when it's 500 feet away from you and moving closer at $16 \mathrm{ft} / \mathrm{se}$ .

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Problem 7470

Find the derivative of $(4 e^{2 x}+12 e^{-8 x})^{12}$ .

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Problem 7471

Find the terminal velocity of a 202 lb skydiver given $v(t)=\frac{202}{1.1}\left(1-e^{-24.2 t / 202}\right)$ and time to reach 74% of it.

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Problem 7472

Find the second derivative of $\sin(4x^4)$ . Calculate: $\frac{d^{2}}{d x^{2}} \sin \left(4 x^{4}\right) =$

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Problem 7473

Find the third derivative of $(14-6 x)^{6}$ . What is $\frac{d^{3}}{d x^{3}}(14-6 x)^{6}=?$

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Problem 7474

Find the tangent line to $y=e^{2 x}$ that is parallel to $y=5 x+15$ . Give your answer as $y=m x+b$ , with exact $m$ and $b$ .

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Problem 7475

Calculate the derivative of $f(g(x))$ at $x=6$ using the given values. Answer as a whole number: $\left.\frac{d}{d x} f(g(x))\right|_{x=6} ^{-}=$

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Problem 7476

Evaluate the series: $\sum_{n=0}^{\infty} \frac{e^{n}}{\sqrt{n !}}$ .

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Problem 7477

Find the derivative $y^{\prime}$ of the function $y=\frac{\sqrt{2 x}}{\sqrt[4]{x}}$ .

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Problem 7478

Calculate the sum: $\sum_{n=1}^{\infty} \frac{n !}{100^{n}}$ .

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Problem 7479

Find the slope of secant lines from $P\left(x, 4 x^{2}\right)$ to $Q\left(x+\delta x, 4(x+\delta x)^{2}\right)$ as $\delta x \rightarrow 0$ . Investigate for $x=1$ .

See Solution

Problem 7480

Calculate $\frac{d P}{d h}$ using the Chain Rule for $T=15.04-0.00649 h$ and $P=101.29\left(\frac{T+273.1}{288.08}\right)^{5.256}$ .

See Solution

Problem 7481

Calculate $\frac{d P}{d h}$ using the Chain Rule for the formulas $T=15.04-0.00649 h$ and $P=101.29\left(\frac{T+273.1}{288.08}\right)^{5.256}$ .

See Solution

Problem 7482

Find the value of the series: $\sum_{n=0}^{\infty} \frac{\sqrt{(2 n) !}}{n !}$ .

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Problem 7483

Test these series for convergence:
1. $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt{n}}$
2. $\sum_{n=1}^{\infty} \frac{(-2)^{n}}{n^{2}}$
3. $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{2}}$
4. $\sum_{n=1}^{\infty} \frac{(-3)^{n}}{n !}$
5. $\sum_{n=2}^{\infty} \frac{(-1)^{n}}{\ln n}$
6. $\sum_{n=1}^{\infty} \frac{(-1)^{n} n}{n+5}$
7. $\sum_{n=0}^{\infty} \frac{(-1)^{n} n}{1+n^{2}}$
8. $\sum_{n=1}^{\infty} \frac{(-1)^{n} \sqrt{10 n}}{n+2}$

See Solution

Problem 7484

Use the ratio test to determine convergence or divergence for these series:
1. $\sum_{n=1}^{\infty} \frac{2^{n}}{n^{2}}$
2. $\sum_{n=0}^{\infty} \frac{3^{n}}{2^{2 n}}$
3. $\sum_{n=0}^{\infty} \frac{n !}{(2 n) !}$
4. $\sum_{n=0}^{\infty} \frac{5^{n}(n !)^{2}}{(2 n) !}$
5. $\sum_{n=1}^{\infty} \frac{10^{n}}{(n !)^{2}}$
6. $\sum_{n=1}^{\infty} \frac{n !}{100^{n}}$
7. $\sum_{n=0}^{\infty} \frac{3^{2 n}}{2^{3 n}}$
8. $\sum_{n=0}^{\infty} \frac{e^{n}}{\sqrt{n !}}$
9. $\sum_{n=0}^{\infty} \frac{(n !)^{3} e^{3 n}}{(3 n) !}$
10. $\sum_{n=0}^{\infty} \frac{100^{n}}{n^{200}}$
11. $\sum_{n=0}^{\infty} \frac{n !(2 n) !}{(3 n) !}$
12. $\sum_{n=0}^{\infty} \frac{\sqrt{(2 n) !}}{n !}$

See Solution

Problem 7485

Find the interval of convergence for these power series and check the endpoints:
1. $\sum_{n=0}^{\infty}(-1)^{n} x^{n}$
4. $\sum_{n=1}^{\infty} \frac{x^{2 n}}{2^{n} n^{2}}$
7. $\sum_{n=1}^{\infty} \frac{x^{3 n}}{n}$
10. $\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{2 n}}{(2 n)^{3 / 2}}$
13. $\sum_{n=1}^{\infty} \frac{n(-x)^{n}}{n^{2}+1}$
16. $\sum_{n=1}^{\infty} \frac{(x-1)^{n}}{2^{n}}$
2. $\sum_{n=0}^{\infty} \frac{(2 x)^{n}}{3^{n}}$
5. $\sum_{n=1}^{\infty} \frac{x^{n}}{(n !)^{2}}$
8. $\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{n}}{\sqrt{n}}$
11. $\sum_{n=1}^{\infty} \frac{1}{n}\left(\frac{x}{5}\right)^{n}$
14. $\sum_{n=1}^{\infty} \frac{n}{n+1}\left(\frac{x}{3}\right)^{n}$
17. $\sum_{n=1}^{\infty} \frac{(-1)^{n}(x+1)^{n}}{n}$
3. $\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{n}}{n(n+1)}$
6. $\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{n}}{(2 n) !}$
9. $\sum_{n=1}^{\infty}(-1)^{n} n^{3} x^{n}$
12. $\sum_{n=1}^{\infty} n(-2 x)^{n}$
15. $\sum_{n=1}^{\infty} \frac{(x-2)^{n}}{3^{n}}$
18. $\sum_{n=1}^{\infty} \frac{(-2)^{n}(2 x+1)^{n}}{n^{2}}$

See Solution

Problem 7486

Find point $B$ on the curve $2y=3x^3-7x^2+4x$ . Then find point $N$ where normals at $O$ and $A$ intersect, and calculate area of $\triangle OAN$ .

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Problem 7487

Find the first few terms of the Maclaurin series for these functions:
20. $e^{x} \sin x$
21. $\tan ^{2} x$
22. $\frac{e^{x}}{1-x}$
23. $\frac{1}{1+x+x^{2}}$
24. $\sec x$
25. $\frac{2 x}{e^{2 x}-1}$
26. $\frac{1}{\sqrt{\cos x}}$
27. $e^{\sin x}$
28. $\sin [\ln (1+x)]$
29. $\sqrt{1+\ln (1+x)}$
30. $\sqrt{\frac{1-x}{1+x}}$
31. $\cos \left(e^{x}-1\right)$
32. $\ln \left(1+x e^{x}\right)$
33. $\frac{1-\sin x}{1-x}$
34. $\ln \left(2-e^{-x}\right)$
35. $\frac{x}{\sin x}$
36. $\int_{0}^{u} \frac{\sin x d x}{\sqrt{1-x^{2}}}$

See Solution

Problem 7488

Find the derivative $\frac{d v}{d s}$ for the function $v(s)=\tan^{2}(\sin^{3}(s))$ .

See Solution

Problem 7489

Find the derivative of $v(s)=\tan^{2}(\sin^{3}(s))$ .

See Solution

Problem 7490

Find the derivative $\frac{d s}{d t}$ for $s(t)=\left(\frac{1-\cos (2 t)}{1+\cos (2 t)}\right)^{4}$ .

See Solution

Problem 7491

Find the whole number value of $f^{\prime}(0.25)$ for the function $f(x)=\frac{1}{x^{2} \sqrt{x}}$ .

See Solution

Problem 7492

Find the derivatives: (a) $h'(4)$ for $h(x)=f(g(x))$ , (b) $h'(2)$ for $h(x)=g(f(x))$ , (c) $h'(1)$ for $h(x)=\sqrt{g(x)}$ .

See Solution

Problem 7493

Find the derivative $\frac{d v}{d s}$ for the function $v(s)=\tan^{2}(\sin^{3}(s))$ .

See Solution

Problem 7494

Bestimmen Sie die Werte für a, sodass der Graph $G_{f_{0}}$ rechtsgekrümmt ist für: a) $f_{a}(x)=a \cdot(2 x-3)^{4}$ , b) $f_{a}(x)=\sqrt{a^{2} x}$ , c) $f_{a}(x)=\frac{a}{x^{2}}$ .

See Solution

Problem 7495

Find the derivatives: (a) $h^{\prime}(4)$ for $h(x)=f(g(x))$ (b) $h^{\prime}(2)$ for $h(x)=g(f(x))$ (c) $h^{\prime}(1)$ for $h(x)=\sqrt{g(x)}$ (d) $h^{\prime}(4)$ for $h(x)=(f(x))^{3/2}$ (e) $h^{\prime}(3)$ for $h(x)=f(x)g(x)$ (f) $h^{\prime}(2)$ for $h(x)=\frac{f(x)}{g(x)}$

See Solution

Problem 7496

Find the derivative $\frac{d y}{d x}$ for: a) $y=(x^{2}-6 x)^{-10}$ , b) $y=\sqrt{(1+3 x^{2})^{5}}$ .

See Solution

Problem 7497

Finde die Extremstellen und bestimme Minimum oder Maximum für die Funktionen: a) $f(x)=2x^{3}-1,5x^{2}-9x$ , b) $g(x)=x^{2}-4\sqrt{x}$ , c) $h(x)=\frac{1}{4}x^{4}-2x^{2}-2$ , d) $k(x)=x^{4}-\frac{4}{x}$ , e) $l(x)=-x^{4}+8x^{2}$ , f) $m(x)=2x-\frac{1}{x^{2}}$ .

See Solution

Problem 7498

Gegeben ist die konstante Funktion $f(x)=c$ . Erklären Sie, warum das VZW- und $f^{\prime \prime}$ -Kriterium keine Extremstellen finden.

See Solution

Problem 7499

Berechnen Sie die 1. und 2. Ableitung von $f$ und geben Sie $D_{f}, D_{f^{\prime}}$ und $D_{f^{\prime \prime}}$ an für: a) $f(x)=2 \sin (x)+x^{-3}$ , b) $f(t)=-4 \sqrt{t}-3 t$ , c) $f(t)=t^{4}-\sqrt[4]{t}$ , d) $f(s)=\frac{2}{\sqrt{s}}-3 s^{5}$ , e) $f(x)=-\cos (x)+\frac{1}{x^{2}}$ , f) $f(x)=\sqrt[3]{x}+3 \cos (x)$ .

See Solution

Problem 7500

Show that the gradient of the curve $y=2 x^{3}-2 x^{2}+5 x+8$ is always non-negative.

See Solution

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Calculus

Problem 7401

Find the derivative of the function f(x)=cos⁡(5x−1)sin⁡(5x+3)f(x)=\cos(5x-1)\sin(5x+3)f(x)=cos(5x−1)sin(5x+3). What is f′(x)f^{\prime}(x)f′(x)?

Problem 7402

Find the derivative dydx\frac{d y}{d x}dxdy​ for $y=\frac{e^{x^{4}}}{\sqrt{3-x^{4}}$.

Problem 7403

Find the derivative of −8sin⁡2(−4x5)-8 \sin^{2}(-4 x^{5})−8sin2(−4x5) using the chain rule twice.

Problem 7404

Find the derivative of the function f(x)=(−3x2−1x+6)sin⁡(4x)f(x)=(-3 x^{2}-1 x+6) \sin (4 x)f(x)=(−3x2−1x+6)sin(4x). What is f′(x)f^{\prime}(x)f′(x)?

Problem 7405

Find the derivative of the function f(x)=cos⁡(−3x−6)sin⁡(3x−4)f(x)=\cos (-3 x-6) \sin (3 x-4)f(x)=cos(−3x−6)sin(3x−4). What is f′(x)f^{\prime}(x)f′(x)?

Problem 7406

Calculate the difference quotient for f(x)=4x2+3xf(x) = 4x^2 + 3xf(x)=4x2+3x using f(x+h)−f(x)h\frac{f(x + h) - f(x)}{h}hf(x+h)−f(x)​.

Problem 7407

Find if there exists a value of 0<x<50 < x < 50<x<5 such that the average rate of change of fff as xxx approaches 0 equals 0.

Problem 7408

Find the derivative of the implicit function given by (x2+y2)2=y2x2−xy(x^{2}+y^{2})^{2}=y^{2}x^{2}-xy(x2+y2)2=y2x2−xy.

Problem 7409

Una población PPP crece según dP/dt=0.5P−0.00125P2dP/dt=0.5P-0.00125P^2dP/dt=0.5P−0.00125P2. Halla P(16)P(16)P(16) para P(0)=40,500,100P(0)=40, 500, 100P(0)=40,500,100 y compara los resultados.

Problem 7410

Given f(3)=5f(3)=5f(3)=5, f′(3)=2f^{\prime}(3)=2f′(3)=2, find D′(3)D^{\prime}(3)D′(3) for D(x)=x2f(x)D(x)=x^{2} f(x)D(x)=x2f(x). Use derivative rules.

Problem 7411

Find the derivative C′(2)C^{\prime}(2)C′(2) for C(x)=f(x)g(x)C(x)=\frac{f(x)}{g(x)}C(x)=g(x)f(x)​ where f(x)=xf(x) = xf(x)=x and g(x)=−12x+2g(x) = -\frac{1}{2}x +2g(x)=−21​x+2.

Problem 7412

Problem 7413

If f(5)=4f(5)=4f(5)=4, f′(5)=−2f'(5)=-2f′(5)=−2, g(5)=3g(5)=3g(5)=3, g′(5)=7g'(5)=7g′(5)=7, then is A′(5)=−14A'(5)=-14A′(5)=−14 for A(x)=f(x)g(x)A(x)=f(x)g(x)A(x)=f(x)g(x)? True or False?

Problem 7414

Find b>1b > 1b>1 such that the average rate of change of f(x)=e2xf(x) = e^{2x}f(x)=e2x over [1,b][1, b][1,b] equals 20. What equation to use?

Problem 7415

Find when the displacement s=t3−3t2s=t^{3}-3t^{2}s=t3−3t2 is 0 for t>0t>0t>0 and the distance traveled in the first 4 seconds.

Problem 7416

Find the global minimum and maximum of f(x,y)=4x3+4x2y+5y2f(x, y)=4 x^{3}+4 x^{2} y+5 y^{2}f(x,y)=4x3+4x2y+5y2 for x,y≥0x, y \geq 0x,y≥0 and x+y≤1x+y \leq 1x+y≤1.

Problem 7417

Find the global minimum and maximum of f(x,y)=4x3−8yf(x, y)=4 x^{3}-8 yf(x,y)=4x3−8y for 0≤x,y≤10 \leq x, y \leq 10≤x,y≤1. fmin⁡= f_{\min }= fmin​= fmax⁡= f_{\max }= fmax​=

Problem 7418

Find max and min of f(x,y,z)=yz+xyf(x, y, z)=y z+x yf(x,y,z)=yz+xy with constraints y2+z2=4y^{2}+z^{2}=4y2+z2=4 and xy=5x y=5xy=5. Max value is Min value is.

Problem 7419

Find the derivative of y=(6−x9)7y=(6-x^{9})^{7}y=(6−x9)7 with respect to xxx: dydx=\frac{d y}{d x}=dxdy​=

Problem 7420

Find the limit: lim⁡x→0x(13+6x)\lim _{x \rightarrow 0} x\left(13+\frac{6}{x}\right)limx→0​x(13+x6​).

Problem 7421

Find the derivative of the function F(x)=(4x+7)3(x2−6x+7)4F(x)=(4 x+7)^{3}(x^{2}-6 x+7)^{4}F(x)=(4x+7)3(x2−6x+7)4. What is F′(x)F^{\prime}(x)F′(x)?

Problem 7422

Evaluate: a lim⁡x→05\lim _{x \rightarrow 0} 5limx→0​5

Problem 7423

Find the limit: lim⁡x→3(x+4)\lim _{x \rightarrow 3}(x+4)limx→3​(x+4).

Problem 7424

Problem 7425

Set up a definite integral for the volume of the solid formed by rotating y=x−1y=\sqrt{x-1}y=x−1​, y=0y=0y=0, x=5x=5x=5 about the xxx-axis.

Problem 7426

Find the first and second derivatives of f(x)=2x(2x6−x)f(x)=2x\left(\frac{2}{x^{6}}-x\right)f(x)=2x(x62​−x).

Problem 7427

Finde die Tangentengleichung t1t_{1}t1​ an f(x)=13x2f(x)=\frac{1}{3} x^{2}f(x)=31​x2 bei x0=−2x_{0}=-2x0​=−2 und die orthogonale Tangente.

Problem 7428

Calculate the first and second derivatives of g(x)=12(x)3+5x4g(x)=12(\sqrt{x})^{3}+5 \sqrt[4]{x}g(x)=12(x​)3+54x​.

Problem 7429

Find the first and second derivatives of h(x)=2ln⁡(x5/2)h(x)=2 \ln \left(x^{5 / 2}\right)h(x)=2ln(x5/2).

Problem 7430

Find critical points and absolute max/min of f(x)=3x12−x52f(x)=3 x^{\frac{1}{2}}-x^{\frac{5}{2}}f(x)=3x21​−x25​ on [0,9][0,9][0,9]. Critical point(s): x=\mathrm{x}=x=

Problem 7431

Find critical points and absolute max/min of f(x)=3x12−x52f(x)=3 x^{\frac{1}{2}}-x^{\frac{5}{2}}f(x)=3x21​−x25​ on [0,9][0,9][0,9]. Critical point(s): x=x=x=

Problem 7432

Find critical points and absolute max/min of f(x)=2x12−x52f(x)=2 x^{\frac{1}{2}}-x^{\frac{5}{2}}f(x)=2x21​−x25​ on [0,25][0,25][0,25]. Critical points: x=x=x=

Problem 7433

Differentiate h(x)=e−axcos⁡(bx)h(x)=e^{-a x} \cos (b x)h(x)=e−axcos(bx) using the product rule. Find h′(x)=h^{\prime}(x)= h′(x)=

Problem 7434

Find the tangent line of y=e4xy=e^{4 x}y=e4x parallel to y=7x+17y=7 x+17y=7x+17. Give the equation in the form y=mx+by=m x+by=mx+b.

Problem 7435

Find the critical points of f(x)=12xx2+36f(x)=\frac{12 x}{x^{2}+36}f(x)=x2+3612x​ and its derivative f′(x)f'(x)f′(x).

Problem 7436

Find critical points of f(x)=6x−7xf(x)=6 \sqrt{x}-7 xf(x)=6x​−7x on the interval [0,4][0,4][0,4].

Problem 7437

Find the critical points of the function f(x)=xx2+25f(x)=\frac{x}{x^{2}+25}f(x)=x2+25x​.

Problem 7438

Find the critical points of the function f(x)=x2x+22f(x)=x^{2} \sqrt{x+22}f(x)=x2x+22​.

Problem 7439

Find the derivative of y=e6x+4y = e^{\sqrt{6x + 4}}y=e6x+4​. What is y′y'y′?

Problem 7440

Find the derivative of h(x)=7x3exx4−x2h(x) = \frac{7 x^{3} e^{x}}{x^{4}-x^{2}}h(x)=x4−x27x3ex​ using the quotient rule. What is h′(x)h^{\prime}(x)h′(x)?

Problem 7441

Find the derivative of the function $f(x)=\cos(5x-1)\sin(5x+3)$ . What is $f^{\prime}(x)$ ?

Find the derivative $\frac{d y}{d x}$ for $y=\frac{e^{x^{4}}}{\sqrt{3-x^{4}}$.

Find the derivative of $-8 \sin^{2}(-4 x^{5})$ using the chain rule twice.

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

Find the derivative of the function $f(x)=\cos (-3 x-6) \sin (3 x-4)$ . What is $f^{\prime}(x)$ ?

Calculate the difference quotient for $f(x) = 4x^2 + 3x$ using $\frac{f(x + h) - f(x)}{h}$ .

Find if there exists a value of $0 < x < 5$ such that the average rate of change of $f$ as $x$ approaches 0 equals 0.

Find the derivative of the implicit function given by $(x^{2}+y^{2})^{2}=y^{2}x^{2}-xy$ .

Una población $P$ crece según $dP/dt=0.5P-0.00125P^2$ . Halla $P(16)$ para $P(0)=40, 500, 100$ y compara los resultados.

Given $f(3)=5$ , $f^{\prime}(3)=2$ , find $D^{\prime}(3)$ for $D(x)=x^{2} f(x)$ . Use derivative rules.

Find the derivative $C^{\prime}(2)$ for $C(x)=\frac{f(x)}{g(x)}$ where $f(x) = x$ and $g(x) = -\frac{1}{2}x +2$ .

If $f(5)=4$ , $f'(5)=-2$ , $g(5)=3$ , $g'(5)=7$ , then is $A'(5)=-14$ for $A(x)=f(x)g(x)$ ? True or False?

Find $b > 1$ such that the average rate of change of $f(x) = e^{2x}$ over $[1, b]$ equals 20. What equation to use?

Find when the displacement $s=t^{3}-3t^{2}$ is 0 for $t>0$ and the distance traveled in the first 4 seconds.

Find the global minimum and maximum of $f(x, y)=4 x^{3}+4 x^{2} y+5 y^{2}$ for $x, y \geq 0$ and $x+y \leq 1$ .

Find the global minimum and maximum of $f(x, y)=4 x^{3}-8 y$ for $0 \leq x, y \leq 1$ . $f_{\min }=$ $f_{\max }=$

Find max and min of $f(x, y, z)=y z+x y$ with constraints $y^{2}+z^{2}=4$ and $x y=5$ . Max value is Min value is.

Find the derivative of $y=(6-x^{9})^{7}$ with respect to $x$ : $\frac{d y}{d x}=$

Find the limit: $\lim _{x \rightarrow 0} x\left(13+\frac{6}{x}\right)$ .

Find the derivative of the function $F(x)=(4 x+7)^{3}(x^{2}-6 x+7)^{4}$ . What is $F^{\prime}(x)$ ?

Evaluate: a $\lim _{x \rightarrow 0} 5$

Find the limit: $\lim _{x \rightarrow 3}(x+4)$ .

Set up a definite integral for the volume of the solid formed by rotating $y=\sqrt{x-1}$ , $y=0$ , $x=5$ about the $x$ -axis.

Find the first and second derivatives of $f(x)=2x\left(\frac{2}{x^{6}}-x\right)$ .

Finde die Tangentengleichung $t_{1}$ an $f(x)=\frac{1}{3} x^{2}$ bei $x_{0}=-2$ und die orthogonale Tangente.

Calculate the first and second derivatives of $g(x)=12(\sqrt{x})^{3}+5 \sqrt[4]{x}$ .

Find the first and second derivatives of $h(x)=2 \ln \left(x^{5 / 2}\right)$ .

Find critical points and absolute max/min of $f(x)=3 x^{\frac{1}{2}}-x^{\frac{5}{2}}$ on $[0,9]$ . Critical point(s): $\mathrm{x}=$

Find critical points and absolute max/min of $f(x)=3 x^{\frac{1}{2}}-x^{\frac{5}{2}}$ on $[0,9]$ . Critical point(s): $x=$

Find critical points and absolute max/min of $f(x)=2 x^{\frac{1}{2}}-x^{\frac{5}{2}}$ on $[0,25]$ . Critical points: $x=$

Differentiate $h(x)=e^{-a x} \cos (b x)$ using the product rule. Find $h^{\prime}(x)=$

Find the tangent line of $y=e^{4 x}$ parallel to $y=7 x+17$ . Give the equation in the form $y=m x+b$ .

Find the critical points of $f(x)=\frac{12 x}{x^{2}+36}$ and its derivative $f'(x)$ .

Find critical points of $f(x)=6 \sqrt{x}-7 x$ on the interval $[0,4]$ .

Find the critical points of the function $f(x)=\frac{x}{x^{2}+25}$ .

Find the critical points of the function $f(x)=x^{2} \sqrt{x+22}$ .

Find the derivative of $y = e^{\sqrt{6x + 4}}$ . What is $y'$ ?

Find the derivative of $h(x) = \frac{7 x^{3} e^{x}}{x^{4}-x^{2}}$ using the quotient rule. What is $h^{\prime}(x)$ ?

Set up a definite integral to find the area between the curves $y=12-x^{2}$ and $y=x^{2}-6$ .

Find the absolute maximum of $f(x)=x^{3}-6x^{2}$ on $[-2,6]$ . What is the max value and its location? A or B?

Find the absolute extreme values of $f(x)=(x-3)^{\frac{4}{3}}$ on the interval $[-5,5]$ . What are they?

Find the absolute maximum of $f(x)=x^{3}-57 x^{2}$ on $[-19,57]$ . Where is it located? A or B?

Find the storm's speed at 3:00 PM, given $s(t)=25+25t-6t^{2}$ for $t$ hours after noon.

Find the absolute extrema of $f(x)=(x-3)^{\frac{4}{3}}$ on the interval $[-5,5]$ .

Find the first and second derivatives of $k(x)=b^{\log_b(x)}$ , where $b>1$ .

Find the stone's velocity after it rises 8 feet, given height $h(t)=5t-\frac{1}{2}t^{2}$ .

Find the critical point(s) of the function $f(x)=-4 \sqrt{x}+3 x$ on the interval $[0,4]$ .

Find the absolute max and min of $f(x)=x^{3}-6x^{2}$ on $[-2,6]$ . What are their values and locations?

Find $\frac{d P}{d h}$ using the Chain Rule for $T=15.04-0.00649 h$ and $P=101.29\left(\frac{T+273.1}{288.08}\right)^{5.256}$ .

Find the absolute extreme values of $f(x)=5 \cos ^{2} x$ on the interval $[0, \pi]$ .

Find the derivative of the composite function $f(g(x))$ at $x = 6$ using the given values. Provide a whole number answer.

Determine how fast the area of a square changes as side length $x$ changes when $x=3 \mathrm{~cm}$ .

Find functions $f(x)$ and $g(x)$ such that $y=(2x+3\sin(x))^{5}=f(g(x))$ , then compute the derivative using the Chain Rule.

Find the derivative of $y=\cos^{9}(\theta+5)$ using the General Power Rule or Chain Rule.

Find the absolute extremes of $f(x)=5 \cos ^{2} x$ on $[0, \pi]$ . Choose A, B, C, or D and fill in values.

Find the absolute extreme values of $f(x)=\cos 5x$ on $[-\frac{\pi}{6}, \frac{\pi}{3}]$ . What are they?

Find the absolute extreme values of $f(x)=-4 \csc x$ on the interval $\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right]$ .

A stone is launched from a cliff $384 \mathrm{ft}$ high at $32 \mathrm{ft/s}$ . When does it reach max height? Find $s'$ . $s' =$

Find the absolute extreme values of $f(x)=\cos 5x$ on the interval $\left[-\frac{\pi}{6}, \frac{\pi}{3}\right]$ .

A stone is launched from a cliff 128 ft high at 32 ft/s. Find when it reaches max height and the derivative $s'$ .

Find the absolute extreme values of $f(x)=-4 \csc x$ on the interval $\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right]$ .

A stone is launched from a cliff $192 \mathrm{ft}$ high at $64 \mathrm{ft/s}$ . When does it reach max height? Find $s'$ . $s' =$

A stone is launched from a cliff at $192 \mathrm{ft}$ with a speed of $64 \mathrm{ft/s}$ . Find when it reaches max height using $s=-16 t^{2}+64 t+192$ .

Find the derivative of $15 \sin(x^{7}) \cos(x^{7})$ .

Find the slope $m$ of the tangent line to the curve $y=\frac{x+3}{x-6}$ at the point $(5,-8)$ and its equation.

Find the elevator's descent speed when it's 500 feet away from you and moving closer at $16 \mathrm{ft} / \mathrm{se}$ .

Find the derivative of $(4 e^{2 x}+12 e^{-8 x})^{12}$ .

Find the terminal velocity of a 202 lb skydiver given $v(t)=\frac{202}{1.1}\left(1-e^{-24.2 t / 202}\right)$ and time to reach 74% of it.

Find the second derivative of $\sin(4x^4)$ . Calculate: $\frac{d^{2}}{d x^{2}} \sin \left(4 x^{4}\right) =$

Find the third derivative of $(14-6 x)^{6}$ . What is $\frac{d^{3}}{d x^{3}}(14-6 x)^{6}=?$

Find the tangent line to $y=e^{2 x}$ that is parallel to $y=5 x+15$ . Give your answer as $y=m x+b$ , with exact $m$ and $b$ .