Calculus

Problem 15601

Find the marginal cost from the average cost function $A C(Q)=4 Q+6+6.2 / Q$ at $Q=3$ . Round to the nearest integer!

See Solution

Problem 15602

Finde die Stelle $x_{0}$ , wo der Graph von $f(x)=x^{2}+2x-3$ einen Steigungswinkel von $45^{\circ}$ hat.

See Solution

Problem 15603

Find the first derivative of $f(x)=\left(\frac{x-1}{3 x}\right)^{5}$ . Which option is correct?

See Solution

Problem 15604

Find the marginal product of labor for $Y=0.1 L \cdot 10 + 3 L^{2} \cdot 10 - 0.1 L^{3} \cdot 10$ . What is the domain for $L$ ?

See Solution

Problem 15605

Berechne die Fläche unter der Funktion $f(x)=\frac{1}{3}x+7$ im Intervall [2, 5].

See Solution

Problem 15606

Find how equilibrium price $p$ and quantity $Q$ of coffee change with cocoa price $p_{c}$ . Calculate $\frac{d p}{d p_{c}}=$ and $\frac{d Q}{d p_{c}}=$ (round to 3 decimals).

See Solution

Problem 15607

Evaluate the integral: $\int\left(x^{3}-3\right) d x$

See Solution

Problem 15608

Find the first derivative of $f(x)=3 x^{3} \ln (x)$ . Choose the correct option from the list.

See Solution

Problem 15609

Zeigen Sie, dass für $b>0$ gilt: $\int_{0}^{b} x^{3} d x=\frac{1}{4} \cdot b^{4}$ und verwenden Sie die Summe der Kubikzahlen.

See Solution

Problem 15610

Evaluate the integral: $\int \sqrt{3 x} \, dx$

See Solution

Problem 15611

Evaluate the integral: $\int(10 x^{4}+6 x-1) \, dx$

See Solution

Problem 15612

Evaluate the integral: $\int \cos \frac{2}{3} x \, dx$

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Problem 15613

Evaluate the integral: $\int \frac{d x}{5-2 x}$

See Solution

Problem 15614

Evaluate the integral: $\int 4 x^{5} \, dx$

See Solution

Problem 15615

Evaluate the integral of the square root function: $\int \sqrt{x} \mathrm{dx}$ .

See Solution

Problem 15616

State Newton's method formula and find $x_{1}$ and $x_{2}$ using $f(x)=x^{2}-12$ and $x_{0}=3$ .

See Solution

Problem 15617

Evaluate the integral: $\int \sin 3 x \, dx$ .

See Solution

Problem 15618

Write Newton's method formula and compute $x_{1}$ and $x_{2}$ using $f(x)=4-\tan(4x)$ and $x_{0}=1$ .

See Solution

Problem 15619

Calculate the definite integral: $\int_{1}^{4} \frac{d x}{x^{2}}.$

See Solution

Problem 15620

Evaluate the integral $\int_{0}^{\frac{1}{2}}\left(\sum_{n=0}^{\infty}\left(\begin{array}{c}n+3 \\ n\end{array}\right) x^{n}\right) \mathrm{d} x$ .

See Solution

Problem 15621

Find the limit of the sequence $a_{n}=\frac{1 !+3 !+5 !+\cdots+(2 n-1) !}{2 !+4 !+6 !+\cdots+(2 n) !}$ as $n \to \infty$ .

See Solution

Problem 15622

Ein Modellhelikopter steigt. Gegeben ist $h(t)=-t^{2}-4t$ für $0 \leq t \leq 3$ .
a) Wann steigt er am schnellsten?
b) Wann ändert sich die Höhe nicht?
c) Weitere mathematische Aussagen zum Flugverlauf?

See Solution

Problem 15623

Approximate a root of the function $f(x)=3 x^{2}-\sqrt{x}-4$ using Newton's method starting with $x_{0}=2$ .

See Solution

Problem 15624

Find the price elasticity of demand for $Q=15.6-0.5p$ at $p=7.2 \,€$ . Type your answer as $\epsilon=$ solution.

See Solution

Problem 15625

Fiona's speed is given by $v(t)=\frac{2}{3} t^{3}-6 t^{2}+13 t+109$ . Find $v(0)$ and acceleration at $t=5$ .

See Solution

Problem 15626

Find the limit of the sequence $a_{n}=\frac{1 !+3 !+\cdots+(2 n-1) !}{2 !+4 !+\cdots+(2 n) !}$ as $n$ approaches infinity using the Stolz-Cesàro theorem.

See Solution

Problem 15627

Fiona's speed is given by $v(t)=\frac{2}{3} t^{3}-6 t^{2}+13 t+109$ .
(a) Find her speed at $t=0$ . (b) Calculate her acceleration at $t=5$ . (c) Determine when she reaches max speed in the first 4 minutes.

See Solution

Problem 15628

Find the limit: $\lim _{n \rightarrow \infty}\left(\frac{4 n^{3}+4 n+2}{4 n^{3}+2 n}\right)^{3 n^{2}}$ .

See Solution

Problem 15629

Evaluate the limit: $\lim _{n \rightarrow \infty} \frac{1+\frac{1}{\sqrt{2}}+\cdots+\frac{1}{\sqrt{n}}}{\sqrt{n}}$

See Solution

Problem 15630

Find the average value of $f(x)=\sqrt{x}(4-x)$ on $[0,4]$ . Options: (A) $\frac{7}{3}$ , (B) $\frac{21}{5}$ , (C) $\frac{32}{15}$ , (D) $\frac{35}{4}$ .

See Solution

Problem 15631

Aufgabe 4: Bakterienwachstum: a) Beschreibe die Folge, b) Wie viele Bakterien nach 1 Stunde?, c) Wann sind es 1 Million?
Aufgabe 5: Grenzwerte: a) Untersuche Konvergenz von (i) $a_{n}=\frac{100}{n}$ , (ii) $d_{n}=2n-(\frac{1}{2})^{n}$ . b) Zeige $g=1$ für $a_{n}=\frac{2n+1}{2n-1}$ . c) Berechne Grenzwerte von (i) $a_{n}=\frac{n^{5}-n^{4}}{6n^{5}}$ , (ii) $a_{n}=\frac{n^{5}+3n^{2}}{4n^{5}-3n^{3}}$ .
Aufgabe 6: Funktionenverhalten: a) Unterschiede zwischen Folge und Funktion? b) Grenzverhalten für $x \rightarrow \infty$ und $x \rightarrow -\infty$ für $f(x)=7x^{5}-x^{6}$ , $f(x)=\frac{1+x}{x^{2}+3}$ , $f(x)=4x+3,2^{x}$ .

See Solution

Problem 15632

Evaluate the integral $\int \frac{\sin (\ln (x))}{x} d x$ .

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Problem 15633

\ $1500 is invested at 10\% interest compounded continuously. Find the balance after 12 years using$ A=1500 e^{(0.10)(12)}$.

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Problem 15634

Calculate the flow rate through the triangular region with vertices (1,0,0), (0,1,0), (0,0,1) for $\mathbf{v}=4 \mathbf{k}$ . $\iint_{T} v \cdot d S=\square \mathrm{m}^{3} / \mathrm{s}$

See Solution

Problem 15635

Untersuchen Sie die Konvergenz der Folgen: (i) $a_{n}=\frac{100}{n}$ , (ii) $d_{n}=2 n-\left(\frac{1}{2}\right)^{n}$ . Zeigen Sie, dass $g=1$ für $a_{n}=\frac{2 n+1}{2 n-1}$ gilt. Berechnen Sie die Grenzwerte: (i) $a_{n}=\frac{n^{5}-n^{4}}{6 n^{5}}$ , (ii) $a_{n}=\frac{n^{5}+3 n^{2}}{4 n^{5}-3 n^{3}}$ .

See Solution

Problem 15636

Evaluate the limit as $x$ approaches 0: $\lim_{x \rightarrow 0} \frac{x - \sin x}{x^2}$ .

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Problem 15637

Find the heat flow rate out of a sphere (radius 1) in a copper cube with $K=400 \mathrm{~kW}/(\mathrm{m} \cdot \mathrm{K})$ .

See Solution

Problem 15638

Untersuchen Sie das Grenzverhalten der Funktionen für $x \rightarrow \infty$ und $x \rightarrow -\infty$ : a. $f(x)=7 x^{5}-x^{6}$ b. $f(x)=\frac{1+x}{x^{2}+3}$ c. $f(x)=4 x+3,2^{x}$ Nennen Sie die Quadranten, in denen die Funktionen beginnen und enden.

See Solution

Problem 15639

Find the magnetic flux $\Phi(t)$ through a rectangle of size $7 \times 2$ m at $d=0.5$ m above a wire with current $i(t)=t(10-t)$ . Use $\Phi = \int B \, dA$ and express in terms of $\mu_{0}$ and $I$ .

See Solution

Problem 15640

Find the flow rate across a net in a river with velocity $\mathbf{v}=\langle x-y, z+y+2, z^{2}\rangle$ and net $y=1-x^{2}-z^{2}, y \geq 0$ . Calculate $\iint_{S} \mathbf{v} \cdot d \mathbf{S}.$

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Problem 15641

Bestimmen Sie die Stammfunktion von $f(x)=\frac{1}{x^{5}}$ .

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Problem 15642

Find velocity and acceleration functions for each position function $s(t)$ . Simplify before differentiating.
a) $s(t)=5+7 t-8 t^{3}$ b) $s(t)=(2 t+3)(4-5 t)$ c) $s(t)=-(t+2)(3 t^{2}-t+5)$ d) $s(t)=\frac{-2 t^{4}-t^{3}+8 t^{2}}{4 t^{2}}$

See Solution

Problem 15643

Bestimme den Inhalt der Flächenstücke A: Halbe Fläche unter $f(x) = -0.5x^2 + 2x$ von 2 bis 4 und Fläche von 4 bis 5.

See Solution

Problem 15644

Find the derivative of $f(x)=-\frac{1}{10}(x^{2}-9)^{2}$ .

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Problem 15645

Calculate the area under the curve $f(x)=-0.5x^2+2x$ from $x=2$ to $x=5$ . What is the total area?

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Problem 15646

Berechne die mittlere Geschwindigkeit in m/s für die Zeitintervalle $[0 ; 2]$ , $[2 ; 4]$ und $[4 ; 6]$ mit $s(t)=0,25 t^{2}$ .

See Solution

Problem 15647

How high is the bridge if a rock dropped hits the water in 3.2 seconds? Use $h = \frac{1}{2} g t^2$ with $g \approx 9.8 \, \text{m/s}^2$ .

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Problem 15648

Find the carrying capacity of the logistic growth model $f(x)=\frac{160}{1+8 e^{-2 x}}$ .

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Problem 15649

a) Berechne die mittlere Geschwindigkeit des Zuges zwischen den Haltestellen. Wo ist die Geschwindigkeit am höchsten? b) Gib die Formel für die mittlere Geschwindigkeit $\bar{v}(t_{1}, t_{2})$ im Intervall $[t_{1}, t_{2}]$ an.

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Problem 15650

Estimate the area under $f(x)=x^{2}+6$ from $x=0$ to $x=4$ using left, midpoint, and right Riemann sums with $n=4$ .

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Problem 15651

A rocket is shot upward with $34.5 \mathrm{~m/s}$ from $3.2 \mathrm{~m}$ . Find velocity, max height time, height, return time, and impact velocity.

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Problem 15652

Given the function $f(x)=x^{3}+12 x^{2}-9$ , find its domain, critical numbers, intervals of increase/decrease, and local extrema.

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Problem 15653

Find $f^{\prime}(x)$ for $f(x)=x^{2} \ln x$ and identify the critical points. List them or state none.

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Problem 15654

Position function: $s(t)=2 t^{3}-15 t^{2}+36 t+10$ . Find velocity, acceleration, rest times, direction, and distance in 7s.

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Problem 15655

Given the function $f(x)=6(x-4)^{2/3}+2$ , find critical values, intervals of increase/decrease, and local max/min.

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Problem 15656

Find the antiderivative of $\int(\sqrt[6]{x}+\sqrt[7]{x}) d x = \square$

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Problem 15657

Find the antiderivative of each function with $\mathrm{C} = 0$ and verify by differentiation: (a) $g(x)=-7 x^{-8}$ , (b) $h(x)=x^{-8}$ .

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Problem 15658

Find the right Riemann sum for $f(x)=x^{2}$ over $[1,4]$ using 6 equal subintervals. Round to the nearest thousandth.

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Problem 15659

Solve the initial value problem: $\frac{d^{2} y}{d x^{2}}=8-12 x, y'(0)=5, y(0)=7$ . Find the integral for $\frac{d y}{d x}$ .

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Problem 15660

Estimate the area under $f(x)=x^{2}$ from $x=1$ to $x=3$ using 2 and then 4 rectangles. Area with 2 rectangles: $\square$ .

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Problem 15661

Find the area between the curve $y=x+2$ and the x-axis from $x=0$ to $x=1$ using limits.

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Problem 15662

Find the second derivative of $f(x)=4 x^{4}-4 x^{3}-4 x^{2}+2$ , then calculate $f^{\prime \prime}(0)$ and $f^{\prime \prime}(4)$ .

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Problem 15663

Approximate the area under $y=\sqrt{x}$ from $x=0$ to $x=4$ using upper and lower sums with $n=4$ partitions.

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Problem 15664

Calculate the surface integral $\iint_{S} \mathbf{F} \cdot d \mathbf{S}$ for $\mathbf{F}=\langle 4 e^{z}, 2 z, 2 x\rangle$ over the given surface. Round to three decimal places.

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Problem 15665

Find the second derivative of the function $f(x)=4 e^{-x^{2}}$ , then calculate $f^{\prime \prime}(0)$ and $f^{\prime \prime}(2)$ .

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Problem 15666

Select the correct statements about the second derivative: A. Finds intercepts of the first derivative. B. Gives $y$ -values for the first derivative. C. Represents the slope of the tangent line to the derivative at $x$ . D. Determines the rate of change of the first derivative. E. Found by setting the first derivative to zero.

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Problem 15667

Bestimmen Sie das Integral $\int_{-2}^{2}(0,5 t+0,5) d t$ mit Dreiecks- und Rechtecksflächen. Flächen: $A_{1}=0,25$ , $A_{2}=0,25$ , $A_{3}=1$ , $A_{4}=1$ .

See Solution

Problem 15668

Find $\lim _{x \rightarrow \infty} \frac{x^{3}}{\sqrt{x^{2}+4}}$ . What is the limit? A. $-\infty$ B. 0 C. 4 D. $\infty$

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Problem 15669

Find $\frac{d y}{d x}$ if $y=f(x^{2})$ and $f^{\prime}(x)=\sqrt{5 x-1}$ . Options: a. $2 x \sqrt{5 x^{2}-1}$ b. $\sqrt{5 x-1}$ c. $2 x \sqrt{5 x-1}$ d. $\sqrt{5 x^{2}-1}$ .

See Solution

Problem 15670

If $y=f\left(x^{2}\right)$ and $f^{\prime}(x)=\sqrt{5 x-1}$ , find $\frac{d y}{d x}$ . Options: a. $2 x \sqrt{5 x^{2}-1}$ , b. $\sqrt{5 x-1}$ , c. $2 x \sqrt{5 x-1}$ , d. $\sqrt{5 x^{2}-1}$ .

See Solution

Problem 15671

Find the velocity of an object dropped after 2, 5, and 8 seconds using $s(t)=-16t^{2}$ , and determine the acceleration.

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Problem 15672

Approximate the area under $f(x)=x^{2}+4x$ from $0$ to $4$ using the midpoint formula with $n=4$ .

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Problem 15673

Find the limit: $\lim _{y \rightarrow 3} \frac{\frac{1}{y}-\frac{1}{3}}{y-3}$ .

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Problem 15674

Find the linearization $L(x)$ of $f(x)=\ln(x^{2})$ at $x=e$ , then estimate $f(3)$ and explain the approximation's reasonableness.

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Problem 15675

A scientist has 180 mg of a radioactive substance. After 13 hours, 90 mg remains. How much is left after 18 hours? Give your answer to one decimal place.

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Problem 15676

Find the sum of the series $3600, 900, 225, \ldots$ ; what is $S_{\infty}$ ?

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Problem 15677

Leiten Sie die Funktion $f$ ab und vereinfachen Sie, wenn möglich: a) $f(x)=2 x+e^{2 x}$ , b) $f(x)=e^{2 x}-5 \cdot e^{\frac{1}{5} x}$ , c) $f(x)=x^{2}+e^{3 x-1}$ , d) $f(x)=20 \cdot e^{0,1 x}+x+5$ , e) $f(x)=\frac{1}{3} x^{3}-2 \cdot e^{-0,25 x}$ , f) $f(x)=2 x^{2}-e^{-x+3}+x^{3}$ .

See Solution

Problem 15678

Find the sum of the infinite series: 10, 20, 40, ... What is $S_{\infty}$ ?

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Problem 15679

Bestimme die Ableitung und klammere aus für die Funktionen: a) $f(x)=x e^{x}$ , b) $f(x)=(x-3)e^{x}$ , c) $f(x)=x^{2}e^{x}$ , d) $f(x)=xe^{x}-x^{2}$ , e) $f(x)=\sin(x)e^{x}$ , f) $f(x)=(x^{2}-5)e^{x}$ , g) $f(x)=x^{4}e^{x}-7$ , h) $f(x)=5x-xe^{x}$ .

See Solution

Problem 15680

Find the limit: $\lim _{h \rightarrow 0} \frac{(7+h)^{3}-7^{3}}{(7+h)^{2}-7^{2}}$ .

See Solution

Problem 15681

David has 80 yards of fencing for a rectangle. (a) Find area $A$ as a function of width $W$ . (b) What width $W$ maximizes area? (c) What is the maximum area?

See Solution

Problem 15682

Finden Sie eine Stammfunktion und berechnen Sie das Integral $\int_{-1}^{3}(4 x^{3}-2 x^{2}+1) \, dx$ . Erklären Sie den Begriff "eine" Stammfunktion.

See Solution

Problem 15683

Find the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{x+2}-\sqrt{2}}{x}$ .

See Solution

Problem 15684

Bestimme die Ableitungen von: I. $f(x)=\sin x^{2} \cdot e^{x+2}$ , II. $f(x)=\sqrt{2 x-2}$ , III. $f(x)=\frac{e^{x^{2}+2 x}}{-2}$ .

See Solution

Problem 15685

Zeichne den Graphen von $f(x)=x^{5}+1$ für $0 \leq x \leq 2$ und berechne den Flächeninhalt zwischen $f$ und der $x$ -Achse.

See Solution

Problem 15686

Approximate $(26.5)^{\frac{1}{3}}$ using linearization of the cube root function at $a=27$ . Answer to three decimal places.

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Problem 15687

Finde eine Stammfunktion für die Integrale: I. $\int e^{2 x} \cdot e^{2 x+2} d x$ , II. $\int_{-1}^{2} \frac{x}{\sqrt{2+x^{2}}} d x$ , III. $\int \alpha \cdot \cos (3-7 x) d x$ .

See Solution

Problem 15688

Differentiate implicitly to find $\frac{d^{2} y}{d x^{2}}$ for the equation $y^{2}-x y+x^{2}=8$ .

See Solution

Problem 15689

Gegeben sind die Funktionen $f(x)=x^{3}-6 x^{2}+8 x$ und $p(x)=x^{2}-2 x$ .
a) Berechne den Flächeninhalt zwischen $f$ und der $x$ -Achse im Intervall $I=[1 ; 5]$ und vergleiche mit dem Integral von $f$ .
b) Bestimme den Inhalt der Fläche zwischen $f$ und $p$ unter Angabe einer Stammfunktion.

See Solution

Problem 15690

Find the second derivative $C_{xx}$ of the function $C(x, y)=2xy+\frac{72000}{x}$ at $y=40$ and $x=30$ .

See Solution

Problem 15691

Evaluate the integral $\int_{1}^{e^{8}} \frac{\ln ^{2}\left(x^{2}\right)}{x} dx$ and simplify the integrand. Use the substitution $u=\square$ .

See Solution

Problem 15692

Maximize area of a rectangular field with 800 ft of fencing, subdivided into 3 plots. Find dimensions as reduced fractions.

See Solution

Problem 15693

Find the first derivative of the function $R(x)=(30-2x)(50+20x)$ .

See Solution

Problem 15694

Evaluate the integral $\int_{-\frac{3}{2}}^{\frac{1}{2}}\left(x^{3}-3 x\right) d x$ and discuss the area above/below the $x$ -axis.

See Solution

Problem 15695

Find the linearization $L_{1}(x)$ of $f(x)=5 x^{3}+\frac{1}{x^{2}}$ at $x=1$ . Use two decimal places for constants.

See Solution

Problem 15696

Find the acceleration of a speedboat from $t=0$ (speed 10) to $t=2$ (speed 20), assuming a straight line graph.

See Solution

Problem 15697

Find the integral of $x^{10} e^{x^{11}}$ with respect to $x$ .

See Solution

Problem 15698

Find the minimum average cost for $C(x)=\frac{4}{3} x^{2}+1200$ where $1 \leq x \leq 100$ .

See Solution

Problem 15699

Berechne den Flächeninhalt zwischen $f(x)=x^{3}-6x^{2}+8x$ und der $x$ -Achse im Intervall $I=[1;5]$ und vergleiche mit dem Integral von $f$ .

See Solution

Problem 15700

Evaluate the integral from 3 to 4: $\int_{3}^{4} \frac{t^{5}-5 t^{2}}{t^{4}} d t$ .

See Solution

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Calculus

Problem 15601

Find the marginal cost from the average cost function AC(Q)=4Q+6+6.2/QA C(Q)=4 Q+6+6.2 / QAC(Q)=4Q+6+6.2/Q at Q=3Q=3Q=3. Round to the nearest integer!

Problem 15602

Finde die Stelle x0x_{0}x0​, wo der Graph von f(x)=x2+2x−3f(x)=x^{2}+2x-3f(x)=x2+2x−3 einen Steigungswinkel von 45∘45^{\circ}45∘ hat.

Problem 15603

Find the first derivative of f(x)=(x−13x)5f(x)=\left(\frac{x-1}{3 x}\right)^{5}f(x)=(3xx−1​)5. Which option is correct?

Problem 15604

Find the marginal product of labor for Y=0.1L⋅10+3L2⋅10−0.1L3⋅10Y=0.1 L \cdot 10 + 3 L^{2} \cdot 10 - 0.1 L^{3} \cdot 10Y=0.1L⋅10+3L2⋅10−0.1L3⋅10. What is the domain for LLL?

Problem 15605

Berechne die Fläche unter der Funktion f(x)=13x+7f(x)=\frac{1}{3}x+7f(x)=31​x+7 im Intervall [2, 5].

Problem 15606

Find how equilibrium price ppp and quantity QQQ of coffee change with cocoa price pcp_{c}pc​. Calculate dpdpc=\frac{d p}{d p_{c}}=dpc​dp​= and dQdpc=\frac{d Q}{d p_{c}}=dpc​dQ​= (round to 3 decimals).

Problem 15607

Evaluate the integral: ∫(x3−3)dx\int\left(x^{3}-3\right) d x∫(x3−3)dx

Problem 15608

Find the first derivative of f(x)=3x3ln⁡(x)f(x)=3 x^{3} \ln (x)f(x)=3x3ln(x). Choose the correct option from the list.

Problem 15609

Zeigen Sie, dass für b>0b>0b>0 gilt: ∫0bx3dx=14⋅b4\int_{0}^{b} x^{3} d x=\frac{1}{4} \cdot b^{4}∫0b​x3dx=41​⋅b4 und verwenden Sie die Summe der Kubikzahlen.

Problem 15610

Evaluate the integral: ∫3x dx\int \sqrt{3 x} \, dx∫3x​dx

Problem 15611

Evaluate the integral: ∫(10x4+6x−1) dx\int(10 x^{4}+6 x-1) \, dx∫(10x4+6x−1)dx

Problem 15612

Evaluate the integral: ∫cos⁡23x dx\int \cos \frac{2}{3} x \, dx∫cos32​xdx

Problem 15613

Evaluate the integral: ∫dx5−2x\int \frac{d x}{5-2 x}∫5−2xdx​

Problem 15614

Evaluate the integral: ∫4x5 dx\int 4 x^{5} \, dx∫4x5dx

Problem 15615

Evaluate the integral of the square root function: ∫xdx\int \sqrt{x} \mathrm{dx}∫x​dx.

Problem 15616

State Newton's method formula and find x1x_{1}x1​ and x2x_{2}x2​ using f(x)=x2−12f(x)=x^{2}-12f(x)=x2−12 and x0=3x_{0}=3x0​=3.

Problem 15617

Evaluate the integral: ∫sin⁡3x dx\int \sin 3 x \, dx∫sin3xdx.

Problem 15618

Write Newton's method formula and compute x1x_{1}x1​ and x2x_{2}x2​ using f(x)=4−tan⁡(4x)f(x)=4-\tan(4x)f(x)=4−tan(4x) and x0=1x_{0}=1x0​=1.

Problem 15619

Calculate the definite integral: ∫14dxx2.\int_{1}^{4} \frac{d x}{x^{2}}.∫14​x2dx​.

Problem 15620

Evaluate the integral ∫012(∑n=0∞(n+3n)xn)dx\int_{0}^{\frac{1}{2}}\left(\sum_{n=0}^{\infty}\left(\begin{array}{c}n+3 \\ n\end{array}\right) x^{n}\right) \mathrm{d} x∫021​​(∑n=0∞​(n+3n​)xn)dx.

Problem 15621

Find the limit of the sequence an=1!+3!+5!+⋯+(2n−1)!2!+4!+6!+⋯+(2n)!a_{n}=\frac{1 !+3 !+5 !+\cdots+(2 n-1) !}{2 !+4 !+6 !+\cdots+(2 n) !}an​=2!+4!+6!+⋯+(2n)!1!+3!+5!+⋯+(2n−1)!​ as n→∞n \to \inftyn→∞.

Problem 15622

Ein Modellhelikopter steigt. Gegeben ist h(t)=−t2−4th(t)=-t^{2}-4th(t)=−t2−4t für 0≤t≤30 \leq t \leq 30≤t≤3. a) Wann steigt er am schnellsten? b) Wann ändert sich die Höhe nicht? c) Weitere mathematische Aussagen zum Flugverlauf?

Problem 15623

Approximate a root of the function f(x)=3x2−x−4f(x)=3 x^{2}-\sqrt{x}-4f(x)=3x2−x​−4 using Newton's method starting with x0=2x_{0}=2x0​=2.

Problem 15624

Find the price elasticity of demand for Q=15.6−0.5pQ=15.6-0.5pQ=15.6−0.5p at p=7.2 €p=7.2 \,€p=7.2€. Type your answer as ϵ=\epsilon=ϵ= solution.

Problem 15625

Fiona's speed is given by v(t)=23t3−6t2+13t+109v(t)=\frac{2}{3} t^{3}-6 t^{2}+13 t+109v(t)=32​t3−6t2+13t+109. Find v(0)v(0)v(0) and acceleration at t=5t=5t=5.

Problem 15626

Find the limit of the sequence an=1!+3!+⋯+(2n−1)!2!+4!+⋯+(2n)!a_{n}=\frac{1 !+3 !+\cdots+(2 n-1) !}{2 !+4 !+\cdots+(2 n) !}an​=2!+4!+⋯+(2n)!1!+3!+⋯+(2n−1)!​ as nnn approaches infinity using the Stolz-Cesàro theorem.

Problem 15627

Fiona's speed is given by v(t)=23t3−6t2+13t+109v(t)=\frac{2}{3} t^{3}-6 t^{2}+13 t+109v(t)=32​t3−6t2+13t+109. (a) Find her speed at t=0t=0t=0. (b) Calculate her acceleration at t=5t=5t=5. (c) Determine when she reaches max speed in the first 4 minutes.

Problem 15628

Find the limit: lim⁡n→∞(4n3+4n+24n3+2n)3n2\lim _{n \rightarrow \infty}\left(\frac{4 n^{3}+4 n+2}{4 n^{3}+2 n}\right)^{3 n^{2}}limn→∞​(4n3+2n4n3+4n+2​)3n2.

Problem 15629

Evaluate the limit: lim⁡n→∞1+12+⋯+1nn\lim _{n \rightarrow \infty} \frac{1+\frac{1}{\sqrt{2}}+\cdots+\frac{1}{\sqrt{n}}}{\sqrt{n}}limn→∞​n​1+2​1​+⋯+n​1​​

Problem 15630

Find the average value of f(x)=x(4−x)f(x)=\sqrt{x}(4-x)f(x)=x​(4−x) on [0,4][0,4][0,4]. Options: (A) 73\frac{7}{3}37​, (B) 215\frac{21}{5}521​, (C) 3215\frac{32}{15}1532​, (D) 354\frac{35}{4}435​.

Problem 15631

Problem 15632

Evaluate the integral ∫sin⁡(ln⁡(x))xdx\int \frac{\sin (\ln (x))}{x} d x∫xsin(ln(x))​dx.

Problem 15633

\1500isinvestedat10%interestcompoundedcontinuously.Findthebalanceafter12yearsusing1500 is invested at 10\% interest compounded continuously. Find the balance after 12 years using 1500isinvestedat10%interestcompoundedcontinuously.Findthebalanceafter12yearsusingA=1500 e^{(0.10)(12)}$.

Problem 15634

Calculate the flow rate through the triangular region with vertices (1,0,0), (0,1,0), (0,0,1) for v=4k\mathbf{v}=4 \mathbf{k}v=4k. ∬Tv⋅dS=□m3/s\iint_{T} v \cdot d S=\square \mathrm{m}^{3} / \mathrm{s}∬T​v⋅dS=□m3/s

Problem 15635

Problem 15636

Evaluate the limit as xxx approaches 0: lim⁡x→0x−sin⁡xx2\lim_{x \rightarrow 0} \frac{x - \sin x}{x^2}limx→0​x2x−sinx​.

Problem 15637

Find the heat flow rate out of a sphere (radius 1) in a copper cube with K=400 kW/(m⋅K)K=400 \mathrm{~kW}/(\mathrm{m} \cdot \mathrm{K})K=400 kW/(m⋅K).

Problem 15638

Problem 15639

Find the magnetic flux Φ(t)\Phi(t)Φ(t) through a rectangle of size 7×27 \times 27×2 m at d=0.5d=0.5d=0.5 m above a wire with current i(t)=t(10−t)i(t)=t(10-t)i(t)=t(10−t). Use Φ=∫B dA\Phi = \int B \, dAΦ=∫BdA and express in terms of μ0\mu_{0}μ0​ and III.

Problem 15640

Find the flow rate across a net in a river with velocity v=⟨x−y,z+y+2,z2⟩\mathbf{v}=\langle x-y, z+y+2, z^{2}\ranglev=⟨x−y,z+y+2,z2⟩ and net y=1−x2−z2,y≥0y=1-x^{2}-z^{2}, y \geq 0y=1−x2−z2,y≥0. Calculate ∬Sv⋅dS.\iint_{S} \mathbf{v} \cdot d \mathbf{S}.∬S​v⋅dS.

Problem 15641

Bestimmen Sie die Stammfunktion von f(x)=1x5f(x)=\frac{1}{x^{5}}f(x)=x51​.

Find the marginal cost from the average cost function $A C(Q)=4 Q+6+6.2 / Q$ at $Q=3$ . Round to the nearest integer!

Finde die Stelle $x_{0}$ , wo der Graph von $f(x)=x^{2}+2x-3$ einen Steigungswinkel von $45^{\circ}$ hat.

Find the first derivative of $f(x)=\left(\frac{x-1}{3 x}\right)^{5}$ . Which option is correct?

Find the marginal product of labor for $Y=0.1 L \cdot 10 + 3 L^{2} \cdot 10 - 0.1 L^{3} \cdot 10$ . What is the domain for $L$ ?

Berechne die Fläche unter der Funktion $f(x)=\frac{1}{3}x+7$ im Intervall [2, 5].

Find how equilibrium price $p$ and quantity $Q$ of coffee change with cocoa price $p_{c}$ . Calculate $\frac{d p}{d p_{c}}=$ and $\frac{d Q}{d p_{c}}=$ (round to 3 decimals).

Evaluate the integral: $\int\left(x^{3}-3\right) d x$

Find the first derivative of $f(x)=3 x^{3} \ln (x)$ . Choose the correct option from the list.

Zeigen Sie, dass für $b>0$ gilt: $\int_{0}^{b} x^{3} d x=\frac{1}{4} \cdot b^{4}$ und verwenden Sie die Summe der Kubikzahlen.

Evaluate the integral: $\int \sqrt{3 x} \, dx$

Evaluate the integral: $\int(10 x^{4}+6 x-1) \, dx$

Evaluate the integral: $\int \cos \frac{2}{3} x \, dx$

Evaluate the integral: $\int \frac{d x}{5-2 x}$

Evaluate the integral: $\int 4 x^{5} \, dx$

Evaluate the integral of the square root function: $\int \sqrt{x} \mathrm{dx}$ .

State Newton's method formula and find $x_{1}$ and $x_{2}$ using $f(x)=x^{2}-12$ and $x_{0}=3$ .

Evaluate the integral: $\int \sin 3 x \, dx$ .

Write Newton's method formula and compute $x_{1}$ and $x_{2}$ using $f(x)=4-\tan(4x)$ and $x_{0}=1$ .

Calculate the definite integral: $\int_{1}^{4} \frac{d x}{x^{2}}.$

Evaluate the integral $\int_{0}^{\frac{1}{2}}\left(\sum_{n=0}^{\infty}\left(\begin{array}{c}n+3 \\ n\end{array}\right) x^{n}\right) \mathrm{d} x$ .

Find the limit of the sequence $a_{n}=\frac{1 !+3 !+5 !+\cdots+(2 n-1) !}{2 !+4 !+6 !+\cdots+(2 n) !}$ as $n \to \infty$ .

Ein Modellhelikopter steigt. Gegeben ist $h(t)=-t^{2}-4t$ für $0 \leq t \leq 3$ .
a) Wann steigt er am schnellsten?
b) Wann ändert sich die Höhe nicht?
c) Weitere mathematische Aussagen zum Flugverlauf?

Approximate a root of the function $f(x)=3 x^{2}-\sqrt{x}-4$ using Newton's method starting with $x_{0}=2$ .

Find the price elasticity of demand for $Q=15.6-0.5p$ at $p=7.2 \,€$ . Type your answer as $\epsilon=$ solution.

Fiona's speed is given by $v(t)=\frac{2}{3} t^{3}-6 t^{2}+13 t+109$ . Find $v(0)$ and acceleration at $t=5$ .

Find the limit of the sequence $a_{n}=\frac{1 !+3 !+\cdots+(2 n-1) !}{2 !+4 !+\cdots+(2 n) !}$ as $n$ approaches infinity using the Stolz-Cesàro theorem.

Fiona's speed is given by $v(t)=\frac{2}{3} t^{3}-6 t^{2}+13 t+109$ .
(a) Find her speed at $t=0$ . (b) Calculate her acceleration at $t=5$ . (c) Determine when she reaches max speed in the first 4 minutes.

Find the limit: $\lim _{n \rightarrow \infty}\left(\frac{4 n^{3}+4 n+2}{4 n^{3}+2 n}\right)^{3 n^{2}}$ .

Evaluate the limit: $\lim _{n \rightarrow \infty} \frac{1+\frac{1}{\sqrt{2}}+\cdots+\frac{1}{\sqrt{n}}}{\sqrt{n}}$

Find the average value of $f(x)=\sqrt{x}(4-x)$ on $[0,4]$ . Options: (A) $\frac{7}{3}$ , (B) $\frac{21}{5}$ , (C) $\frac{32}{15}$ , (D) $\frac{35}{4}$ .

Evaluate the integral $\int \frac{\sin (\ln (x))}{x} d x$ .

\ $1500 is invested at 10\% interest compounded continuously. Find the balance after 12 years using$ A=1500 e^{(0.10)(12)}$.

Calculate the flow rate through the triangular region with vertices (1,0,0), (0,1,0), (0,0,1) for $\mathbf{v}=4 \mathbf{k}$ . $\iint_{T} v \cdot d S=\square \mathrm{m}^{3} / \mathrm{s}$

Evaluate the limit as $x$ approaches 0: $\lim_{x \rightarrow 0} \frac{x - \sin x}{x^2}$ .

Find the heat flow rate out of a sphere (radius 1) in a copper cube with $K=400 \mathrm{~kW}/(\mathrm{m} \cdot \mathrm{K})$ .

Find the magnetic flux $\Phi(t)$ through a rectangle of size $7 \times 2$ m at $d=0.5$ m above a wire with current $i(t)=t(10-t)$ . Use $\Phi = \int B \, dA$ and express in terms of $\mu_{0}$ and $I$ .

Find the flow rate across a net in a river with velocity $\mathbf{v}=\langle x-y, z+y+2, z^{2}\rangle$ and net $y=1-x^{2}-z^{2}, y \geq 0$ . Calculate $\iint_{S} \mathbf{v} \cdot d \mathbf{S}.$

Bestimmen Sie die Stammfunktion von $f(x)=\frac{1}{x^{5}}$ .

Bestimme den Inhalt der Flächenstücke A: Halbe Fläche unter $f(x) = -0.5x^2 + 2x$ von 2 bis 4 und Fläche von 4 bis 5.

Find the derivative of $f(x)=-\frac{1}{10}(x^{2}-9)^{2}$ .

Calculate the area under the curve $f(x)=-0.5x^2+2x$ from $x=2$ to $x=5$ . What is the total area?

Berechne die mittlere Geschwindigkeit in m/s für die Zeitintervalle $[0 ; 2]$ , $[2 ; 4]$ und $[4 ; 6]$ mit $s(t)=0,25 t^{2}$ .

How high is the bridge if a rock dropped hits the water in 3.2 seconds? Use $h = \frac{1}{2} g t^2$ with $g \approx 9.8 \, \text{m/s}^2$ .

Find the carrying capacity of the logistic growth model $f(x)=\frac{160}{1+8 e^{-2 x}}$ .

a) Berechne die mittlere Geschwindigkeit des Zuges zwischen den Haltestellen. Wo ist die Geschwindigkeit am höchsten? b) Gib die Formel für die mittlere Geschwindigkeit $\bar{v}(t_{1}, t_{2})$ im Intervall $[t_{1}, t_{2}]$ an.

Estimate the area under $f(x)=x^{2}+6$ from $x=0$ to $x=4$ using left, midpoint, and right Riemann sums with $n=4$ .

A rocket is shot upward with $34.5 \mathrm{~m/s}$ from $3.2 \mathrm{~m}$ . Find velocity, max height time, height, return time, and impact velocity.

Given the function $f(x)=x^{3}+12 x^{2}-9$ , find its domain, critical numbers, intervals of increase/decrease, and local extrema.

Find $f^{\prime}(x)$ for $f(x)=x^{2} \ln x$ and identify the critical points. List them or state none.

Position function: $s(t)=2 t^{3}-15 t^{2}+36 t+10$ . Find velocity, acceleration, rest times, direction, and distance in 7s.

Given the function $f(x)=6(x-4)^{2/3}+2$ , find critical values, intervals of increase/decrease, and local max/min.

Find the antiderivative of $\int(\sqrt[6]{x}+\sqrt[7]{x}) d x = \square$

Find the antiderivative of each function with $\mathrm{C} = 0$ and verify by differentiation: (a) $g(x)=-7 x^{-8}$ , (b) $h(x)=x^{-8}$ .

Find the right Riemann sum for $f(x)=x^{2}$ over $[1,4]$ using 6 equal subintervals. Round to the nearest thousandth.

Solve the initial value problem: $\frac{d^{2} y}{d x^{2}}=8-12 x, y'(0)=5, y(0)=7$ . Find the integral for $\frac{d y}{d x}$ .

Estimate the area under $f(x)=x^{2}$ from $x=1$ to $x=3$ using 2 and then 4 rectangles. Area with 2 rectangles: $\square$ .

Find the area between the curve $y=x+2$ and the x-axis from $x=0$ to $x=1$ using limits.

Find the second derivative of $f(x)=4 x^{4}-4 x^{3}-4 x^{2}+2$ , then calculate $f^{\prime \prime}(0)$ and $f^{\prime \prime}(4)$ .

Approximate the area under $y=\sqrt{x}$ from $x=0$ to $x=4$ using upper and lower sums with $n=4$ partitions.

Calculate the surface integral $\iint_{S} \mathbf{F} \cdot d \mathbf{S}$ for $\mathbf{F}=\langle 4 e^{z}, 2 z, 2 x\rangle$ over the given surface. Round to three decimal places.

Find the second derivative of the function $f(x)=4 e^{-x^{2}}$ , then calculate $f^{\prime \prime}(0)$ and $f^{\prime \prime}(2)$ .

Bestimmen Sie das Integral $\int_{-2}^{2}(0,5 t+0,5) d t$ mit Dreiecks- und Rechtecksflächen. Flächen: $A_{1}=0,25$ , $A_{2}=0,25$ , $A_{3}=1$ , $A_{4}=1$ .

Find $\lim _{x \rightarrow \infty} \frac{x^{3}}{\sqrt{x^{2}+4}}$ . What is the limit? A. $-\infty$ B. 0 C. 4 D. $\infty$

Find $\frac{d y}{d x}$ if $y=f(x^{2})$ and $f^{\prime}(x)=\sqrt{5 x-1}$ . Options: a. $2 x \sqrt{5 x^{2}-1}$ b. $\sqrt{5 x-1}$ c. $2 x \sqrt{5 x-1}$ d. $\sqrt{5 x^{2}-1}$ .

If $y=f\left(x^{2}\right)$ and $f^{\prime}(x)=\sqrt{5 x-1}$ , find $\frac{d y}{d x}$ . Options: a. $2 x \sqrt{5 x^{2}-1}$ , b. $\sqrt{5 x-1}$ , c. $2 x \sqrt{5 x-1}$ , d. $\sqrt{5 x^{2}-1}$ .

Find the velocity of an object dropped after 2, 5, and 8 seconds using $s(t)=-16t^{2}$ , and determine the acceleration.

Approximate the area under $f(x)=x^{2}+4x$ from $0$ to $4$ using the midpoint formula with $n=4$ .

Find the limit: $\lim _{y \rightarrow 3} \frac{\frac{1}{y}-\frac{1}{3}}{y-3}$ .

Find the linearization $L(x)$ of $f(x)=\ln(x^{2})$ at $x=e$ , then estimate $f(3)$ and explain the approximation's reasonableness.