Calculus

Problem 6401

For each function, determine the signs of $f'(x)$ and $f''(x)$ over the given interval.
(a) $f(x)$ values: 7.1, 7.9, 8.8, 9.8, 10.9, 12.1 (b) $f(x)$ values: 10.1, 8.6, 7.0, 5.3, 3.5, 1.6 (c) $f(x)$ values: 3000, 3024, 3049, 3075, 3102, 3130

See Solution

Problem 6402

Find the derivative $f^{\prime}(4)$ for the function $f(x)=\frac{1}{x}-\frac{2}{\sqrt{x}}$ as a single simplified fraction.

See Solution

Problem 6403

True or false: Prove or provide a counterexample for each statement about sequences and integrals.

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Problem 6404

Test the following series for convergence or divergence: (a) $\sum_{n=1}^{\infty} \frac{2+\cos n}{\sqrt{1+n}}$ (b) $\sum_{n=1}^{\infty}(2^{1/n}-2^{1/(n+1)})$ (c) $\sum_{n=1}^{\infty} \sqrt[n]{2}-1$ (d) $\sum_{n=1}^{\infty} \frac{\sqrt{n^{2}-1}}{n^{3}+2n+5}$ (e) $\sum_{n=2}^{\infty} \frac{1}{(\ln n)^{\ln n}}$ (f) $\sum_{n=1}^{\infty} \frac{1}{n+\sqrt{n} \sin n}$ (g) $\sum_{n=1}^{\infty} \frac{5^{n}-2^{n}}{7^{n}-6^{n}}$ (h) $\sum_{n=3}^{\infty} \frac{1}{n \ln n \ln (\ln n)}$

See Solution

Problem 6405

Find the production level $x$ that minimizes the cost function $C(x)=x^{2}-140x+7900$ and determine the minimum cost.

See Solution

Problem 6406

Find the time for a sphere to fall from height $h=6r$ to $h=2r$ in a viscous fluid using $\frac{d h}{d t}=-\frac{\alpha}{r} h$ .

See Solution

Problem 6407

An object is thrown up at $14 \mathrm{~m/s}$ with height $s=-0.7 t^{2}+14 t$ . Find velocity, highest point, height, ground strike time, strike velocity, and speed intervals.

See Solution

Problem 6408

Find the average growth rate from 1991 to 2001 using $p(t)=-0.28 t^{2}+111 t+6423$ . Also, find growth rates in $1995$ and $2001$ . Graph $p^{\prime}$ for $0 \leq t \leq 10$ and analyze.

See Solution

Problem 6409

Determine the vertical, horizontal, and oblique asymptotes of the function $T(x)=\frac{x^{3}}{x^{4}-16}$ .

See Solution

Problem 6410

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $h=0$ with the function $f(x)=-4 x^{2}-4 x+8$ .

See Solution

Problem 6411

Find $\int_{1}^{4} f\left(\frac{8}{x}\right) d x$ given that $\int_{2}^{8} \frac{f(x)}{x^{2}} d x=6$ .

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Problem 6412

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=3 x^{2}-5 x-8$ , where $h \neq 0$ .

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Problem 6413

Find the limit of $\frac{f(x+h)-f(x)}{h}$ for $f(x)=2 x^{2}-3 x-5$ where $h \neq 0$ .

See Solution

Problem 6414

Find the limit of $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=2 x^{2}-3 x-5$ , where $h \neq 0$ .

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Problem 6415

Finde den Wendepunkt der Funktion $f(x)=\frac{1}{3} x^{3}-2 x^{2}+3 x-1$ .

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Problem 6416

Bestimme die Intervalle der Monotonie für die Funktion $f(x)=-\frac{1}{6} x^{3}+\frac{3}{4} x^{2}+1$ und prüfe die Aussagen zu $f'$ .

See Solution

Problem 6417

Bestimmen Sie die mittlere Fläche $f(t)$ im Intervall $[0; t]$ für $f(t)=\frac{1000}{1+499 \cdot 2^{-t}}$ und $t \rightarrow \infty$ .

See Solution

Problem 6418

Finde den Wendepunkt der Funktion $f(x)=x^{4}$ .

See Solution

Problem 6419

Bestimmen Sie die mittlere Temperatur von $T(t)=10+8 \cdot \sin \left(\frac{\pi}{12} t\right)$ zwischen 9 und 21 Uhr.

See Solution

Problem 6420

Finde Nullstellen von $f(x)=(x+3)e^{-x}$ , berechne Ableitungen, untersuche Extrempunkte und skizziere den Graphen. Bestimme Tangente und Normale bei $A(0 \mid 3)$ .

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Problem 6421

Bestimmen Sie den Flächeninhalt zwischen der Funktion $f(x)=x^{3}+x^{2}-2 x$ und der $x$ -Achse im Intervall $I=[-2 ; 1]$ .

See Solution

Problem 6422

Gegeben ist die Funktion $f(x)=(x+3) \cdot e^{-x}$ . Finde Nullstellen, Ableitungen, Hoch-/Tiefpunkte und skizziere den Graphen. Berechne die Tangente und Normale im Punkt $A(0 \mid 3)$ .

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Problem 6423

Find the derivatives of these functions: a) $f(x)=\sin (3x+3)$ , d) $f(x)=-4\sin(\pi-x)$ .

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Problem 6424

Zeichne den Graphen von $f$ und bestimme $I_{0}(x) = \int_{0}^{x} f(t) dt$ für die Funktionen a) bis d).

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Problem 6425

Find the limit: $\lim _{x \rightarrow 2} \frac{0.5 x}{2-x}$ .

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Problem 6426

Find the absolute extrema of $f(x)=6 x^{2}-48 x$ on the interval $[0,7]$ . Provide your answer as $(x, f(x))$ .

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Problem 6427

Calculate the integral $f(x)=\int_{0}^{4} \sqrt{x} \, dx$ .

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Problem 6428

Find the absolute extrema of $f(x)=4 x^{3}+42 x^{2}+144 x$ on $[-8,5]$ . Give your answer as $(x, f(x))$ .

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Problem 6429

Finde die Extrempunkte der Funktion $h(x) = -\frac{2}{55}x^4 + \frac{25}{88}x^2 - \frac{11}{160}$ .

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Problem 6430

Berechnen Sie die Untersumme $U_n$ und Obersumme $O_n$ für $f(x)=x-1$ auf $I=[0, 1]$ im Grenzwert gegen $\infty$ .

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Problem 6431

Evaluate the integral: $\int n \cdot x^{2 n-1} \, dx$

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Problem 6432

Gegeben ist die Funktion $f(x)=\frac{1}{9}(3 x+2)^{3}$ . Bestimme: a) Steigung bei $P(2 \mid f(2))$ , b) Punkte mit waagerechter Tangente, c) Punkte mit Tangente der Steigung 1 und deren Gleichungen.

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Problem 6433

Simplify $f(x)=\frac{\tan (x)-1}{\sec (x)}$ using $\sin(x)$ and $\cos(x)$ , then find its derivative.

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Problem 6434

Find the velocity and acceleration from $x(t) = 4\sin(t)$ , then evaluate at $t=2\pi/3$ and determine the direction.

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Problem 6435

Find the derivative of $y=x \sqrt[3]{1+x^{2}}$ with respect to $x$ .

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Problem 6436

Bestimme die Tangentengleichung der Funktion $f(x)=\left(\frac{1}{4}\right)^{x}$ am Punkt $A(-1, 4)$ .

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Problem 6437

Evaluate the integral: $\int \frac{x+1}{x^{2}+2 x+7} d x$ using substitution with $u=x^{2}+2 x+7$ .

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Problem 6438

Find the derivative of $y=\frac{\sin x \cos x \tan^{3} x}{\sqrt{x}}$ with respect to $x$ .

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Problem 6439

Bestimme den Flächeninhalt zwischen dem Graphen der Funktionen und der $x$ -Achse für die angegebenen Intervalle.

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Problem 6440

Find the derivative of $y=\frac{\sin x \cos x \tan^{3} x}{\sqrt{x}}$ using logarithmic differentiation.

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Problem 6441

Evaluate the integral: $\int_{0}^{1}(3 x+5)^{2} \mathrm{~d} x$ and show it equals $\int_{5}^{8} u^{2} \frac{1}{3} \mathrm{~d} u$ .

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Problem 6442

Bestimmen Sie, ob die Folge $a_{n}$ konvergent oder divergent ist. Geben Sie den Grenzwert an, falls konvergent. a) $a_{n}=\frac{1}{n^{2}}$ , b) $a_{n}=-n$ , c) $a_{n}=n+\frac{1}{n}$ , d) $a_{n}=\frac{4 n}{n+2}$ , e) $a_{n}=\sqrt{n}$ , f) $a_{n}=\frac{(-1)^{n}}{n^{2}}$ , g) $a_{n}=\frac{n}{2}-\frac{2}{n}$ , h) $a_{n}=n \cdot(-1)^{n}$ , i) $a_{n}=\frac{\sqrt{n}}{n+1}$ , j) $a_{n}=\sin n$ .

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Problem 6443

Find the derivative of $\log_{x} 2$ with respect to $x$ : $\frac{d}{d x}\left[\log_{x} 2\right]$ .

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Problem 6444

Find the derivative of $\log_{x} e$ with respect to $x$ .

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Problem 6445

Evaluate the integral: $\int x\left(2 x^{2}+2\right)^{1 / 2} d x$

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Problem 6446

Find the limits: (a) $\lim _{\Delta x \rightarrow 0} \frac{\ln \left(e^{2}+\Delta x\right)-2}{\Delta x}$ , (b) $\lim _{w \rightarrow 1} \frac{\ln w}{w-1}$ .

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Problem 6447

Find the derivative $\frac{dy}{dx}$ for $\sin(x^{2}y^{2})=x$ using implicit differentiation.

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Problem 6448

Find $d y / d x$ using implicit differentiation for the following equations:
11. $x^{2}+y^{2}=100$
12. $x^{3}+y^{3}=3xy^{2}$
13. $x^{2}y+3xy^{3}-x=3$
14. $x^{3}y^{2}-5x^{2}y+x=1$
15. $\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}=1$
16. $x^{2}=\frac{x+y}{x-y}$
17. $\sin(x^{2}y^{2})=x$
18. $\cos(xy^{2})=y$
19. $\tan^{3}(xy^{2}+y)=x$
20. $\frac{xy^{3}}{1+\sec y}=1+y^{4}$

See Solution

Problem 6449

Untersuchen Sie die Funktionen $f(x)=e^{0,5x}$ und $g(x)=e^{1,5-0,25x}$ : Graphen skizzieren, Ableitungen finden, Schnittpunkt und Winkel bestimmen, Tangente ermitteln, Fläche A berechnen.

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Problem 6450

Bestimme die Grenzwerte: a) $\lim _{x \rightarrow 5} \frac{x^{2}-25}{x-5}$ , b) $\lim _{x \rightarrow 3} \frac{3 x^{2}-27}{x-3}$ , c) $\lim _{x \rightarrow 1} \frac{x^{3}-x}{x-1}$ , d) $\lim _{x \rightarrow-2} \frac{x^{4}-16}{x+2}$ .

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Problem 6451

Solve the integral $\int x \ln x \mathrm{~d} x$ using integration by parts.

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Problem 6452

Verify the linear approximation at $a=0$ : $(1+3x)^{-4} \approx 1-12x$ . Find $x$ where it's accurate within 0.1. $x \in$

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Problem 6453

Estimate $e^{0.01}$ using linear approximation or differentials.

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Problem 6454

Estimate the paint volume (in $m^3$ ) for a $0.06 \mathrm{~cm}$ thick coat on a hemispherical dome with a $42 \mathrm{~m}$ diameter. Round to two decimal places.

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Problem 6455

Vereinfachen Sie $f(s)=\frac{2}{\sqrt{s}}-3 s^{5}$ und bestimmen Sie die Ableitung.

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Problem 6456

Calculate the balance after 3 years for an investment of \$2,500 at 12\% interest, compounded continuously.

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Problem 6457

Untersuchen Sie die Kurvenschar $f_{a}(x)=x+a \cdot e^{-x}$ auf Extrema, Wendepunkte und zeichnen Sie die Graphen für $a=1, 0.5, -1$ .

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Problem 6458

Find the second derivative of $f(x)=\sqrt[3]{x}+3\cos(x)$ .

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Problem 6459

Beschreibe die Funktion $f(x)=20 \cdot(1-e^{-x})+0,5 \cdot x$ . a) Trainingseffekt für $x=5,0$ Minuten. b) Wann übersteigt der Trainingseffekt $17 \mathrm{kcal}/$ Minute?

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Problem 6460

Find the relation between $\frac{d A}{d t}$ , $\frac{d l}{d t}$ , and $\frac{d w}{d t}$ for $l=9$ and $w=10$ .

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Problem 6461

Find the equation for $\frac{d A}{d t}$ in terms of $\frac{d l}{d t}$ and $\frac{d w}{d t}$ for $l=9$ and $w=10$ .

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Problem 6462

Find the relation between $\frac{d A}{d t}$ , $\frac{d l}{d t}$ , and $\frac{d w}{d t}$ for $l=9$ , $w=10$ .

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Problem 6463

Find the relation between $\frac{d A}{d t}$ , $\frac{d l}{d t}$ , and $\frac{d w}{d t}$ for $l=18$ , $w=13$ .

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Problem 6464

Radioaktivität: Gegeben ist $N(t)=1000 \cdot e^{-0,069 t}$ . Bestimme den Anfangsbestand, den Bestand nach 1 Tag und den täglichen Zerfall in %.

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Problem 6465

A ball is dropped. What is its acceleration in $\mathrm{m} / \mathrm{s}^{2}$ while falling, assuming no air resistance?

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Problem 6466

Find the equation for $\frac{d A}{d t}$ in terms of $\frac{d l}{d t}$ and $\frac{d w}{d t}$ when $l=8$ and $w=4$ .

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Problem 6467

Find $\frac{d z}{d t}$ given $z^{4}=x^{3}+y^{2}$ , $\frac{d x}{d t}=-1$ , $\frac{d y}{d t}=3$ , at $(x, y)=(0,4)$ .

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Problem 6468

Find the equation for $\frac{d A}{d t}$ in terms of $\frac{d l}{d t}$ and $\frac{d w}{d t}$ for $l=19$ and $w=14$ .

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Problem 6469

Beurteilen Sie Naels Aussage zum Grenzwert der Funktion $f(x) = 0,001 x^{4}-0,11 x^{2}+3$ im Unendlichen.

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Problem 6470

A woman 5 ft tall walks away from an 18 ft pole at 8 ft/s. Find the speed of her shadow's tip when she's 12 ft from the pole.

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Problem 6471

Find the derivative of $y=\sin(4x^3)+\sin^5(3x^2)$ and simplify the answer.

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Problem 6472

Understand upper/lower sums, antiderivatives, definite integrals, area between functions, and solve initial value problems.

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Problem 6473

An airplane at $12000 \mathrm{ft}$ flies away from a man $5000 \mathrm{ft}$ from a tower. If it moves at $400 \mathrm{ft/s}$ , find the rate the distance increases when over the tower.

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Problem 6474

Calculate the amount in an account after 10 years with a \$24,000 deposit at 7.25% continuous interest. Choices: \$46,414.20, \$48,326.40, \$49,553.54, \$47,897.10.

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Problem 6475

Find the derivative of $y = \left(\frac{5x+7}{3-2x}\right)^4$ and simplify the answer.

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Problem 6476

Gegeben ist die Funktion $f(x)=\frac{1}{3} x^{3}-x$ . Untersuche das Verhalten, die Symmetrie, Extremwerte und Wendepunkte sowie Tangenten.

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Problem 6477

Gegeben ist $f_{e}(x)=\frac{4}{9} t^{2} x^{3}+t x^{2}+x$ . a) Zeigen Sie, dass alle Kurven im Ursprung berühren. Bestimmen Sie die Tangente. b) Finden Sie die Kurven $P_{t}$ mit Steigung 1. c) Zeichnen Sie Wendetangenten für drei $t$ -Werte und beschreiben Sie Ihre Beobachtungen.

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Problem 6478

Bestimmen Sie die Obersumme und Untersumme für:
a) $f(x)=x+1$ auf $I=[0 ; 1]$
b) $f(x)=2-x$ auf $I=[0 ; 2]$
c) $f(x)=\frac{1}{2} x^{2}$ auf $I=[0 ; 1]$

See Solution

Problem 6479

Differentiate $y=\sqrt{x} \tan (\sqrt{x})$ and simplify the result.

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Problem 6480

Find the gradient of $f(x, y, z)=(x-2 z) e^{y}$ at the origin and state its components.

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Problem 6481

Find the derivative of $y=\left(x^{3}+5\right)^{4} \sec (2 x)$ and simplify it.

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Problem 6482

Find the expressions that compute the rate of change of $f$ at $a$ with respect to $\mathbf{h}$ . Options include:
1. $[D f]_{a} \frac{\mathbf{h}}{\|\mathbf{b}\|}$
2. $(\nabla f)_{a} \cdot \mathbf{h}$
3. $(\nabla f) \cdot h$
4. $(\nabla f)_{a} \cdot \frac{\mathbf{h}}{\|\mathbf{h}\|}$
5. $[D f]_{a} h$
6. $[D f] \cdot \frac{\mathbf{h}}{\|\mathbf{b}\|}$
7. $[D f] \cdot \mathbf{h}$

See Solution

Problem 6483

Find the tangent plane equation to the surface $3 x^{2}-x y+2 z^{2}=4$ at the point $(1,1,1)$ .

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Problem 6484

Find the derivative of $y=\frac{\cos(x^{3})}{(3x^{2}+1)^{5}}$ and simplify it.

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Problem 6485

If $z=e^{x y}-x^{2} y$ , find the differential $d z$ . Choose the correct expression for $d z$ .

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Problem 6486

Identify the incorrect Taylor series expansions about $x=0$ from the following list (ignore convergence radii).

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Problem 6487

Expand $f(x, y)=e^{x^{2} y} \sin(3y)$ using Taylor series. Find the coefficient of $x^{4} y^{5}$ . Provide a decimal.

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Problem 6488

Find the Hessian matrix, $D^{2} f$ , of the function $f(x, y)=x^{3}-3xy^{2}+x^{2}-xy+2y+5x-17$ .

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Problem 6489

Expand $f(x, y)=e^{x^{2} y} \sin (3 y)$ using Taylor series. Find the coefficient of the $x^{4} y^{5}$ term. Provide a decimal.

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Problem 6490

If $z=e^{x y}-x^{2} y$ , find the differential $d z$ . What is $d z$ equal to?

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Problem 6491

Find the derivative of $y=(3 x^{2}-1)^{3}(2 x+5)^{2}$ and simplify the expression.

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Problem 6492

Find the derivative of $y=\sqrt{\frac{x^{2}+5}{x^{2}-5}}$ and simplify your answer.

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Problem 6493

Find the derivative of $y=\frac{x^{2}+3}{\sqrt{2 x-5}}$ and simplify it.

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Problem 6494

Evaluate the integral $\int x \ln x \, dx$ using integration by parts with $u' = x$ and $v = \ln x$ .

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Problem 6495

Berechne die Stammfunktion von $f(x)=3 x^{2}+7 x$ mit dem Punkt $P(2, 16)$ .

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Problem 6496

Untersuchen Sie das Verhalten der Funktionen für $x \to \pm \infty$ :
a) $f(x)=3 x^{4}+2 x^{3}+x^{2}+10 x-3$
b) $f(x)=2^{x-2}$

See Solution

Problem 6497

Find the derivative of $y=\sqrt{\sec^{3} 5x} + \sec \sqrt{5x}$ and simplify using chain rule and secant rules.

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Problem 6498

Use integration by parts to solve the integral $\int x e^{x} \, dx$ .

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Problem 6499

Evaluate the integral: $\int x e^{x} \, dx$ and show that it equals $x e^{x} - e^{x} + c$ .

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Problem 6500

Bestimmen Sie die Steigung und den Steigungswinkel von $f$ an den Stellen $x_{0}$ : a) $f(x)=2 x^{3}$ , $x_{0}=-1$ ; b) $f(x)=-2 \sqrt{x}$ , $x_{0}=3$ ; c) $f(x)=-x^{2}-5 x$ , $x_{0}=0$ ; d) $f(x)=-\frac{4}{x^{\prime}}$ , $x_{0}=-2$ ; e) $f(x)=(2 x+1)^{2}$ , $x_{0}=-\frac{1}{2}$ ; f) $f(x)=\frac{1}{2 x^{2}}$ , $x_{0}=-3$ .

See Solution

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Calculus

Problem 6401

For each function, determine the signs of f′(x)f'(x)f′(x) and f′′(x)f''(x)f′′(x) over the given interval. (a) f(x)f(x)f(x) values: 7.1, 7.9, 8.8, 9.8, 10.9, 12.1 (b) f(x)f(x)f(x) values: 10.1, 8.6, 7.0, 5.3, 3.5, 1.6 (c) f(x)f(x)f(x) values: 3000, 3024, 3049, 3075, 3102, 3130

Problem 6402

Find the derivative f′(4)f^{\prime}(4)f′(4) for the function f(x)=1x−2xf(x)=\frac{1}{x}-\frac{2}{\sqrt{x}}f(x)=x1​−x​2​ as a single simplified fraction.

Problem 6403

True or false: Prove or provide a counterexample for each statement about sequences and integrals.

Problem 6404

Problem 6405

Find the production level xxx that minimizes the cost function C(x)=x2−140x+7900C(x)=x^{2}-140x+7900C(x)=x2−140x+7900 and determine the minimum cost.

Problem 6406

Find the time for a sphere to fall from height h=6rh=6rh=6r to h=2rh=2rh=2r in a viscous fluid using dhdt=−αrh\frac{d h}{d t}=-\frac{\alpha}{r} hdtdh​=−rα​h.

Problem 6407

An object is thrown up at 14 m/s14 \mathrm{~m/s}14 m/s with height s=−0.7t2+14ts=-0.7 t^{2}+14 ts=−0.7t2+14t. Find velocity, highest point, height, ground strike time, strike velocity, and speed intervals.

Problem 6408

Find the average growth rate from 1991 to 2001 using p(t)=−0.28t2+111t+6423p(t)=-0.28 t^{2}+111 t+6423p(t)=−0.28t2+111t+6423. Also, find growth rates in 199519951995 and 200120012001. Graph p′p^{\prime}p′ for 0≤t≤100 \leq t \leq 100≤t≤10 and analyze.

Problem 6409

Determine the vertical, horizontal, and oblique asymptotes of the function T(x)=x3x4−16T(x)=\frac{x^{3}}{x^{4}-16}T(x)=x4−16x3​.

Problem 6410

Calculate the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for h=0h=0h=0 with the function f(x)=−4x2−4x+8f(x)=-4 x^{2}-4 x+8f(x)=−4x2−4x+8.

Problem 6411

Find ∫14f(8x)dx\int_{1}^{4} f\left(\frac{8}{x}\right) d x∫14​f(x8​)dx given that ∫28f(x)x2dx=6\int_{2}^{8} \frac{f(x)}{x^{2}} d x=6∫28​x2f(x)​dx=6.

Problem 6412

Calculate the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for the function f(x)=3x2−5x−8f(x)=3 x^{2}-5 x-8f(x)=3x2−5x−8, where h≠0h \neq 0h=0.

Problem 6413

Find the limit of f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=2x2−3x−5f(x)=2 x^{2}-3 x-5f(x)=2x2−3x−5 where h≠0h \neq 0h=0.

Problem 6414

Find the limit of f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for the function f(x)=2x2−3x−5f(x)=2 x^{2}-3 x-5f(x)=2x2−3x−5, where h≠0h \neq 0h=0.

Problem 6415

Finde den Wendepunkt der Funktion f(x)=13x3−2x2+3x−1f(x)=\frac{1}{3} x^{3}-2 x^{2}+3 x-1f(x)=31​x3−2x2+3x−1.

Problem 6416

Bestimme die Intervalle der Monotonie für die Funktion f(x)=−16x3+34x2+1f(x)=-\frac{1}{6} x^{3}+\frac{3}{4} x^{2}+1f(x)=−61​x3+43​x2+1 und prüfe die Aussagen zu f′f'f′.

Problem 6417

Bestimmen Sie die mittlere Fläche f(t)f(t)f(t) im Intervall [0;t][0; t][0;t] für f(t)=10001+499⋅2−tf(t)=\frac{1000}{1+499 \cdot 2^{-t}}f(t)=1+499⋅2−t1000​ und t→∞t \rightarrow \inftyt→∞.

Problem 6418

Finde den Wendepunkt der Funktion f(x)=x4f(x)=x^{4}f(x)=x4.

Problem 6419

Bestimmen Sie die mittlere Temperatur von T(t)=10+8⋅sin⁡(π12t)T(t)=10+8 \cdot \sin \left(\frac{\pi}{12} t\right)T(t)=10+8⋅sin(12π​t) zwischen 9 und 21 Uhr.

Problem 6420

Finde Nullstellen von f(x)=(x+3)e−xf(x)=(x+3)e^{-x}f(x)=(x+3)e−x, berechne Ableitungen, untersuche Extrempunkte und skizziere den Graphen. Bestimme Tangente und Normale bei A(0∣3)A(0 \mid 3)A(0∣3).

Problem 6421

Bestimmen Sie den Flächeninhalt zwischen der Funktion f(x)=x3+x2−2xf(x)=x^{3}+x^{2}-2 xf(x)=x3+x2−2x und der xxx-Achse im Intervall I=[−2;1]I=[-2 ; 1]I=[−2;1].

Problem 6422

Gegeben ist die Funktion f(x)=(x+3)⋅e−xf(x)=(x+3) \cdot e^{-x}f(x)=(x+3)⋅e−x. Finde Nullstellen, Ableitungen, Hoch-/Tiefpunkte und skizziere den Graphen. Berechne die Tangente und Normale im Punkt A(0∣3)A(0 \mid 3)A(0∣3).

Problem 6423

Find the derivatives of these functions: a) f(x)=sin⁡(3x+3)f(x)=\sin (3x+3)f(x)=sin(3x+3), d) f(x)=−4sin⁡(π−x)f(x)=-4\sin(\pi-x)f(x)=−4sin(π−x).

Problem 6424

Zeichne den Graphen von fff und bestimme I0(x)=∫0xf(t)dtI_{0}(x) = \int_{0}^{x} f(t) dtI0​(x)=∫0x​f(t)dt für die Funktionen a) bis d).

Problem 6425

Find the limit: lim⁡x→20.5x2−x\lim _{x \rightarrow 2} \frac{0.5 x}{2-x}limx→2​2−x0.5x​.

Problem 6426

Find the absolute extrema of f(x)=6x2−48xf(x)=6 x^{2}-48 xf(x)=6x2−48x on the interval [0,7][0,7][0,7]. Provide your answer as (x,f(x))(x, f(x))(x,f(x)).

Problem 6427

Calculate the integral f(x)=∫04x dxf(x)=\int_{0}^{4} \sqrt{x} \, dxf(x)=∫04​x​dx.

Problem 6428

Find the absolute extrema of f(x)=4x3+42x2+144xf(x)=4 x^{3}+42 x^{2}+144 xf(x)=4x3+42x2+144x on [−8,5][-8,5][−8,5]. Give your answer as (x,f(x))(x, f(x))(x,f(x)).

Problem 6429

Finde die Extrempunkte der Funktion h(x)=−255x4+2588x2−11160h(x) = -\frac{2}{55}x^4 + \frac{25}{88}x^2 - \frac{11}{160}h(x)=−552​x4+8825​x2−16011​.

Problem 6430

Berechnen Sie die Untersumme UnU_nUn​ und Obersumme OnO_nOn​ für f(x)=x−1f(x)=x-1f(x)=x−1 auf I=[0,1]I=[0, 1]I=[0,1] im Grenzwert gegen ∞\infty∞.

Problem 6431

Evaluate the integral: ∫n⋅x2n−1 dx\int n \cdot x^{2 n-1} \, dx∫n⋅x2n−1dx

Problem 6432

Gegeben ist die Funktion f(x)=19(3x+2)3f(x)=\frac{1}{9}(3 x+2)^{3}f(x)=91​(3x+2)3. Bestimme: a) Steigung bei P(2∣f(2))P(2 \mid f(2))P(2∣f(2)), b) Punkte mit waagerechter Tangente, c) Punkte mit Tangente der Steigung 1 und deren Gleichungen.

Problem 6433

Simplify f(x)=tan⁡(x)−1sec⁡(x)f(x)=\frac{\tan (x)-1}{\sec (x)}f(x)=sec(x)tan(x)−1​ using sin⁡(x)\sin(x)sin(x) and cos⁡(x)\cos(x)cos(x), then find its derivative.

Problem 6434

Find the velocity and acceleration from x(t)=4sin⁡(t)x(t) = 4\sin(t)x(t)=4sin(t), then evaluate at t=2π/3t=2\pi/3t=2π/3 and determine the direction.

Problem 6435

Find the derivative of y=x1+x23y=x \sqrt[3]{1+x^{2}}y=x31+x2​ with respect to xxx.

Problem 6436

Bestimme die Tangentengleichung der Funktion f(x)=(14)xf(x)=\left(\frac{1}{4}\right)^{x}f(x)=(41​)x am Punkt A(−1,4)A(-1, 4)A(−1,4).

Problem 6437

Evaluate the integral: ∫x+1x2+2x+7dx\int \frac{x+1}{x^{2}+2 x+7} d x∫x2+2x+7x+1​dx using substitution with u=x2+2x+7u=x^{2}+2 x+7u=x2+2x+7.

Problem 6438

Find the derivative of y=sin⁡xcos⁡xtan⁡3xxy=\frac{\sin x \cos x \tan^{3} x}{\sqrt{x}}y=x​sinxcosxtan3x​ with respect to xxx.

Problem 6439

Bestimme den Flächeninhalt zwischen dem Graphen der Funktionen und der xxx-Achse für die angegebenen Intervalle.

Problem 6440

Find the derivative of y=sin⁡xcos⁡xtan⁡3xxy=\frac{\sin x \cos x \tan^{3} x}{\sqrt{x}}y=x​sinxcosxtan3x​ using logarithmic differentiation.

For each function, determine the signs of $f'(x)$ and $f''(x)$ over the given interval.
(a) $f(x)$ values: 7.1, 7.9, 8.8, 9.8, 10.9, 12.1 (b) $f(x)$ values: 10.1, 8.6, 7.0, 5.3, 3.5, 1.6 (c) $f(x)$ values: 3000, 3024, 3049, 3075, 3102, 3130

Find the derivative $f^{\prime}(4)$ for the function $f(x)=\frac{1}{x}-\frac{2}{\sqrt{x}}$ as a single simplified fraction.

Find the production level $x$ that minimizes the cost function $C(x)=x^{2}-140x+7900$ and determine the minimum cost.

Find the time for a sphere to fall from height $h=6r$ to $h=2r$ in a viscous fluid using $\frac{d h}{d t}=-\frac{\alpha}{r} h$ .

An object is thrown up at $14 \mathrm{~m/s}$ with height $s=-0.7 t^{2}+14 t$ . Find velocity, highest point, height, ground strike time, strike velocity, and speed intervals.

Find the average growth rate from 1991 to 2001 using $p(t)=-0.28 t^{2}+111 t+6423$ . Also, find growth rates in $1995$ and $2001$ . Graph $p^{\prime}$ for $0 \leq t \leq 10$ and analyze.

Determine the vertical, horizontal, and oblique asymptotes of the function $T(x)=\frac{x^{3}}{x^{4}-16}$ .

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $h=0$ with the function $f(x)=-4 x^{2}-4 x+8$ .

Find $\int_{1}^{4} f\left(\frac{8}{x}\right) d x$ given that $\int_{2}^{8} \frac{f(x)}{x^{2}} d x=6$ .

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=3 x^{2}-5 x-8$ , where $h \neq 0$ .

Find the limit of $\frac{f(x+h)-f(x)}{h}$ for $f(x)=2 x^{2}-3 x-5$ where $h \neq 0$ .

Find the limit of $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=2 x^{2}-3 x-5$ , where $h \neq 0$ .

Finde den Wendepunkt der Funktion $f(x)=\frac{1}{3} x^{3}-2 x^{2}+3 x-1$ .

Bestimme die Intervalle der Monotonie für die Funktion $f(x)=-\frac{1}{6} x^{3}+\frac{3}{4} x^{2}+1$ und prüfe die Aussagen zu $f'$ .

Bestimmen Sie die mittlere Fläche $f(t)$ im Intervall $[0; t]$ für $f(t)=\frac{1000}{1+499 \cdot 2^{-t}}$ und $t \rightarrow \infty$ .

Finde den Wendepunkt der Funktion $f(x)=x^{4}$ .

Bestimmen Sie die mittlere Temperatur von $T(t)=10+8 \cdot \sin \left(\frac{\pi}{12} t\right)$ zwischen 9 und 21 Uhr.

Finde Nullstellen von $f(x)=(x+3)e^{-x}$ , berechne Ableitungen, untersuche Extrempunkte und skizziere den Graphen. Bestimme Tangente und Normale bei $A(0 \mid 3)$ .

Bestimmen Sie den Flächeninhalt zwischen der Funktion $f(x)=x^{3}+x^{2}-2 x$ und der $x$ -Achse im Intervall $I=[-2 ; 1]$ .

Gegeben ist die Funktion $f(x)=(x+3) \cdot e^{-x}$ . Finde Nullstellen, Ableitungen, Hoch-/Tiefpunkte und skizziere den Graphen. Berechne die Tangente und Normale im Punkt $A(0 \mid 3)$ .

Find the derivatives of these functions: a) $f(x)=\sin (3x+3)$ , d) $f(x)=-4\sin(\pi-x)$ .

Zeichne den Graphen von $f$ und bestimme $I_{0}(x) = \int_{0}^{x} f(t) dt$ für die Funktionen a) bis d).

Find the limit: $\lim _{x \rightarrow 2} \frac{0.5 x}{2-x}$ .

Find the absolute extrema of $f(x)=6 x^{2}-48 x$ on the interval $[0,7]$ . Provide your answer as $(x, f(x))$ .

Calculate the integral $f(x)=\int_{0}^{4} \sqrt{x} \, dx$ .

Find the absolute extrema of $f(x)=4 x^{3}+42 x^{2}+144 x$ on $[-8,5]$ . Give your answer as $(x, f(x))$ .

Finde die Extrempunkte der Funktion $h(x) = -\frac{2}{55}x^4 + \frac{25}{88}x^2 - \frac{11}{160}$ .

Berechnen Sie die Untersumme $U_n$ und Obersumme $O_n$ für $f(x)=x-1$ auf $I=[0, 1]$ im Grenzwert gegen $\infty$ .

Evaluate the integral: $\int n \cdot x^{2 n-1} \, dx$

Gegeben ist die Funktion $f(x)=\frac{1}{9}(3 x+2)^{3}$ . Bestimme: a) Steigung bei $P(2 \mid f(2))$ , b) Punkte mit waagerechter Tangente, c) Punkte mit Tangente der Steigung 1 und deren Gleichungen.

Simplify $f(x)=\frac{\tan (x)-1}{\sec (x)}$ using $\sin(x)$ and $\cos(x)$ , then find its derivative.

Find the velocity and acceleration from $x(t) = 4\sin(t)$ , then evaluate at $t=2\pi/3$ and determine the direction.

Find the derivative of $y=x \sqrt[3]{1+x^{2}}$ with respect to $x$ .

Bestimme die Tangentengleichung der Funktion $f(x)=\left(\frac{1}{4}\right)^{x}$ am Punkt $A(-1, 4)$ .

Evaluate the integral: $\int \frac{x+1}{x^{2}+2 x+7} d x$ using substitution with $u=x^{2}+2 x+7$ .

Find the derivative of $y=\frac{\sin x \cos x \tan^{3} x}{\sqrt{x}}$ with respect to $x$ .

Bestimme den Flächeninhalt zwischen dem Graphen der Funktionen und der $x$ -Achse für die angegebenen Intervalle.

Find the derivative of $y=\frac{\sin x \cos x \tan^{3} x}{\sqrt{x}}$ using logarithmic differentiation.

Evaluate the integral: $\int_{0}^{1}(3 x+5)^{2} \mathrm{~d} x$ and show it equals $\int_{5}^{8} u^{2} \frac{1}{3} \mathrm{~d} u$ .

Find the derivative of $\log_{x} 2$ with respect to $x$ : $\frac{d}{d x}\left[\log_{x} 2\right]$ .

Find the derivative of $\log_{x} e$ with respect to $x$ .

Evaluate the integral: $\int x\left(2 x^{2}+2\right)^{1 / 2} d x$

Find the limits: (a) $\lim _{\Delta x \rightarrow 0} \frac{\ln \left(e^{2}+\Delta x\right)-2}{\Delta x}$ , (b) $\lim _{w \rightarrow 1} \frac{\ln w}{w-1}$ .

Find the derivative $\frac{dy}{dx}$ for $\sin(x^{2}y^{2})=x$ using implicit differentiation.

Untersuchen Sie die Funktionen $f(x)=e^{0,5x}$ und $g(x)=e^{1,5-0,25x}$ : Graphen skizzieren, Ableitungen finden, Schnittpunkt und Winkel bestimmen, Tangente ermitteln, Fläche A berechnen.

Solve the integral $\int x \ln x \mathrm{~d} x$ using integration by parts.

Verify the linear approximation at $a=0$ : $(1+3x)^{-4} \approx 1-12x$ . Find $x$ where it's accurate within 0.1. $x \in$

Estimate $e^{0.01}$ using linear approximation or differentials.

Estimate the paint volume (in $m^3$ ) for a $0.06 \mathrm{~cm}$ thick coat on a hemispherical dome with a $42 \mathrm{~m}$ diameter. Round to two decimal places.

Vereinfachen Sie $f(s)=\frac{2}{\sqrt{s}}-3 s^{5}$ und bestimmen Sie die Ableitung.

Untersuchen Sie die Kurvenschar $f_{a}(x)=x+a \cdot e^{-x}$ auf Extrema, Wendepunkte und zeichnen Sie die Graphen für $a=1, 0.5, -1$ .

Find the second derivative of $f(x)=\sqrt[3]{x}+3\cos(x)$ .

Beschreibe die Funktion $f(x)=20 \cdot(1-e^{-x})+0,5 \cdot x$ . a) Trainingseffekt für $x=5,0$ Minuten. b) Wann übersteigt der Trainingseffekt $17 \mathrm{kcal}/$ Minute?

Find the relation between $\frac{d A}{d t}$ , $\frac{d l}{d t}$ , and $\frac{d w}{d t}$ for $l=9$ and $w=10$ .

Find the equation for $\frac{d A}{d t}$ in terms of $\frac{d l}{d t}$ and $\frac{d w}{d t}$ for $l=9$ and $w=10$ .

Find the relation between $\frac{d A}{d t}$ , $\frac{d l}{d t}$ , and $\frac{d w}{d t}$ for $l=9$ , $w=10$ .

Find the relation between $\frac{d A}{d t}$ , $\frac{d l}{d t}$ , and $\frac{d w}{d t}$ for $l=18$ , $w=13$ .

Radioaktivität: Gegeben ist $N(t)=1000 \cdot e^{-0,069 t}$ . Bestimme den Anfangsbestand, den Bestand nach 1 Tag und den täglichen Zerfall in %.

A ball is dropped. What is its acceleration in $\mathrm{m} / \mathrm{s}^{2}$ while falling, assuming no air resistance?

Find the equation for $\frac{d A}{d t}$ in terms of $\frac{d l}{d t}$ and $\frac{d w}{d t}$ when $l=8$ and $w=4$ .

Find $\frac{d z}{d t}$ given $z^{4}=x^{3}+y^{2}$ , $\frac{d x}{d t}=-1$ , $\frac{d y}{d t}=3$ , at $(x, y)=(0,4)$ .

Find the equation for $\frac{d A}{d t}$ in terms of $\frac{d l}{d t}$ and $\frac{d w}{d t}$ for $l=19$ and $w=14$ .

Beurteilen Sie Naels Aussage zum Grenzwert der Funktion $f(x) = 0,001 x^{4}-0,11 x^{2}+3$ im Unendlichen.

Find the derivative of $y=\sin(4x^3)+\sin^5(3x^2)$ and simplify the answer.

An airplane at $12000 \mathrm{ft}$ flies away from a man $5000 \mathrm{ft}$ from a tower. If it moves at $400 \mathrm{ft/s}$ , find the rate the distance increases when over the tower.