Calculus

Problem 11701

Travel D miles is $T(x)=60 D(60+x)^{-1}$ . Approximate time for 77 miles at 58 mph using $L(x)$ ; find exact time too.

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Problem 11702

Identify the integral for the volume of the cylinder defined by $x^{2}+y^{2}=4$ and $0 \leq z \leq 2$ .

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Problem 11703

Find the linear approximation of $f(x)=\frac{x}{x+1}$ at $a=1$ , estimate $f(0.9)$ , and compute the percent error.

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Problem 11704

Find the line for the linear approximation of $f(x)=\sec x$ at $a=0$ , estimate $f(-0.02)$ , and calculate the percent error.

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Problem 11705

Find the tangent line equations for the function $f(x)=e^{2x}$ at the point where $x=2$ .

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Problem 11706

Given the demand function $q=D(p)=445-p$ , find: a) elasticity, b) elasticity at $p=88$ (elastic, inelastic, or unit), c) $p$ for max total revenue.

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Problem 11707

Find the maximum height and time to reach it for $h(t)=-3t^{2}+120t$ . Also, how long is the rocket above $900 \mathrm{~m}$ ?

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Problem 11708

Find the linear approximation of $f(x)=\frac{2}{x+1}$ at $a=0$ , estimate $\frac{2}{1.1}$ , and compute percent error.

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Problem 11709

Find the line equation for $f(x)=2 e^{x}$ at $a=0$ , estimate $f(0.03)$ , and compute percent error.

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Problem 11710

Find the value of $x=a$ where the linearization of $f(x)=\ln x$ is $y=1+\frac{1}{e}(x-e)$ .

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Problem 11711

Find the following limits: (a) $\lim _{x \rightarrow \infty} \frac{9 \sqrt{x}+15 x^{-2}}{6 x-14}$ (b) $\lim _{x \rightarrow-\infty} \frac{7+10 \sqrt[5]{x}}{7-10 \sqrt[5]{x}}$ (c) $\lim _{x \rightarrow \infty} \frac{12 x^{5 / 3}-x^{1 / 3}+3}{x^{8 / 5}+5 x+12 \sqrt{x}}$ (d) $\lim _{x \rightarrow-\infty} \frac{10 \sqrt[5]{x}-8 x+12}{7 x+4 x^{2 / 3}-10}$

See Solution

Problem 11712

Find the marginal revenue function from $R(x)=100 x e^{-0.006 x}$ . What units maximize revenue? Use calculus.

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Problem 11713

Find the quantity $q$ for max profit, price $p$ , and max profit $P$ given: $C(q)=70+5q e^{-0.02 q}$ and $p=20 e^{-0.02 q}$ .

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Problem 11714

Find the value of $\lim _{x \rightarrow 3} \frac{x^{3} f(x)-54}{g(x)-1}$ given $f(3)=2$ , $g(3)=1$ . Choices: (A) -108 (B) -54 (C) -27 (D) nonexistent.

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Problem 11715

Find the total distance traveled by the particle with position $x(t)=2 t^{3}+3 t^{2}-36 t+50$ from $t=0$ to $t=5$ .

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Problem 11716

Find the relative extrema of power usage described by $P(t)=-0.005899 t^{3}+0.1925 t^{2}-1.077 t+18.08$ for $0 \leq t \leq 24$ .

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Problem 11717

Maximize profit $P(x)=\ln(-x^{3}+6x^{2}+135x+1)$ for $0 \leq x \leq 10$ . Find optimal $x$ and max profit.

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Problem 11718

A conical tank (diameter 10 ft, height 15 ft) has water leaking at 12 ft³/hr. Volume is $27 \pi$ ft³. Find height change rate.

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Problem 11719

Find the training session length $t$ that maximizes the rating $R(t)=\frac{18 t}{t^{2}+81}$ .

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Problem 11720

Given advertising spending (in thousands) and revenue (in thousands), find a cubic polynomial that models the data, its derivatives, and analyze the results.

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Problem 11721

Maximize profit $P(x)=\ln(-x^{3}+9x^{2}+21x+1)$ for $0 \leq x \leq 10$ . Find optimal $x$ and max profit.

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Problem 11722

Which statement ensures the tangent line at $x=5$ overestimates $f(5.25)$ ? (A) $f$ is decreasing on $[5, 5.25]$ . (B) $f$ is increasing on $[5, 5.25]$ . (C) $f$ is concave down on $[5, 5.25]$ . (D) $f$ is concave up on $[5, 5.25]$ .

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Problem 11723

A car is 0.5 miles from a sign 0.25 miles east of an intersection. Is the distance to the intersection increasing or decreasing?

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Problem 11724

Approximate the area under $f(x)=0.03 x^{4}-1.69 x^{2}+91$ from $x=3$ to $x=11$ using 4 left endpoints.

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Problem 11725

Find the relative extrema of power usage given by $P(t)=-0.005899 t^{3}+0.1925 t^{2}-1.077 t+18.08$ for $0 \leq t \leq 24$ .

See Solution

Problem 11726

Find the intervals where the function $y=\frac{4}{(9+x)^{2}}$ is increasing and decreasing, with $x \neq -9$ .

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Problem 11727

Calculate the volume of the solid formed by rotating the area between $y=(1-x^2)^{1/2}$ and $y=((x/2)^{1/2})(1-x)$ from $x=0$ to $x=1$ around $y=0$ .

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Problem 11728

Evaluate the integral $\int_{R} \frac{2 x+2-y}{4} d x d y$ over the region $R=\{(x, y): 0 \leq x \leq 1,0 \leq y \leq 2, x+y \geq 1\}$ .

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Problem 11729

Find where the function $y=x^{3}-4 x^{2}+4 x+1$ is increasing, decreasing, concave up, and concave down using derivatives.

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Problem 11730

Evaluate the integral from -2 to 3 of the function $(6x + 3)$ .

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Problem 11731

Find the total distance a particle travels on the $x$ -axis from $t=0$ to $t=3$ given $x(t)=2t^{3}-6t^{2}+15$ .

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Problem 11732

Analyze the function $y=15 x e^{-x}$ for intervals of increase, decrease, concavity up, and concavity down using the first and second derivative tests.

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Problem 11733

Analyze the function $y=15 x e^{-x}$ for intervals of increase, decrease, concavity up, and concavity down.

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Problem 11734

Find $f(x)$ if $\int f(x) d x=(x+1)^{3} e^{x}+\ln (|x|)+C$ .

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Problem 11735

A 15-foot ladder leans against a wall. It slides down at 2 ft/s. Find how fast $\theta$ changes when $\theta = \frac{\pi}{4}$ .

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Problem 11736

Find $k$ such that $\int x^{4} e^{x^{5}} dx = k e^{x^{5}} + C$ .

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Problem 11737

Given $f(2)=-5$ , $g(2)=3$ , $f^{\prime}(2)=-4$ , $g^{\prime}(2)=2$ . Find $h^{\prime}(2)$ for: (a) $h(x)=3 f(x)-4 g(x)$ (b) $h(x)=f(x) g(x)$ (c) $h(x)=\frac{f(x)}{g(x)}$ (d) $h(x)=\frac{g(x)}{1+f(x)}$

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Problem 11738

Given $f(2)=-3, g(2)=5, f^{\prime}(2)=-1, g^{\prime}(2)=4$ , find $h^{\prime}(2)$ for: (a) $h(x)=3 f(x)-4 g(x)$ , (b) $h(x)=f(x) g(x)$ , (c) $h(x)=\frac{f(x)}{g(x)}$ , (d) $h(x)=\frac{g(x)}{1+f(x)}$ .

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Problem 11739

Find $f(3)$ given that $f^{\prime}(t)=6 t^{2}+e^{t}$ and $f(0)=6$ .

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Problem 11740

Find local maxima, minima, and intervals of increase/decrease for $y=x^{3}-12 x+6, x \in \mathbb{R}$ .

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Problem 11741

How long did King Kong take to fall $380 \mathrm{~m}$ ? What was his velocity just before landing?

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Problem 11742

Compute the integral of the function: $\int\left(\frac{1}{x^{7}}+\frac{1}{x-7}+e^{-8 x}\right) d x$

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Problem 11743

Prove that if $f(x)=\cos (x)$ , then $f^{\prime}(x)=-\sin (x)$ using the derivative definition.

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Problem 11744

Determine where the function $f(x)=x^{4}-50 x^{2}+625$ is increasing, decreasing, and find local extrema.

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Problem 11745

Find the average rate of change of $f(x)=-x^{2}+3x+2$ from $x=-2$ to $x=2$ . Simplify your answer.

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Problem 11746

Find the formula for the $z$ -coordinate of the centroid for the region defined by $x^{2}+y^{2}+z^{2} \leq 4$ and $z \geq 0$ .

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Problem 11747

A stream with insecticide at $9 \mathrm{~g/m}^{3}$ flows into a pond at $25 \mathrm{~m}^{3}/$ day with $2000 \mathrm{~m}^{3}$ volume.
(a) Let $y(t)$ be the insecticide amount (grams) in the pond. Write and solve the differential equation for $y(t)$ with initial condition. (b) What happens to the insecticide concentration after a long time? (c) How many days until the concentration reaches $7 \mathrm{~g/m}^{3}$ ?

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Problem 11748

Find $\frac{d y}{d t}$ at $x=0$ for $y=\frac{x+2}{x^{2}+1}$ , given $\frac{d x}{d t}=-5.2$ .

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Problem 11749

Find the absolute extreme values of $f(x)=2 x^{3}-27 x^{2}+84 x$ on the interval $[1,8]$ .

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Problem 11750

A winch pulls a 12m pipe vertically at -0.7 m/s. Find horizontal and vertical change rates when $y=4$ . Round to three decimals.

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Problem 11751

Find the dimensions of a cylindrical container with volume $5 \mathrm{~m}^{3}$ that minimizes material cost: top/bottom at \$6/m², side at \$9/m².

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Problem 11752

Find the dimensions of a cylindrical container with volume $5 \mathrm{~m}^{3}$ that minimizes material costs of \$6/m² for top/bottom and \$9/m² for the side.

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Problem 11753

Compute the future value after 6 years for income $f(t)=8000 e^{0.04 t}$ at $6.2\%$ interest using $\int_{0}^{6} f(t) e^{0.062 t} dt$ . Find the present value using $\int_{0}^{6} f(t) e^{-0.062 t} dt$ .

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Problem 11754

Find the second derivative of $f(x) = e^{x}(2x + x^{2} - 4)$ .

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Problem 11755

Find the area between $f(x)=x^{3}$ and $g(x)=2 k x^{2}-k^{2} x$ for all real $k$ . Calculate this area.

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Problem 11756

Invest \ $11,000 for 2 years. Which is better:$ 7\% $monthly or$ 6.94\%$ continuous compounding?

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Problem 11757

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

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Problem 11758

Find the first and second derivatives of $f_{k}(x)=x(x-k)e^{-x}$ .

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Problem 11759

Determine the inflection point of $f_{k}^{\prime \prime}(x)=e^{x}(2+4x+x^{2}-k)$ in terms of $k$ .

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Problem 11760

Find the derivative of $f(y)=\left(\frac{1}{y^{2}}-\frac{3}{y^{4}}\right)\left(y+5y^{3}\right)$ .

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Problem 11761

Find the derivative of $f(z)=(1-e^{z})(z+e^{2})$ .

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Problem 11762

Analyze the function $f(z)=(1-e^{z})(z+e^{z})$ to find its derivatives or zeros.

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Problem 11763

Find the derivative of $y=\frac{t^{3}+3t}{t^{2}-4t+3}$ .

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Problem 11764

Determine if the function $f(x)=\left\{\begin{array}{l}\sin (\pi x) \text { for } x \leq 2 \\ \pi(x-2) \text { for } x>2\end{array}\right.$ is continuous or differentiable at $x=1$ and $x=2$ .

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Problem 11765

Ordnen Sie die Grenzwertterme den passenden Grenzwerten zu:
1. $\lim _{x \rightarrow \infty} \frac{2 x}{x-2}$
2. $\lim _{x \rightarrow 2} \frac{2 x^{2}-8}{x-2}$
Wählen Sie aus: -1, 0, 4.5, 2, 8, $\infty$ .

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Problem 11766

Untersuchen Sie die Grenzwerte von links und rechts für die Funktionen: a) $f(x)=\frac{1}{x}$ bei $x_{0}=0$ b) $f(x)=\left\{\begin{array}{ll}x-1, & x \leq 0 \\ x^{2}, & x>0\end{array}\right.$ bei $x_{0}=0$ c) $f(x)=\left\{\begin{array}{l}x^{2}+1, & x \leq 1 \\ 3-x, & x>1\end{array}\right.$ bei $x_{0}=1$

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Problem 11767

Untersuchen Sie die Funktion $f_a(x) = x^3$ für $a > 0$ : Bestimmen Sie Extrema, Wendepunkte und Nullstellen.

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Problem 11768

Untersuchen Sie die Funktion $f_{a}(x)=\frac{a-1}{3} x^{3}-a x$ auf:
1. Definitionsbereich
2. Nullstellen
3. Extremstellen
4. Wendepunkte
5. Verhalten für $x \rightarrow \pm \infty$

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Problem 11769

Finde den Wendepunkt der Funktion $f_a(x)=(x-a) \cdot e^{0.5 x}$ .

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Problem 11770

Leiten Sie die Funktion $f(x)=\frac{1}{\sqrt{n}+2} x^{\sqrt{n}+2}$ ab.

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Problem 11771

Determine if the function $f(x)=\left\{\begin{array}{l}\sin (\pi x) \text{ for } x \leq 2 \\ \pi(x-2) \text{ for } x>2\end{array}\right.$ is continuous or differentiable at each point.

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Problem 11772

Find $f(x)$ from the equation $\frac{d y}{d x}=f^{\prime}(x)=3 x^{2}$ with $f(2)=7$ .

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Problem 11773

Gegeben ist die Funktion $f(x)=x^{2}-2x$ . Zeichne den Graphen für $-2 \leq x \leq 4$ und berechne die mittlere Steigung in den Intervallen $[-2; 0]$ , $[0; 3]$ und $[-1; 3]$ . Nenne zwei Intervalle mit mittlerer Steigung 1 und 2.

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Problem 11774

Find the tangent and normal line equations for the curve $y=x^{4}+2 e^{x}$ at the point $(0,2)$ .

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Problem 11775

Sketch the area between $y=13x-x^{2}$ and $y=10x$ , then find the enclosed area.

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Problem 11776

Given the function $f(x)=2 x^{3}+3 x^{2}-336 x$ , find where it's increasing, decreasing, local min/max, and inflection point.

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Problem 11777

Find the first derivative of $f(x)=x^{5/3} \ln(x)$ and determine where it has horizontal tangents. $x=$

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Problem 11778

Find the max/min values, zeroes, increasing/decreasing intervals, and $y$ -intercept for: 5. $g(x)=2 x^{2}-5 x-9$ , 6. $h(x)=0.5 x^{3}-7 x^{2}+8$ , 7. $j(x)=x^{4}-x^{3}-x^{2}-1$ .

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Problem 11779

Find $f^{\prime \prime}(x)$ from $f^{\prime}(x)=8 \cos x-8 \sin x$ and solve $\tan x=$ when it equals 0.

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Problem 11780

Let $f(x)=x+\frac{1+\ln x}{x}$ . Find $\lim_{x \to 0} f(x)$ , $\lim_{x \to +\infty} f(x)$ , and show $y=x$ is an asymptote for the curve.

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Problem 11781

Given the graph of the derivative $f'$ for $0 \leq x \leq 9$ :
(i) (a) Where is $f$ increasing? $(0,4) \cup(6,8)$ (b) Where are local maxima for $f$ ? $x=4,8$ Where is the local minimum? $x=6$ (c) Where is $f$ concave upward? $(0,1) \cup(2,3) \cup(5,7)$ Where is $f$ concave downward? $(1,2) \cup(3,5) \cup(7,9)$ (d) What are the inflection points of $f$ ? $x=$

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Problem 11782

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

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Problem 11783

Cliff divers in Acapulco jump from $39.8 \mathrm{~m}$ . Find their speed upon hitting the water, rounded to 1 decimal place.

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Problem 11784

Soit la fonction $f(x)=x+\frac{1+\ln x}{x}$ . Trouvez $\lim _{x \rightarrow 0} f(x)$ et $\lim _{x \rightarrow \infty} f(x)$ pour déterminer une asymptote.

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Problem 11785

Deutschland hat 55,5% Nadelbäume. Wachstumsrate der Fichte: $f(x)=0,3 x \cdot e^{-0,1 x}$ .
a) Beschreibe das Wachstum der Fichte. b) Prognostiziere die Höhe nach 20 Jahren. c) Zeige, dass $F(x)=-3(x+10) \cdot e^{-0,1 x}+c$ eine Stammfunktion ist. d) Bestimme, wann eine Fichte mit 8 m gefällt werden muss. e) Passe die Funktion für verlangsamtes Wachstum an.

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Problem 11786

Select a graph of $y=f(x)$ based on these derivative properties:
- $x<-2$ : $y'>0, y''<0$ - $-2$ : $y'=0, y''<0$ - $-2<x<0$ : $y'<0, y''<0$ - $0$ : $y'<0, y''=0$ - $0<x<2$ : $y'<0, y''>0$ - $2$ : $y'=0, y''>0$ - $x>2$ : $y'>0, y''>0$

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Problem 11787

求一只重 $889 \mathrm{~g}$ 的猎鹰在自由落体中达到 $73 \mathrm{~m} / \mathrm{s}$ 所需的最小高度。

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Problem 11788

Given functions $f(x)=(x-1)(\ln (x)-1)$ and $g(x)=\ln (x)-\frac{1}{x}$ , find limits, derivatives, and variations for $g$ and $f$ .

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Problem 11789

Which function has the largest integral value on $[a, b]$ : $f(x)$ , $f(x)-1$ , $-f(x)-3$ , $-3 f(x)$ , or $2 f(x)$ ?

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Problem 11790

Find the dimensions of a box with a square base and open top that has a volume of $32 \mathrm{~cm}^{3}$ and minimizes material.

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Problem 11791

Analyze the function with derivative $f^{\prime}(x)=(x-3)^{2}(x+4)$ : a. When is $f$ increasing or decreasing? b. Where are the local maxima and minima?

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Problem 11792

Find the tangent line $y=m x+b$ to $f(x)=\frac{5 \sin x}{2 \sin x+4 \cos x}$ at $a=\frac{\pi}{3}$ , with $m=\square$ , $b=\square$ .

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Problem 11793

Find the tangent line to $y=f(x)$ at $a=\pi/3$ where $f(x)=\frac{5 \sin x}{2 \sin x+4 \cos x}$ , with $m=1$ and $b=\square$ .

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Problem 11794

Find critical numbers, intervals of increase/decrease/constancy, and local extrema for $f(x)=\sqrt[3]{x^{2}+4 x}$ .

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Problem 11795

Bestimmen Sie die Extremstellen und den Wendepunkt der Funktion $f_{a}(x)=x^{3}-3 a^{2} x+2 a^{3}$ .

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Problem 11796

Find the derivative of the function $\frac{-12}{(x-6)^{2}}$ .

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Problem 11797

Bestimmen Sie den Wendepunkt der Funktion $f_{a}(x)=x^{3}-3 a^{2} x+2 a^{3}$ .

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Problem 11798

Bestimme den Parameter $a > 2$ , sodass $s(t)=1500 t^{2}$ für $t$ im Intervall [2; a] den Wert 6 erreicht. Wie lange dauert das und was ist die mittlere Geschwindigkeit?

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Problem 11799

Calculate the integral: $\int\left(\frac{12}{\sqrt{x}}+12 \sqrt{x}\right) dx$ .

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Problem 11800

Calculate the integral of $(6x + 5)^{2}$ with respect to $x$ .

See Solution

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Calculus

Problem 11701

Travel D miles is T(x)=60D(60+x)−1T(x)=60 D(60+x)^{-1}T(x)=60D(60+x)−1. Approximate time for 77 miles at 58 mph using L(x)L(x)L(x); find exact time too.

Problem 11702

Identify the integral for the volume of the cylinder defined by x2+y2=4x^{2}+y^{2}=4x2+y2=4 and 0≤z≤20 \leq z \leq 20≤z≤2.

Problem 11703

Find the linear approximation of f(x)=xx+1f(x)=\frac{x}{x+1}f(x)=x+1x​ at a=1a=1a=1, estimate f(0.9)f(0.9)f(0.9), and compute the percent error.

Problem 11704

Find the line for the linear approximation of f(x)=sec⁡xf(x)=\sec xf(x)=secx at a=0a=0a=0, estimate f(−0.02)f(-0.02)f(−0.02), and calculate the percent error.

Problem 11705

Find the tangent line equations for the function f(x)=e2xf(x)=e^{2x}f(x)=e2x at the point where x=2x=2x=2.

Problem 11706

Given the demand function q=D(p)=445−pq=D(p)=445-pq=D(p)=445−p, find: a) elasticity, b) elasticity at p=88p=88p=88 (elastic, inelastic, or unit), c) ppp for max total revenue.

Problem 11707

Find the maximum height and time to reach it for h(t)=−3t2+120th(t)=-3t^{2}+120th(t)=−3t2+120t. Also, how long is the rocket above 900 m900 \mathrm{~m}900 m?

Problem 11708

Find the linear approximation of f(x)=2x+1f(x)=\frac{2}{x+1}f(x)=x+12​ at a=0a=0a=0, estimate 21.1\frac{2}{1.1}1.12​, and compute percent error.

Problem 11709

Find the line equation for f(x)=2exf(x)=2 e^{x}f(x)=2ex at a=0a=0a=0, estimate f(0.03)f(0.03)f(0.03), and compute percent error.

Problem 11710

Find the value of x=ax=ax=a where the linearization of f(x)=ln⁡xf(x)=\ln xf(x)=lnx is y=1+1e(x−e)y=1+\frac{1}{e}(x-e)y=1+e1​(x−e).

Problem 11711

Problem 11712

Find the marginal revenue function from R(x)=100xe−0.006xR(x)=100 x e^{-0.006 x}R(x)=100xe−0.006x. What units maximize revenue? Use calculus.

Problem 11713

Find the quantity qqq for max profit, price ppp, and max profit PPP given: C(q)=70+5qe−0.02qC(q)=70+5q e^{-0.02 q}C(q)=70+5qe−0.02q and p=20e−0.02qp=20 e^{-0.02 q}p=20e−0.02q.

Problem 11714

Find the value of lim⁡x→3x3f(x)−54g(x)−1\lim _{x \rightarrow 3} \frac{x^{3} f(x)-54}{g(x)-1}limx→3​g(x)−1x3f(x)−54​ given f(3)=2f(3)=2f(3)=2, g(3)=1g(3)=1g(3)=1. Choices: (A) -108 (B) -54 (C) -27 (D) nonexistent.

Problem 11715

Find the total distance traveled by the particle with position x(t)=2t3+3t2−36t+50x(t)=2 t^{3}+3 t^{2}-36 t+50x(t)=2t3+3t2−36t+50 from t=0t=0t=0 to t=5t=5t=5.

Problem 11716

Find the relative extrema of power usage described by P(t)=−0.005899t3+0.1925t2−1.077t+18.08P(t)=-0.005899 t^{3}+0.1925 t^{2}-1.077 t+18.08P(t)=−0.005899t3+0.1925t2−1.077t+18.08 for 0≤t≤240 \leq t \leq 240≤t≤24.

Problem 11717

Maximize profit P(x)=ln⁡(−x3+6x2+135x+1)P(x)=\ln(-x^{3}+6x^{2}+135x+1)P(x)=ln(−x3+6x2+135x+1) for 0≤x≤100 \leq x \leq 100≤x≤10. Find optimal xxx and max profit.

Problem 11718

A conical tank (diameter 10 ft, height 15 ft) has water leaking at 12 ft³/hr. Volume is 27π27 \pi27π ft³. Find height change rate.

Problem 11719

Find the training session length ttt that maximizes the rating R(t)=18tt2+81R(t)=\frac{18 t}{t^{2}+81}R(t)=t2+8118t​.

Problem 11720

Given advertising spending (in thousands) and revenue (in thousands), find a cubic polynomial that models the data, its derivatives, and analyze the results.

Problem 11721

Maximize profit P(x)=ln⁡(−x3+9x2+21x+1)P(x)=\ln(-x^{3}+9x^{2}+21x+1)P(x)=ln(−x3+9x2+21x+1) for 0≤x≤100 \leq x \leq 100≤x≤10. Find optimal xxx and max profit.

Problem 11722

Problem 11723

A car is 0.5 miles from a sign 0.25 miles east of an intersection. Is the distance to the intersection increasing or decreasing?

Problem 11724

Approximate the area under f(x)=0.03x4−1.69x2+91f(x)=0.03 x^{4}-1.69 x^{2}+91f(x)=0.03x4−1.69x2+91 from x=3x=3x=3 to x=11x=11x=11 using 4 left endpoints.

Problem 11725

Find the relative extrema of power usage given by P(t)=−0.005899t3+0.1925t2−1.077t+18.08P(t)=-0.005899 t^{3}+0.1925 t^{2}-1.077 t+18.08P(t)=−0.005899t3+0.1925t2−1.077t+18.08 for 0≤t≤240 \leq t \leq 240≤t≤24.

Problem 11726

Find the intervals where the function y=4(9+x)2y=\frac{4}{(9+x)^{2}}y=(9+x)24​ is increasing and decreasing, with x≠−9x \neq -9x=−9.

Problem 11727

Calculate the volume of the solid formed by rotating the area between y=(1−x2)1/2y=(1-x^2)^{1/2}y=(1−x2)1/2 and y=((x/2)1/2)(1−x)y=((x/2)^{1/2})(1-x)y=((x/2)1/2)(1−x) from x=0x=0x=0 to x=1x=1x=1 around y=0y=0y=0.

Problem 11728

Evaluate the integral ∫R2x+2−y4dxdy\int_{R} \frac{2 x+2-y}{4} d x d y∫R​42x+2−y​dxdy over the region R={(x,y):0≤x≤1,0≤y≤2,x+y≥1}R=\{(x, y): 0 \leq x \leq 1,0 \leq y \leq 2, x+y \geq 1\}R={(x,y):0≤x≤1,0≤y≤2,x+y≥1}.

Problem 11729

Find where the function y=x3−4x2+4x+1y=x^{3}-4 x^{2}+4 x+1y=x3−4x2+4x+1 is increasing, decreasing, concave up, and concave down using derivatives.

Problem 11730

Evaluate the integral from -2 to 3 of the function (6x+3)(6x + 3)(6x+3).

Problem 11731

Find the total distance a particle travels on the xxx-axis from t=0t=0t=0 to t=3t=3t=3 given x(t)=2t3−6t2+15x(t)=2t^{3}-6t^{2}+15x(t)=2t3−6t2+15.

Problem 11732

Analyze the function y=15xe−xy=15 x e^{-x}y=15xe−x for intervals of increase, decrease, concavity up, and concavity down using the first and second derivative tests.

Problem 11733

Analyze the function y=15xe−xy=15 x e^{-x}y=15xe−x for intervals of increase, decrease, concavity up, and concavity down.

Problem 11734

Find f(x)f(x)f(x) if ∫f(x)dx=(x+1)3ex+ln⁡(∣x∣)+C\int f(x) d x=(x+1)^{3} e^{x}+\ln (|x|)+C∫f(x)dx=(x+1)3ex+ln(∣x∣)+C.

Problem 11735

A 15-foot ladder leans against a wall. It slides down at 2 ft/s. Find how fast θ\thetaθ changes when θ=π4\theta = \frac{\pi}{4}θ=4π​.

Problem 11736

Find kkk such that ∫x4ex5dx=kex5+C\int x^{4} e^{x^{5}} dx = k e^{x^{5}} + C∫x4ex5dx=kex5+C.

Problem 11737

Problem 11738

Problem 11739

Find f(3)f(3)f(3) given that f′(t)=6t2+etf^{\prime}(t)=6 t^{2}+e^{t}f′(t)=6t2+et and f(0)=6f(0)=6f(0)=6.

Problem 11740

Find local maxima, minima, and intervals of increase/decrease for y=x3−12x+6,x∈Ry=x^{3}-12 x+6, x \in \mathbb{R}y=x3−12x+6,x∈R.

Problem 11741

How long did King Kong take to fall 380 m380 \mathrm{~m}380 m? What was his velocity just before landing?

Problem 11742

Travel D miles is $T(x)=60 D(60+x)^{-1}$ . Approximate time for 77 miles at 58 mph using $L(x)$ ; find exact time too.

Identify the integral for the volume of the cylinder defined by $x^{2}+y^{2}=4$ and $0 \leq z \leq 2$ .

Find the linear approximation of $f(x)=\frac{x}{x+1}$ at $a=1$ , estimate $f(0.9)$ , and compute the percent error.

Find the line for the linear approximation of $f(x)=\sec x$ at $a=0$ , estimate $f(-0.02)$ , and calculate the percent error.

Find the tangent line equations for the function $f(x)=e^{2x}$ at the point where $x=2$ .

Given the demand function $q=D(p)=445-p$ , find: a) elasticity, b) elasticity at $p=88$ (elastic, inelastic, or unit), c) $p$ for max total revenue.

Find the maximum height and time to reach it for $h(t)=-3t^{2}+120t$ . Also, how long is the rocket above $900 \mathrm{~m}$ ?

Find the linear approximation of $f(x)=\frac{2}{x+1}$ at $a=0$ , estimate $\frac{2}{1.1}$ , and compute percent error.

Find the line equation for $f(x)=2 e^{x}$ at $a=0$ , estimate $f(0.03)$ , and compute percent error.

Find the value of $x=a$ where the linearization of $f(x)=\ln x$ is $y=1+\frac{1}{e}(x-e)$ .

Find the marginal revenue function from $R(x)=100 x e^{-0.006 x}$ . What units maximize revenue? Use calculus.

Find the quantity $q$ for max profit, price $p$ , and max profit $P$ given: $C(q)=70+5q e^{-0.02 q}$ and $p=20 e^{-0.02 q}$ .

Find the value of $\lim _{x \rightarrow 3} \frac{x^{3} f(x)-54}{g(x)-1}$ given $f(3)=2$ , $g(3)=1$ . Choices: (A) -108 (B) -54 (C) -27 (D) nonexistent.

Find the total distance traveled by the particle with position $x(t)=2 t^{3}+3 t^{2}-36 t+50$ from $t=0$ to $t=5$ .

Find the relative extrema of power usage described by $P(t)=-0.005899 t^{3}+0.1925 t^{2}-1.077 t+18.08$ for $0 \leq t \leq 24$ .

Maximize profit $P(x)=\ln(-x^{3}+6x^{2}+135x+1)$ for $0 \leq x \leq 10$ . Find optimal $x$ and max profit.

A conical tank (diameter 10 ft, height 15 ft) has water leaking at 12 ft³/hr. Volume is $27 \pi$ ft³. Find height change rate.

Find the training session length $t$ that maximizes the rating $R(t)=\frac{18 t}{t^{2}+81}$ .

Maximize profit $P(x)=\ln(-x^{3}+9x^{2}+21x+1)$ for $0 \leq x \leq 10$ . Find optimal $x$ and max profit.

Approximate the area under $f(x)=0.03 x^{4}-1.69 x^{2}+91$ from $x=3$ to $x=11$ using 4 left endpoints.

Find the relative extrema of power usage given by $P(t)=-0.005899 t^{3}+0.1925 t^{2}-1.077 t+18.08$ for $0 \leq t \leq 24$ .

Find the intervals where the function $y=\frac{4}{(9+x)^{2}}$ is increasing and decreasing, with $x \neq -9$ .

Calculate the volume of the solid formed by rotating the area between $y=(1-x^2)^{1/2}$ and $y=((x/2)^{1/2})(1-x)$ from $x=0$ to $x=1$ around $y=0$ .

Evaluate the integral $\int_{R} \frac{2 x+2-y}{4} d x d y$ over the region $R=\{(x, y): 0 \leq x \leq 1,0 \leq y \leq 2, x+y \geq 1\}$ .

Find where the function $y=x^{3}-4 x^{2}+4 x+1$ is increasing, decreasing, concave up, and concave down using derivatives.

Evaluate the integral from -2 to 3 of the function $(6x + 3)$ .

Find the total distance a particle travels on the $x$ -axis from $t=0$ to $t=3$ given $x(t)=2t^{3}-6t^{2}+15$ .

Analyze the function $y=15 x e^{-x}$ for intervals of increase, decrease, concavity up, and concavity down using the first and second derivative tests.

Analyze the function $y=15 x e^{-x}$ for intervals of increase, decrease, concavity up, and concavity down.

Find $f(x)$ if $\int f(x) d x=(x+1)^{3} e^{x}+\ln (|x|)+C$ .

A 15-foot ladder leans against a wall. It slides down at 2 ft/s. Find how fast $\theta$ changes when $\theta = \frac{\pi}{4}$ .

Find $k$ such that $\int x^{4} e^{x^{5}} dx = k e^{x^{5}} + C$ .

Find $f(3)$ given that $f^{\prime}(t)=6 t^{2}+e^{t}$ and $f(0)=6$ .

Find local maxima, minima, and intervals of increase/decrease for $y=x^{3}-12 x+6, x \in \mathbb{R}$ .

How long did King Kong take to fall $380 \mathrm{~m}$ ? What was his velocity just before landing?

Compute the integral of the function: $\int\left(\frac{1}{x^{7}}+\frac{1}{x-7}+e^{-8 x}\right) d x$

Prove that if $f(x)=\cos (x)$ , then $f^{\prime}(x)=-\sin (x)$ using the derivative definition.

Determine where the function $f(x)=x^{4}-50 x^{2}+625$ is increasing, decreasing, and find local extrema.

Find the average rate of change of $f(x)=-x^{2}+3x+2$ from $x=-2$ to $x=2$ . Simplify your answer.

Find the formula for the $z$ -coordinate of the centroid for the region defined by $x^{2}+y^{2}+z^{2} \leq 4$ and $z \geq 0$ .

Find $\frac{d y}{d t}$ at $x=0$ for $y=\frac{x+2}{x^{2}+1}$ , given $\frac{d x}{d t}=-5.2$ .

Find the absolute extreme values of $f(x)=2 x^{3}-27 x^{2}+84 x$ on the interval $[1,8]$ .

A winch pulls a 12m pipe vertically at -0.7 m/s. Find horizontal and vertical change rates when $y=4$ . Round to three decimals.

Find the dimensions of a cylindrical container with volume $5 \mathrm{~m}^{3}$ that minimizes material cost: top/bottom at \$6/m², side at \$9/m².

Find the dimensions of a cylindrical container with volume $5 \mathrm{~m}^{3}$ that minimizes material costs of \$6/m² for top/bottom and \$9/m² for the side.

Find the second derivative of $f(x) = e^{x}(2x + x^{2} - 4)$ .

Find the area between $f(x)=x^{3}$ and $g(x)=2 k x^{2}-k^{2} x$ for all real $k$ . Calculate this area.

Invest \ $11,000 for 2 years. Which is better:$ 7\% $monthly or$ 6.94\%$ continuous compounding?

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

Find the first and second derivatives of $f_{k}(x)=x(x-k)e^{-x}$ .

Determine the inflection point of $f_{k}^{\prime \prime}(x)=e^{x}(2+4x+x^{2}-k)$ in terms of $k$ .

Find the derivative of $f(y)=\left(\frac{1}{y^{2}}-\frac{3}{y^{4}}\right)\left(y+5y^{3}\right)$ .

Find the derivative of $f(z)=(1-e^{z})(z+e^{2})$ .

Analyze the function $f(z)=(1-e^{z})(z+e^{z})$ to find its derivatives or zeros.

Find the derivative of $y=\frac{t^{3}+3t}{t^{2}-4t+3}$ .

Determine if the function $f(x)=\left\{\begin{array}{l}\sin (\pi x) \text { for } x \leq 2 \\ \pi(x-2) \text { for } x>2\end{array}\right.$ is continuous or differentiable at $x=1$ and $x=2$ .

Ordnen Sie die Grenzwertterme den passenden Grenzwerten zu:
1. $\lim _{x \rightarrow \infty} \frac{2 x}{x-2}$
2. $\lim _{x \rightarrow 2} \frac{2 x^{2}-8}{x-2}$
Wählen Sie aus: -1, 0, 4.5, 2, 8, $\infty$ .

Untersuchen Sie die Funktion $f_a(x) = x^3$ für $a > 0$ : Bestimmen Sie Extrema, Wendepunkte und Nullstellen.

Untersuchen Sie die Funktion $f_{a}(x)=\frac{a-1}{3} x^{3}-a x$ auf:
1. Definitionsbereich
2. Nullstellen
3. Extremstellen
4. Wendepunkte
5. Verhalten für $x \rightarrow \pm \infty$

Finde den Wendepunkt der Funktion $f_a(x)=(x-a) \cdot e^{0.5 x}$ .

Leiten Sie die Funktion $f(x)=\frac{1}{\sqrt{n}+2} x^{\sqrt{n}+2}$ ab.

Determine if the function $f(x)=\left\{\begin{array}{l}\sin (\pi x) \text{ for } x \leq 2 \\ \pi(x-2) \text{ for } x>2\end{array}\right.$ is continuous or differentiable at each point.

Find $f(x)$ from the equation $\frac{d y}{d x}=f^{\prime}(x)=3 x^{2}$ with $f(2)=7$ .

Find the tangent and normal line equations for the curve $y=x^{4}+2 e^{x}$ at the point $(0,2)$ .