Calculus

Problem 12501

Berechne die Gesamtmasse der Glocke (Haube + Körper) mit Dichte $\rho=8,5 \frac{\mathrm{g}}{\mathrm{cm}^{3}}$ .

See Solution

Problem 12502

Untersuchen Sie den Mittelwert der Innendurchmesser des Glockenkörpers und vergleichen Sie ihn mit dem Wert bei $x=3,95$ .

See Solution

Problem 12503

Estimate $f(3.2)$ given $f(3)=2$ and $f^{\prime}(3)=5$ .

See Solution

Problem 12504

Find the $x$ -coordinate of the stationary point of $y=\mathrm{e}^{2 x}(\sin x+3 \cos x)$ in $0 \leq x \leq \pi$ , to 2 decimal places.

See Solution

Problem 12505

Gegeben ist die Funktion $f_{a}(x)=-x^{2}+3 a x-6 a+4$ . a) Zeigen, dass $f_{a}(2)=0$ . b) Bestimmen Sie Extrempunkte in Abhängigkeit von a.

See Solution

Problem 12506

Find the limits: $\lim_{x \rightarrow 3^{-}} g(x)$ , $\lim_{x \rightarrow 3^{+}} g(x)$ , and $\lim_{x \rightarrow 3} g(x)$ .

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Problem 12507

Find the derivative $f^{\prime}(x)$ using the limit definition for $f(x)=x-x^{2}$ . Show all work and simplify.

See Solution

Problem 12508

Find $f$ where $f'(x)=10x-9$ and $f(9)=0$ .

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Problem 12509

Find the derivative $f^{\prime}(x)$ using the limit definition for $f(x)=x-x^{2}$ . Show all work and simplify.

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Problem 12510

Find the function $f$ where $f^{\prime}(x)=\frac{8}{\sqrt{x}}$ and $f(1)=22$ .

See Solution

Problem 12511

Find $f^{\prime}(2)$ for $f(x) = \frac{q(x)-p(x)}{3-(-1)}$ given $p(2)=-1$ , $q(2)=3$ .

See Solution

Problem 12512

A circle inscribed in a square has a circumference increasing at $6 \mathrm{in/sec}$ .
(a) Find the rate of increase of the square's perimeter. (b) When the circle's area is $25 \pi \mathrm{in}^2$ , find the rate of increase of the area between the circle and square.

See Solution

Problem 12513

Bestimmen Sie die Extrempunkte der Funktionen:
1. $f(x)=\frac{1}{2} x^{2}+2 k x+k$
2. $f(x)=x^{3}-3 b^{2} x+2 b^{3}$
3. $f(x)=\frac{1}{4} x^{4}-\frac{2}{4} x^{2}$
Nutzen Sie die Ableitung und setzen Sie sie gleich Null.

See Solution

Problem 12514

Find the demand function given that $D^{\prime}(x)=-\frac{2000}{x^{2}}$ and $D(2)=1006$ .

See Solution

Problem 12515

Find the total cost function $C$ if the marginal cost is $C^{\prime}(x)=x^{3}-4x$ and fixed costs are \$2000.

See Solution

Problem 12516

A circle's radius grows at 7 ft/min. When the radius is 4 ft, find the area change rate. Round to three decimals.

See Solution

Problem 12517

Untersuchen Sie den Mittelwert der Innendurchmesser des Glockenkörpers und vergleichen Sie ihn mit dem Wert bei $x=3,95$ .

See Solution

Problem 12518

A sphere's radius decreases at 5 ft/s. When its volume is 510 ft³, find the volume's rate of change using $V=\frac{4}{3} \pi r^{3}$ . Round to three decimals.

See Solution

Problem 12519

Differentiate $g(x)=x^{\frac{2}{3}}-a^{\frac{2}{3}}$ using the power rule: if $f(x)=x^n$ , then $f'(x)=nx^{n-1}$ .

See Solution

Problem 12520

Untersuchen Sie den Mittelwert der Innendurchmesser des Glockenkörpers und vergleichen Sie ihn mit dem Wert bei $x=3,95$ .

See Solution

Problem 12521

Is there a solution for $g'(x)=16$ with $3<x<5$ using the mean value theorem for $g(x)=2^{x}$ ? Choose A, B, or C.

See Solution

Problem 12522

Can we apply the mean value theorem to $f(x)=\frac{1}{|x|}$ for $2<x<4$ with $f'(x)=-\frac{1}{4}$ ?
A No, not differentiable. B No, average rate isn't $-\frac{1}{4}$ . C Yes, conditions are met.

See Solution

Problem 12523

Differentiate $f(x) = \frac{1}{x^c}$ where $c$ is a constant.

See Solution

Problem 12524

Find $\frac{d y}{d x}$ if $y=\frac{1+x^{2}}{1-x^{2}}$ . Options: a. $-\frac{4 x}{(1-x^{2})^{2}}$ , b. $\frac{4 x}{(1-x^{2})^{2}}$ , c. $-\frac{4 x^{3}}{(1-x^{2})^{2}}$ , d. $\frac{2 x}{1-x^{2}}$ .

See Solution

Problem 12525

Can we apply the mean value theorem to find a solution for $g^{\prime}(x)=\frac{1}{2}$ in $100<x<101$ ?
Choose 1 answer: (A) No, since we don't know if the function is differentiable on that interval. (B) No, since the average rate of change of $g$ over $100 \leq x \leq 101$ isn't $\frac{1}{2}$ . (C) Yes, both conditions for using the mean value theorem have been met.

See Solution

Problem 12526

Find the area under the curve $y=13 x^{2}+2$ from $x=0$ to $x=3$ using a calculator.

See Solution

Problem 12527

Given values of $h$ : $h(-5)=18$ , $h(-4)=13$ , $h(-3)=15$ , $h(-2)=19$ . Is Tom's justification for $h'(c)=2$ in $(-4,-3)$ complete?

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Problem 12528

Is Lucy's justification for $g(c)=0$ in $(8,10)$ complete? Choose: (A) Yes, (B) No, she didn’t show average rate is 0. (C) No, she needed to mention differentiability.

See Solution

Problem 12529

Find the value of $c$ that satisfies the Mean Value Theorem for $h(x)=\sqrt{4x-8}$ on $[3,11]$ . Choose: (A) 3.5, (B) 6, (C) 8, (D) 9.5.

See Solution

Problem 12530

Find the derivative of $y=8 x^{2}+\ln(x^{2})$ .

See Solution

Problem 12531

Find the derivative of $y=18 x^{2} \ln(x^{3})$ .

See Solution

Problem 12532

Eine Schraube fällt aus $h=65 \mathrm{~m}$ . a) Bestimme die Fallzeit $t$ . b) Bestimme die Aufprallgeschwindigkeit $v$ .

See Solution

Problem 12533

Estimate the area between the curves $y=3x$ and $y=0.5x^{2}$ .

See Solution

Problem 12534

A circle inscribed in a square has a circumference increasing at $6 \mathrm{in} / \mathrm{sec}$ .
(a) Find the rate of increase of the square's perimeter. (b) When the circle's area is $25 \pi \mathrm{in}^{2}$ , find the rate of increase in the area between the circle and square.

See Solution

Problem 12535

Find the derivative of $f(x)=5 x^{4} \ln(x^{2})$ .

See Solution

Problem 12536

Graph the curves $y=2x$ and $y=8\sqrt{x}$ . Find the other $\mathrm{x}$ -coordinate where they intersect and the area between them. $\mathrm{x}=$ $\text{Area}=$

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Problem 12537

Calculate the area between the curves $y=x^{1/2}$ and $y=x^{1/3}$ from $x=0$ to $x=1$ .

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Problem 12538

Find the derivative of $f(x)=20 x^{5} \ln (x)$ .

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Problem 12539

Determine if the functions are continuous at the given points: a. $f(x)=\frac{x^{2}-1}{x+1}$ at $x=-1$ b. $f(x)=\begin{cases}2x+3 & \text{if } x \leq 1 \\ 6x-1 & \text{if } x>1\end{cases}$ at $x=1$

See Solution

Problem 12540

Check if these functions are continuous at the given points: a. $f(x)=\frac{x^{2}-1}{x+1}$ at $x=-1$ b. $f(x)=\begin{cases} 2x+3 & \text{if } x \leq 1 \\ 6x-1 & \text{if } x > 1 \end{cases}$ at $x=1$ c. $f(x)=x^{5}-x^{3}$ for $x>1$

See Solution

Problem 12541

Find $\frac{d x}{d t}$ when $x=4$ , $y=3$ , $\frac{d z}{d t}=1$ , and $\frac{d x}{d t}=3 \frac{d y}{d t}$ .

See Solution

Problem 12542

Water drains from a conical tank (height 12 ft, diameter 8 ft) into a cylindrical tank (area $400 \pi \mathrm{ft}^{2}$ ).
(a) Find the volume of the conical tank as a function of $h$ . (b) Determine the volume change rate when $h=3$ . (c) Find the rate of change of depth $y$ in the cylindrical tank when $h=3$ .

See Solution

Problem 12543

Find the antiderivative $F$ of $f(x)=x^{4}-4 x^{-3}-5$ such that $F(1)=1$ .

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Problem 12544

Water drains from a conical tank (height 12 ft, diameter 8 ft) into a cylindrical tank (area $400\pi$ ft²).
(a) Find volume of the conical tank as a function of $h$ . (b) Rate of volume change when $h=3$ ? (c) Rate of change of depth $y$ in the cylindrical tank when $h=3$ ?

See Solution

Problem 12545

Determine if these functions are continuous at the given points: a. $f(x)=\frac{x^{2}-1}{x+1}$ at $x=-1$ b. $f(x)=\left(\begin{array}{l}2 x+3 \\ 6 x-1\end{array}\right.$ if $x>1$ at $x=1$ c. $f(x)=x^{5}-x^{3}$ for all $x$

See Solution

Problem 12546

Find the area between the lines $f(x)=2x$ and $g(x)=4$ from $x=0$ to $x=4$ .

See Solution

Problem 12547

Find the production level $x$ that minimizes the average cost given $c(x)=5 x^{3}-35 x^{2}+5,547 x$ .

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Problem 12548

Find the antiderivative $F$ of $f(x)=x^{5}-4 x^{-3}-3$ such that $F(1)=2$ . What is $F(x)=\square$ ?

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Problem 12549

Find the derivative of $f(x)=\lfloor x\rfloor$ at $x=1$ and $x=\frac{3}{2}$ : compute $f'(1)$ and $f'\left(\frac{3}{2}\right)$ .

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Problem 12550

Find $16 \lim_{t \rightarrow \infty} \frac{y}{t}$ for the equation $y' + 2ty = t^2$ . Options: (1) 8, (2) 9, (3) 7, (4) 6.

See Solution

Problem 12551

A sphere with radius $1 \mathrm{~m}$ is dissolving at $3 \mathrm{~cm} /$ hour. Find the volume change rate in $\mathrm{cm}^{3} / \mathrm{hr}$ .

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Problem 12552

Find the derivative $f'(x)$ of the piecewise function $f(x)=\{x^{2}+2, x \geq 1; 2x+1, x<1\}$ .

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Problem 12553

Approximate the area under $f(x)=\frac{1}{5} x^{3}-x+7$ on $[-1,2]$ using a Midpoint Riemann Sum with $n=3$ .

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Problem 12554

Find the derivative $g^{\prime}(1)$ for the piecewise function $g(x)=\{x^{2}+2, x \geq 1; <3, x<1\}$ .

See Solution

Problem 12555

Analyze how the average rate of change of the function $f(x) = \sqrt{x}$ varies as interval endpoints increase.

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Problem 12556

Find the average rate of change of $f(x)=20(0.5)^{x}$ from $x=-2$ to $x=2$ . Options: $-18.75$ , $18.75$ , $-17.5$ , $17.5$ .

See Solution

Problem 12557

Find the critical numbers of $f(x)=\sqrt{49-x^{2}}$ . Options: a) 0 b) $-7,7$ c) $-7,0,7$ d) 7,0 e) None.

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Problem 12558

Find the interval where $f(x)=x^{\frac{1}{3}}(x^{2}-49)$ is decreasing. Choose from the options given.

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Problem 12559

Calculate the integral: $\int(4 x^{4}+4 x^{14}+4) \, dx$

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Problem 12560

Determine where the function $f(x)=x^{\frac{1}{3}}(x^{2}-21)$ is increasing from the options given.

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Problem 12561

Find the rate of increase for the function $y=1.08(1.16)^{x}$ .

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Problem 12562

Find the limits: a. $\lim_{x \to \infty} \frac{2x^5 - 4x + 10}{10x^5 + x - 1}$ , b. $f(x)=\{x^2 + 1, 2x + 4\}$ for $x < 3$ (limit at 3), c. $\lim_{x \to 1} \frac{x^2 + 3x + 1}{4x^2 - 9}$ .

See Solution

Problem 12563

Find the antiderivative $F$ of $f(x)=x^{3}-3 x^{-4}+2$ such that $F(1)=1$ .

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Problem 12564

Find the limits: a. $\lim _{x \rightarrow \infty} \frac{2 x^{5}-4 x+10}{10 x^{5}+x-1}$ , b. $f(x)=\left\{\begin{array}{c}x^{2}+1 \\ 2 x+4\end{array}\right.$ (limit at 3), c. $\lim _{x \rightarrow 1} \frac{x^{2}+3 x+1}{4 x^{2}-9}$ .

See Solution

Problem 12565

Find local minimum(s) of $f(x)=x^{\frac{1}{3}}\left(x^{2}-7\right)$ using the first derivative test. Choices: a) 1 b) 0,1 c) -1 d) $\infty$ e) None.

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Problem 12566

Find local maxima of $f(x)=x^{\frac{1}{3}}\left(x^{2}-49\right)$ using the first derivative test. Options: a) $\sqrt{7}$ b) 0,7 c) $-\sqrt{7}$ d) $\infty$ e) None.

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Problem 12567

Find the second derivative of $f(x)=\ln(5x)-5x$ . Choices: a) $\frac{1}{x^{2}}$ , b) $\frac{2}{x^{2}}$ , c) $\frac{3}{x^{2}}$ , d) $\frac{-1}{x^{2}}$ , e) None.

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Problem 12568

Find the interval where the graph of $f(x)=\ln(4x)-4x$ is always concave down, given $x > 0$ . Options: a) $(4, \infty)$ b) $(-\infty, \infty)$ c) $(0, \infty)$ d) $(-\infty, 0)$ e) None.

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Problem 12569

Find the absolute max and min of $f(t)=\frac{t^{2}}{t-8}$ on the interval $-5 \leq t \leq -\frac{1}{4}$ .

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Problem 12570

Calculate the arc length of the curve $y=\left(x+\frac{5}{9}\right)^{3 / 2}$ between $x=0$ and $x=3$ .

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Problem 12571

Übung: Differenzialrechnung Bestimmen Sie die 1. und 2. Ableitung der Funktionen: a) $f(x)=6 x^{3}+2 x^{2}-3$ b) $f(x)=5-\frac{3}{x^{2}}$ c) $f(x)=\frac{4}{\sqrt{x}}$ d) $f(x)=\frac{1}{6} x^{3}+\frac{2}{3} x^{2}+\frac{4}{5}$ e) $f_{t}(x)=\frac{5}{2 t} x^{2}-t x+3 t$ f) $f(x)=(3 x^{3}+x^{2})(4 x^{2}-2 x)$

See Solution

Problem 12572

Solve the differential equation $y' + \frac{1}{x} y = e^{x}$ with the initial condition $y(1) = 1$ .

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Problem 12573

Calculate the integral from -1 to 3 of the function $x^{4}-5 x^{3}+8 x^{2}-4 x$ .

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Problem 12574

Find the average of $f(x)=x^{3} \sqrt{1+x^{2}}$ from $0$ to $\sqrt{3}$ , rounded to four decimal places.

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Problem 12575

Calcola l'integrale particolare di $y^{\prime \prime}+y=0$ con $y=\sqrt{2}$ a $x=\frac{\pi}{4}$ e $y=1$ a $x=\frac{\pi}{2}$ .

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Problem 12576

Given the function $g(x)=x^{3}-12 x^{2}+45 x+9$ , find critical points, classify them, and determine intervals of increase/decrease and concavity.

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Problem 12577

Find the function $f(x)$ if $f''(x)=\sqrt{x}(x+1)$ and the tangent line at $(1,0)$ is $y=x-1$ .

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Problem 12578

Given $y=x^{3}+5$ and $\frac{dx}{dt}=2$ when $x=1$ , find $\frac{dy}{dt}$ at $x=1$ . $\frac{dy}{dt}=\square$ .

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Problem 12579

Find $\frac{dy}{dx}$ for $y=\sin^{-1}\left(\frac{1}{x^{4}}\right)$ .

See Solution

Problem 12580

How much will \$6000 grow in 15 years with 6% continuous interest? Use the formula for continuous compounding.

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Problem 12581

How much will you have in an account after 10 years if you deposit \$5000 at a 5% continuous interest rate?

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Problem 12582

Find the derivative of the function $f(x)=\frac{-9}{x^{8}}$ at the point $x=3$ . What is $f^{\prime}(3)$ ?

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Problem 12583

Compare the average rates of change for $f(x)=2 x^{2}$ and $g(x)=3 x^{2}$ over the interval $-3 \leq x \leq 4$ .

See Solution

Problem 12584

Find $f'(2)$ and $f'(4)$ for the function $f(x)=-4 x^{1/6}$ .

See Solution

Problem 12585

Find $f'(1)$ and $f'(6)$ for the function $f(x)=8 x^{1/4}(x^{3}-4)$ .

See Solution

Problem 12586

Differentiate $Y(u) = (u^{-2} + u^{-3})(u^{5} - 5u^{2})$ and find $Y'(u)$ .

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Problem 12587

Find the values of $P_{2}(0)$ , $P_{2}^{\prime}(0)$ , and $P_{2}^{\prime \prime}(0)$ for $P_{2}(x)=1+x+\frac{x^{2}}{2}$ . Then compute $P_{2}(1)$ .

See Solution

Problem 12588

Bestimmen Sie die Extrempunkte von $f_{a}$ in Abhängigkeit von a. Für welches a liegt ein Extrempunkt auf der $x$ -Achse?

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Problem 12589

Finde eine Funktion $F$ mit $F(1)=100$ für die folgenden $f$ :
a) $f(x)=x^{2}$
b) $f(x)=-10$
c) $f(x)=-x$

See Solution

Problem 12590

Finde die Punkte des Graphen von $f(x)=\frac{1}{6}(2x^{3}-3x^{2}-30x+8)$ mit Steigungswinkel a) $45^{\circ}$ , b) $60^{\circ}$ .

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Problem 12591

Bestimmen Sie die Ableitung $f^{\prime}$ für folgende Funktionen: a) $f(x)=x^{3}$ , b) $f(x)=2 x$ , c) $f(t)=3 t^{2}$ , d) $f(x)=\frac{1}{3} x^{3}$ , e) $f(x)=\frac{1}{3} x^{n}$ , f) $f(u)=2 u^{0}$ , g) $f(a)=a \cdot \sin \left(\frac{\pi}{2}\right)$ , h) $f(x)=3^{2}$ , i) $f(x)=x^{n+2}$ , j) $f(p)=p^{n-1}$ , k) $f(t)=0,5 t^{2 n}$ , l) $f(v)=a^{n} v^{n+1}$ .

See Solution

Problem 12592

Bestimmen Sie die Steigung von $f$ an $x_{0}$ für: a) $f(x)=x^{2}, x_{0}=5$ ; b) $f(x)=x^{3}, x_{0}=\sqrt{2}$ ; c) $f(x)=\frac{1}{x^{2}}, x_{0}=-2$ ; d) $f(x)=2\sqrt{x}, x_{0}=0$ .

See Solution

Problem 12593

Find the tangent line equation for $g(x)=\frac{f(x)}{2x}$ at $x=-3$ , given $f(-3)=-6$ and $f'(-3)=-1$ .

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Problem 12594

Untersuchen Sie die Funktion $f(x) = \frac{1}{5}(x^{4}-8x^{3}+18x^{2})$ auf Symmetrie und Nullstellen. Bestimmen Sie die Wendepunktgerade und zeichnen Sie die Graphen.

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Problem 12595

Geben Sie eine Stammfunktion für $f(x)$ an: a) $f(x)=x^{3}$ , b) $f(x)=3x^{3}+x^{2}$ , c) $f(x)=\frac{2}{7}x^{4}+\frac{1}{3}$ , d) $f(x)=\sqrt[5]{x^{7}}$ .

See Solution

Problem 12596

Finde die Stellen, wo $f$ die Steigung $m$ hat und wo $f$ und $g$ parallel sind. a) $f(x)=3x^{2}-4x$ , $m=-1$ b) $f(x)=x-\frac{4}{x^{\prime}}$ , $m=2$ c) $f(x)=1-2\sqrt{x}$ , $m=-\frac{1}{2}$ . $g(x)=-\frac{1}{3}x^{3}+2x^{2}-x$ , $g(x)=\frac{2}{x}+7x$ , $g(x)=-\frac{1}{3}x+4$ .

See Solution

Problem 12597

Berechne die Punkte des Graphen von $f(x)=\frac{1}{6}(2 x^{3}-3 x^{2}-30 x+8)$ mit Steigungswinkel a) $45^{\circ}$ , b) $60^{\circ}$ .

See Solution

Problem 12598

Find the tangent line equation for $g(x)=(2x-4)f(x)$ at $x=6$ , given $f(6)=1$ and $f'(6)=-2$ .

See Solution

Problem 12599

Bestimme die Maße eines offenen Kartons mit quadratischer Grundfläche, um bei $100 \mathrm{~cm}^{2}$ Oberfläche maximales Volumen zu erreichen. Zeige, dass es keine weiteren Maxima gibt.

See Solution

Problem 12600

Find the tangent line equation for $g(x)=3x f(x)$ at $x=-2$ , given $f(-2)=-2$ and $f'(-2)=-3$ .

See Solution

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Calculus

Problem 12501

Berechne die Gesamtmasse der Glocke (Haube + Körper) mit Dichte ρ=8,5gcm3\rho=8,5 \frac{\mathrm{g}}{\mathrm{cm}^{3}}ρ=8,5cm3g​.

Problem 12502

Untersuchen Sie den Mittelwert der Innendurchmesser des Glockenkörpers und vergleichen Sie ihn mit dem Wert bei x=3,95x=3,95x=3,95.

Problem 12503

Estimate f(3.2)f(3.2)f(3.2) given f(3)=2f(3)=2f(3)=2 and f′(3)=5f^{\prime}(3)=5f′(3)=5.

Problem 12504

Find the xxx-coordinate of the stationary point of y=e2x(sin⁡x+3cos⁡x)y=\mathrm{e}^{2 x}(\sin x+3 \cos x)y=e2x(sinx+3cosx) in 0≤x≤π0 \leq x \leq \pi0≤x≤π, to 2 decimal places.

Problem 12505

Gegeben ist die Funktion fa(x)=−x2+3ax−6a+4f_{a}(x)=-x^{2}+3 a x-6 a+4fa​(x)=−x2+3ax−6a+4. a) Zeigen, dass fa(2)=0f_{a}(2)=0fa​(2)=0. b) Bestimmen Sie Extrempunkte in Abhängigkeit von a.

Problem 12506

Find the limits: lim⁡x→3−g(x)\lim_{x \rightarrow 3^{-}} g(x)limx→3−​g(x), lim⁡x→3+g(x)\lim_{x \rightarrow 3^{+}} g(x)limx→3+​g(x), and lim⁡x→3g(x)\lim_{x \rightarrow 3} g(x)limx→3​g(x).

Problem 12507

Find the derivative f′(x)f^{\prime}(x)f′(x) using the limit definition for f(x)=x−x2f(x)=x-x^{2}f(x)=x−x2. Show all work and simplify.

Problem 12508

Find fff where f′(x)=10x−9f'(x)=10x-9f′(x)=10x−9 and f(9)=0f(9)=0f(9)=0.

Problem 12509

Find the derivative f′(x)f^{\prime}(x)f′(x) using the limit definition for f(x)=x−x2f(x)=x-x^{2}f(x)=x−x2. Show all work and simplify.

Problem 12510

Find the function fff where f′(x)=8xf^{\prime}(x)=\frac{8}{\sqrt{x}}f′(x)=x​8​ and f(1)=22f(1)=22f(1)=22.

Problem 12511

Find f′(2)f^{\prime}(2)f′(2) for f(x)=q(x)−p(x)3−(−1)f(x) = \frac{q(x)-p(x)}{3-(-1)}f(x)=3−(−1)q(x)−p(x)​ given p(2)=−1p(2)=-1p(2)=−1, q(2)=3q(2)=3q(2)=3.

Problem 12512

A circle inscribed in a square has a circumference increasing at 6in/sec6 \mathrm{in/sec}6in/sec. (a) Find the rate of increase of the square's perimeter. (b) When the circle's area is 25πin225 \pi \mathrm{in}^225πin2, find the rate of increase of the area between the circle and square.

Problem 12513

Problem 12514

Find the demand function given that D′(x)=−2000x2D^{\prime}(x)=-\frac{2000}{x^{2}}D′(x)=−x22000​ and D(2)=1006D(2)=1006D(2)=1006.

Problem 12515

Find the total cost function CCC if the marginal cost is C′(x)=x3−4xC^{\prime}(x)=x^{3}-4xC′(x)=x3−4x and fixed costs are \$2000.

Problem 12516

A circle's radius grows at 7 ft/min. When the radius is 4 ft, find the area change rate. Round to three decimals.

Problem 12517

Untersuchen Sie den Mittelwert der Innendurchmesser des Glockenkörpers und vergleichen Sie ihn mit dem Wert bei x=3,95x=3,95x=3,95.

Problem 12518

A sphere's radius decreases at 5 ft/s. When its volume is 510 ft³, find the volume's rate of change using V=43πr3V=\frac{4}{3} \pi r^{3}V=34​πr3. Round to three decimals.

Problem 12519

Differentiate g(x)=x23−a23g(x)=x^{\frac{2}{3}}-a^{\frac{2}{3}}g(x)=x32​−a32​ using the power rule: if f(x)=xnf(x)=x^nf(x)=xn, then f′(x)=nxn−1f'(x)=nx^{n-1}f′(x)=nxn−1.

Problem 12520

Untersuchen Sie den Mittelwert der Innendurchmesser des Glockenkörpers und vergleichen Sie ihn mit dem Wert bei x=3,95x=3,95x=3,95.

Problem 12521

Is there a solution for g′(x)=16g'(x)=16g′(x)=16 with 3<x<53<x<53<x<5 using the mean value theorem for g(x)=2xg(x)=2^{x}g(x)=2x? Choose A, B, or C.

Problem 12522

Can we apply the mean value theorem to f(x)=1∣x∣f(x)=\frac{1}{|x|}f(x)=∣x∣1​ for 2<x<42<x<42<x<4 with f′(x)=−14f'(x)=-\frac{1}{4}f′(x)=−41​? A No, not differentiable. B No, average rate isn't −14-\frac{1}{4}−41​. C Yes, conditions are met.

Problem 12523

Differentiate f(x)=1xcf(x) = \frac{1}{x^c}f(x)=xc1​ where ccc is a constant.

Problem 12524

Problem 12525

Problem 12526

Find the area under the curve y=13x2+2y=13 x^{2}+2y=13x2+2 from x=0x=0x=0 to x=3x=3x=3 using a calculator.

Problem 12527

Given values of hhh: h(−5)=18h(-5)=18h(−5)=18, h(−4)=13h(-4)=13h(−4)=13, h(−3)=15h(-3)=15h(−3)=15, h(−2)=19h(-2)=19h(−2)=19. Is Tom's justification for h′(c)=2h'(c)=2h′(c)=2 in (−4,−3)(-4,-3)(−4,−3) complete?

Problem 12528

Is Lucy's justification for g(c)=0g(c)=0g(c)=0 in (8,10)(8,10)(8,10) complete? Choose: (A) Yes, (B) No, she didn’t show average rate is 0. (C) No, she needed to mention differentiability.

Problem 12529

Find the value of ccc that satisfies the Mean Value Theorem for h(x)=4x−8h(x)=\sqrt{4x-8}h(x)=4x−8​ on [3,11][3,11][3,11]. Choose: (A) 3.5, (B) 6, (C) 8, (D) 9.5.

Problem 12530

Find the derivative of y=8x2+ln⁡(x2)y=8 x^{2}+\ln(x^{2})y=8x2+ln(x2).

Problem 12531

Find the derivative of y=18x2ln⁡(x3)y=18 x^{2} \ln(x^{3})y=18x2ln(x3).

Problem 12532

Eine Schraube fällt aus h=65 mh=65 \mathrm{~m}h=65 m. a) Bestimme die Fallzeit ttt. b) Bestimme die Aufprallgeschwindigkeit vvv.

Problem 12533

Estimate the area between the curves y=3xy=3xy=3x and y=0.5x2y=0.5x^{2}y=0.5x2.

Problem 12534

Problem 12535

Find the derivative of f(x)=5x4ln⁡(x2)f(x)=5 x^{4} \ln(x^{2})f(x)=5x4ln(x2).

Problem 12536

Graph the curves y=2xy=2xy=2x and y=8xy=8\sqrt{x}y=8x​. Find the other x\mathrm{x}x-coordinate where they intersect and the area between them. x=\mathrm{x}=x= Area=\text{Area}=Area=

Problem 12537

Calculate the area between the curves y=x1/2y=x^{1/2}y=x1/2 and y=x1/3y=x^{1/3}y=x1/3 from x=0x=0x=0 to x=1x=1x=1.

Problem 12538

Find the derivative of f(x)=20x5ln⁡(x)f(x)=20 x^{5} \ln (x)f(x)=20x5ln(x).

Problem 12539

Problem 12540

Problem 12541

Find dxdt\frac{d x}{d t}dtdx​ when x=4x=4x=4, y=3y=3y=3, dzdt=1\frac{d z}{d t}=1dtdz​=1, and dxdt=3dydt\frac{d x}{d t}=3 \frac{d y}{d t}dtdx​=3dtdy​.

Problem 12542

Problem 12543

Find the antiderivative FFF of f(x)=x4−4x−3−5f(x)=x^{4}-4 x^{-3}-5f(x)=x4−4x−3−5 such that F(1)=1F(1)=1F(1)=1.

Berechne die Gesamtmasse der Glocke (Haube + Körper) mit Dichte $\rho=8,5 \frac{\mathrm{g}}{\mathrm{cm}^{3}}$ .

Untersuchen Sie den Mittelwert der Innendurchmesser des Glockenkörpers und vergleichen Sie ihn mit dem Wert bei $x=3,95$ .

Estimate $f(3.2)$ given $f(3)=2$ and $f^{\prime}(3)=5$ .

Find the $x$ -coordinate of the stationary point of $y=\mathrm{e}^{2 x}(\sin x+3 \cos x)$ in $0 \leq x \leq \pi$ , to 2 decimal places.

Gegeben ist die Funktion $f_{a}(x)=-x^{2}+3 a x-6 a+4$ . a) Zeigen, dass $f_{a}(2)=0$ . b) Bestimmen Sie Extrempunkte in Abhängigkeit von a.

Find the limits: $\lim_{x \rightarrow 3^{-}} g(x)$ , $\lim_{x \rightarrow 3^{+}} g(x)$ , and $\lim_{x \rightarrow 3} g(x)$ .

Find the derivative $f^{\prime}(x)$ using the limit definition for $f(x)=x-x^{2}$ . Show all work and simplify.

Find $f$ where $f'(x)=10x-9$ and $f(9)=0$ .

Find the derivative $f^{\prime}(x)$ using the limit definition for $f(x)=x-x^{2}$ . Show all work and simplify.

Find the function $f$ where $f^{\prime}(x)=\frac{8}{\sqrt{x}}$ and $f(1)=22$ .

Find $f^{\prime}(2)$ for $f(x) = \frac{q(x)-p(x)}{3-(-1)}$ given $p(2)=-1$ , $q(2)=3$ .

A circle inscribed in a square has a circumference increasing at $6 \mathrm{in/sec}$ .
(a) Find the rate of increase of the square's perimeter. (b) When the circle's area is $25 \pi \mathrm{in}^2$ , find the rate of increase of the area between the circle and square.

Find the demand function given that $D^{\prime}(x)=-\frac{2000}{x^{2}}$ and $D(2)=1006$ .

Find the total cost function $C$ if the marginal cost is $C^{\prime}(x)=x^{3}-4x$ and fixed costs are \$2000.

Untersuchen Sie den Mittelwert der Innendurchmesser des Glockenkörpers und vergleichen Sie ihn mit dem Wert bei $x=3,95$ .

A sphere's radius decreases at 5 ft/s. When its volume is 510 ft³, find the volume's rate of change using $V=\frac{4}{3} \pi r^{3}$ . Round to three decimals.

Differentiate $g(x)=x^{\frac{2}{3}}-a^{\frac{2}{3}}$ using the power rule: if $f(x)=x^n$ , then $f'(x)=nx^{n-1}$ .

Untersuchen Sie den Mittelwert der Innendurchmesser des Glockenkörpers und vergleichen Sie ihn mit dem Wert bei $x=3,95$ .

Is there a solution for $g'(x)=16$ with $3<x<5$ using the mean value theorem for $g(x)=2^{x}$ ? Choose A, B, or C.

Can we apply the mean value theorem to $f(x)=\frac{1}{|x|}$ for $2<x<4$ with $f'(x)=-\frac{1}{4}$ ?
A No, not differentiable. B No, average rate isn't $-\frac{1}{4}$ . C Yes, conditions are met.

Differentiate $f(x) = \frac{1}{x^c}$ where $c$ is a constant.

Find the area under the curve $y=13 x^{2}+2$ from $x=0$ to $x=3$ using a calculator.

Given values of $h$ : $h(-5)=18$ , $h(-4)=13$ , $h(-3)=15$ , $h(-2)=19$ . Is Tom's justification for $h'(c)=2$ in $(-4,-3)$ complete?

Is Lucy's justification for $g(c)=0$ in $(8,10)$ complete? Choose: (A) Yes, (B) No, she didn’t show average rate is 0. (C) No, she needed to mention differentiability.

Find the value of $c$ that satisfies the Mean Value Theorem for $h(x)=\sqrt{4x-8}$ on $[3,11]$ . Choose: (A) 3.5, (B) 6, (C) 8, (D) 9.5.

Find the derivative of $y=8 x^{2}+\ln(x^{2})$ .

Find the derivative of $y=18 x^{2} \ln(x^{3})$ .

Eine Schraube fällt aus $h=65 \mathrm{~m}$ . a) Bestimme die Fallzeit $t$ . b) Bestimme die Aufprallgeschwindigkeit $v$ .

Estimate the area between the curves $y=3x$ and $y=0.5x^{2}$ .

Find the derivative of $f(x)=5 x^{4} \ln(x^{2})$ .

Graph the curves $y=2x$ and $y=8\sqrt{x}$ . Find the other $\mathrm{x}$ -coordinate where they intersect and the area between them. $\mathrm{x}=$ $\text{Area}=$

Calculate the area between the curves $y=x^{1/2}$ and $y=x^{1/3}$ from $x=0$ to $x=1$ .

Find the derivative of $f(x)=20 x^{5} \ln (x)$ .

Find $\frac{d x}{d t}$ when $x=4$ , $y=3$ , $\frac{d z}{d t}=1$ , and $\frac{d x}{d t}=3 \frac{d y}{d t}$ .

Find the antiderivative $F$ of $f(x)=x^{4}-4 x^{-3}-5$ such that $F(1)=1$ .

Water drains from a conical tank (height 12 ft, diameter 8 ft) into a cylindrical tank (area $400\pi$ ft²).
(a) Find volume of the conical tank as a function of $h$ . (b) Rate of volume change when $h=3$ ? (c) Rate of change of depth $y$ in the cylindrical tank when $h=3$ ?

Find the area between the lines $f(x)=2x$ and $g(x)=4$ from $x=0$ to $x=4$ .

Find the production level $x$ that minimizes the average cost given $c(x)=5 x^{3}-35 x^{2}+5,547 x$ .

Find the antiderivative $F$ of $f(x)=x^{5}-4 x^{-3}-3$ such that $F(1)=2$ . What is $F(x)=\square$ ?

Find the derivative of $f(x)=\lfloor x\rfloor$ at $x=1$ and $x=\frac{3}{2}$ : compute $f'(1)$ and $f'\left(\frac{3}{2}\right)$ .

Find $16 \lim_{t \rightarrow \infty} \frac{y}{t}$ for the equation $y' + 2ty = t^2$ . Options: (1) 8, (2) 9, (3) 7, (4) 6.

A sphere with radius $1 \mathrm{~m}$ is dissolving at $3 \mathrm{~cm} /$ hour. Find the volume change rate in $\mathrm{cm}^{3} / \mathrm{hr}$ .

Find the derivative $f'(x)$ of the piecewise function $f(x)=\{x^{2}+2, x \geq 1; 2x+1, x<1\}$ .

Approximate the area under $f(x)=\frac{1}{5} x^{3}-x+7$ on $[-1,2]$ using a Midpoint Riemann Sum with $n=3$ .

Find the derivative $g^{\prime}(1)$ for the piecewise function $g(x)=\{x^{2}+2, x \geq 1; <3, x<1\}$ .

Analyze how the average rate of change of the function $f(x) = \sqrt{x}$ varies as interval endpoints increase.

Find the average rate of change of $f(x)=20(0.5)^{x}$ from $x=-2$ to $x=2$ . Options: $-18.75$ , $18.75$ , $-17.5$ , $17.5$ .

Find the critical numbers of $f(x)=\sqrt{49-x^{2}}$ . Options: a) 0 b) $-7,7$ c) $-7,0,7$ d) 7,0 e) None.

Find the interval where $f(x)=x^{\frac{1}{3}}(x^{2}-49)$ is decreasing. Choose from the options given.

Calculate the integral: $\int(4 x^{4}+4 x^{14}+4) \, dx$

Determine where the function $f(x)=x^{\frac{1}{3}}(x^{2}-21)$ is increasing from the options given.

Find the rate of increase for the function $y=1.08(1.16)^{x}$ .

Find the antiderivative $F$ of $f(x)=x^{3}-3 x^{-4}+2$ such that $F(1)=1$ .

Find local minimum(s) of $f(x)=x^{\frac{1}{3}}\left(x^{2}-7\right)$ using the first derivative test. Choices: a) 1 b) 0,1 c) -1 d) $\infty$ e) None.

Find local maxima of $f(x)=x^{\frac{1}{3}}\left(x^{2}-49\right)$ using the first derivative test. Options: a) $\sqrt{7}$ b) 0,7 c) $-\sqrt{7}$ d) $\infty$ e) None.

Find the second derivative of $f(x)=\ln(5x)-5x$ . Choices: a) $\frac{1}{x^{2}}$ , b) $\frac{2}{x^{2}}$ , c) $\frac{3}{x^{2}}$ , d) $\frac{-1}{x^{2}}$ , e) None.

Find the interval where the graph of $f(x)=\ln(4x)-4x$ is always concave down, given $x > 0$ . Options: a) $(4, \infty)$ b) $(-\infty, \infty)$ c) $(0, \infty)$ d) $(-\infty, 0)$ e) None.

Find the absolute max and min of $f(t)=\frac{t^{2}}{t-8}$ on the interval $-5 \leq t \leq -\frac{1}{4}$ .

Calculate the arc length of the curve $y=\left(x+\frac{5}{9}\right)^{3 / 2}$ between $x=0$ and $x=3$ .

Solve the differential equation $y' + \frac{1}{x} y = e^{x}$ with the initial condition $y(1) = 1$ .

Calculate the integral from -1 to 3 of the function $x^{4}-5 x^{3}+8 x^{2}-4 x$ .

Find the average of $f(x)=x^{3} \sqrt{1+x^{2}}$ from $0$ to $\sqrt{3}$ , rounded to four decimal places.