Calculus

Problem 25601

Evaluate the series: $S = \sum_{n=1}^{\infty}(-1)^{n} \frac{1}{n+4}$ .

See Solution

Problem 25602

Define $\alpha$ , $f(x)$ , and $L$ for the limit: $\lim_{x \rightarrow 4}(2x^{2}-7)=25$ .

See Solution

Problem 25603

Untersuche die Stetigkeit der Funktion
$f(x)=\left\{\begin{array}{cc} \cos \frac{1}{x}, & \text { für } x<0 \\ x^{2}+1, & \text { für } x \geq 0 . \end{array}\right.$
an $x_{0}=0$ . Was ist an der Argumentation korrekt oder nicht?

See Solution

Problem 25604

Bestimmen Sie die ersten Ableitungen der Funktionen: a) $f(x)=\frac{x}{10}+\frac{10}{x}$ und skizzieren Sie die Graphen. b) $f(x)=x^{-2}(2 \sin(x)+\cos(x))$ .

See Solution

Problem 25605

Untersuche die Stetigkeit der Funktion $f(x)=\left\{\begin{array}{cc} \cos \frac{1}{x}, & x<0 \\ x^{2}+1, & x \geq 0 \end{array}\right.$ an $x_{0}=0$ . Was ist korrekt an der Argumentation? Was muss allgemein gezeigt werden? Ist das Ergebnis richtig?

See Solution

Problem 25606

Find the number of iPhones, $x$ , to minimize average cost given $C(x)=200000+180x+0.004x^{2}$ . What is the average cost?

See Solution

Problem 25607

Bestimme die Ableitungsfunktion für die folgenden Funktionen: a) $f(x)=2 x+x^{3}$ , b) $f(x)=5 x$ , c) $f(x)=a x^{2}$ , d) $f(x)=a x^{n}$ , e) $f(x)=2 x^{2}+4 x$ , f) $f(x)=\frac{1}{2} x^{2}+5$ , g) $f(x)=2 x^{3}-3 x^{2}+2$ , h) $f(x)=a x^{3}+b x+c$ .

See Solution

Problem 25608

Find the derivative $\frac{d y}{d x}$ using implicit differentiation for the equation $x^{2} y - y^{2} = 36$ .

See Solution

Problem 25609

Find the derivative $\frac{d p}{d q}$ using implicit differentiation for the equation: $p^{2}-p q=9 p^{2} q^{2}$ .

See Solution

Problem 25610

Bestimmen Sie die Ableitungsfunktion $f'(x)$ für $f(x)=3(x-2)^{2}+x$ mit Ableitungsregeln.

See Solution

Problem 25611

Find the slope and equation of the tangent line to the curve $6 x^{2}-y^{2}=x y-6$ at the point $(-1,4)$ .

See Solution

Problem 25612

Find the differential of the functions: 11. $y=e^{5x}$ , 12. $y=\text{[incomplete function]}$ .

See Solution

Problem 25613

Evaluate the integral: $\int \frac{6 x}{\sqrt{4+x^{2}}} d x$

See Solution

Problem 25614

Hanson's parents deposited \$6,966 at 14% continuous interest. He withdrew \$7,760 for tuition. How long was it in the account?

See Solution

Problem 25615

Find the tangent line to $y=e^{x}$ parallel to $2x-y=5$ .

See Solution

Problem 25616

Evaluate the series $30 \sum_{k=1}^{\infty} \frac{(-i)^{k}}{3^{k+1}}$ .

See Solution

Problem 25617

Evaluate $\int_{C} \mathbf{F} \cdot d \mathbf{r}$ using Green's Theorem for $\mathbf{F}=\left(2 y+e^{y}\right) \mathbf{i}+x e^{y} \mathbf{j}$ on circle $(x+1)^{2}+(y-5)^{2}=4$ . Choices: A. $-8 \pi$ , B. $\frac{9}{2} \pi$ , C. $-\frac{3}{2} \pi$ , D. $16 \pi$ , E. $-12 \pi$ .

See Solution

Problem 25618

Calculate the sum: $10 \sum_{k=0}^{\infty} \frac{(-i)^{k}}{3^{k+1}}$ .

See Solution

Problem 25619

Find $f(\pi)$ if $f(0)=4$ and $f^{\prime}(x)=-\sin (x)-e^{-x}+3 x^{2}$ . Choose from the options.

See Solution

Problem 25620

Calculate the integral: $\int x^{2}(x+3) \, dx$

See Solution

Problem 25621

Calculate the integral $\int_{-1}^{0} e^{-x} \, dx$ .

See Solution

Problem 25622

Evaluate the integral $\int_{-\pi}^{\pi} f(x) dx$ where $f(x) = 2x^4$ for $-\pi \leq x < 0$ and $f(x) = 5 \sin(x)$ for $0 \leq x \leq \pi$ .

See Solution

Problem 25623

Calculate the integral $\int_{-1}^{1} \frac{1}{5} e^{\frac{1}{2} x} d x$ .

See Solution

Problem 25624

Calculate the integral: $\int_{1}^{5} \frac{3}{x} d x$

See Solution

Problem 25625

Find the inflection point of the function $g(x) = -4 x^{3} + 5$ .

See Solution

Problem 25626

Find $\int_{4}^{7} f(x) d x$ given $\int_{2}^{7} f(x) d x=12$ and $\int_{2}^{4}(f(x)+5) d x=8$ .

See Solution

Problem 25627

Find the inflection point of $f(x)$ and its inverse $f^{-1}(x)$ where $f(x)=2(x+3)^{3}-1$ .

See Solution

Problem 25628

Bestimmen Sie die Ableitungen von $f(x)=x^{3}$ , $f(x)=4 x^{3}$ , $f(x)=x^{3}+x$ , $f(x)=x^{4}+5$ , $f(x)=2 x^{4}+3 x^{2}$ , $f(x)=3 x^{5}-2 x^{2}+3 x-2$ . Nennen Sie die angewendeten Regeln.

See Solution

Problem 25629

Evaluate the integral $\int \frac{\cos x}{1+\sin ^{2} x} d x$ using $u=\sin x$ . Rewrite and find the result.

See Solution

Problem 25630

Find the inflection points of $g(x) = -4x^3 + 5$ and $g^{-1}(x)$ .

See Solution

Problem 25631

What is the kinetic energy (KE) of a 1 kg rock just before hitting the bottom of a 120 m canyon?

See Solution

Problem 25632

Prove that for the pdf $f(x)=\frac{2 x}{\alpha} e^{-x^{2} / \alpha}$ ( $0<X<\infty$ ), i) $\mu=0.5 \sqrt{\alpha \pi}$ ii) $\sigma^{2}=\alpha(1-\frac{\pi}{4})$ .

See Solution

Problem 25633

Calculate the indefinite integral: $\int \frac{d}{d x}(x^{5}+8 x^{2}+3) d x = \square$

See Solution

Problem 25634

Find the limit: $\lim _{x \rightarrow 0} \frac{\int_{x}^{0} \sqrt{t^{2}+4} d t}{x}$ . What is its value?

See Solution

Problem 25635

Find $g^{\prime}(9)$ for $g(x)=\int_{2}^{\sqrt{x}} f(t) dt$ , given $f(0)=4, f(1)=0, f(3)=6, f(9)=2$ .

See Solution

Problem 25636

Bestimme die Ableitungsfunktionen für: $f_{1}(x)=x^{2}+4$ , $f_{2}(x)=x^{2}+4 x$ , $f_{3}(x)=-3 x^{2}$ , $f_{4}(x)=x^{2}-2 x+1$ , $f_{5}(x)=(x-2)^{2}$ .

See Solution

Problem 25637

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

See Solution

Problem 25638

Gegeben ist $f(x)=\frac{1}{2} x^{3}+2 x^{2}-1$ . Bestimmen Sie das Monotonieverhalten und die Steigung der Tangente in P(1|f(1)).

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Problem 25639

Leiten Sie die Funktion $f$ ab und vereinfachen Sie, wenn möglich. a) $f(x)=x^{6}$ b) $f(x)=\frac{1}{5} x^{5}$ c) $f(x)=7 x$ d) $f(x)=x+1$ e) $f(x)=-\frac{2}{3} x^{4}+x+2$ f) $f(x)=7$ g) $f(x)=-x+2 x^{5}-x^{3}$ h) $f(x)=0,25 x^{4}-x^{3}+x$ i) $f(x)=0$

See Solution

Problem 25640

Compute the integral $\int_{-1}^{\pi} f(x) dx$ where $f(x) = -x^{2}$ for $x \leq 0$ and $f(x) = -\sin x + C$ for $x > 0$ .

See Solution

Problem 25641

Find the average rate of change of $g(x)=-2x^{2}-5x$ from $x=-2$ to $x=3$ . Simplify your answer.

See Solution

Problem 25642

Find $g'(x)$ for $g(x)=\int_{2}^{x^{3}} \sin(t)e^{t} dt$ . Options include $3x^{2}\sin(x^{3})e^{x^{3}}$ and others.

See Solution

Problem 25643

Berechnen Sie die folgenden Integrale: a) $\int_{0}^{2}(x^{3}-4 x) dx$ , b) $\int_{1}^{2}(x^{4}-6 x^{2}+9) dx$ , c) $\int_{-2}^{4} 3 dx$ , d) $\int_{0}^{\pi} \cos (x) dx$ , e) $\int_{0}^{\pi} \sin (x) dx$ , f) $\int_{0}^{\pi}(x+\sin (x)) dx$ , g) $\int_{1}^{9} \frac{1}{\sqrt{x}} dx$ , h) $\int_{1}^{2}(x^{3}-\frac{1}{x^{2}}) dx$ .

See Solution

Problem 25644

Finde a für $g_{a}(2)=6$ , a für $g'_{a}(-2)=0$ und erkläre, wie man Extrempunkte von $g_{a}$ findet.

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Problem 25645

Identify the true identity for $g(x)=\int f(x) d x$ among the following options.

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Problem 25646

Find the steepest slope of $f(x)=\sqrt{x+2}$ on $[0,1]$ , the secant line equation, and compare slopes.

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Problem 25647

Find the surface area of the curve $x=y^{2/3}$ from $(0,0)$ to $(1,1)$ when revolved around the $y$ -axis.

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Problem 25648

Jenna shoots an arrow with an initial velocity of $11 \, \mathrm{m/s}$ from a $2 \, \mathrm{m}$ high platform. Find its velocity and acceleration after $3 \, s$ . $V(t)=h^{\prime \prime}(t)=$

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Problem 25649

Let $f$ be continuous on $[1, \infty)$ with $F(1)=-4$ , $F(0)=1$ , and $\lim_{x \to \infty} F(x)=\infty$ . If $m<0$ , find the value of $\int_{1}^{\infty} f(x^{m}) x^{m-1} dx$ . Choices: $\frac{5}{m}, \frac{4}{m}, 0, \text{diverges}, 5$ .

See Solution

Problem 25650

Berechne die Integrale: b) $\int_{2}^{3}\left(1+\frac{1}{x^{2}}\right) d x$ , c) $\int_{0}^{2} \frac{1}{(x+1)^{2}} d x$ , f) $\int_{-1}^{0} e^{-x} d x$ , g) $\int_{-1}^{1} \frac{1}{5} e^{\frac{1}{2} x} d x$ .

See Solution

Problem 25651

Approximate $\int_{2}^{6} f(x) dx$ using a Riemann sum with 2 rectangles and left endpoints.

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Problem 25652

Evaluate the integral $\int_{0}^{8} \frac{1}{2\left(x^{1 / 3}\right)} d x$ . Does it converge or diverge?

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Problem 25653

For the integral $\int_{14}^{\infty} \frac{\sqrt{x}}{x^{p}} d x$ to converge, which condition on $p$ is TRUE?

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Problem 25654

e) Evaluate $\int_{-0.5}^{0} e^{2x+1} \, dx$ f) Evaluate $\int_{-1}^{0} e^{-x} \, dx$ 4 a) Evaluate $\int_{1}^{5} \frac{3}{x} \, dx$ b) Evaluate $\int_{1}^{2}\left(1+\frac{1}{x}\right) \, dx$

See Solution

Problem 25655

Find critical points and extrema for: A. $f(x)=x^{3}-12 x$ on $[-1,4]$ , B. $f(x)=x^{3}-6 x^{2}+6 x-4$ on $[-1,4]$ , C. $g(x)=\frac{x}{x^{2}+1}$ on $[1,5]$ .

See Solution

Problem 25656

Find the function $f(x)$ given that $f^{\prime}(x)=x^{2}$ and $f(1)=2$ .

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Problem 25657

Estimate the integral $\frac{2 \pi}{3} \int_{0}^{1} y^{2/3} \sqrt{9y^{2/3}+4} \, dy$ .

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Problem 25658

Let $f$ be continuous with $\lim_{x \to 0^+} f(x) = \infty$ and $\lim_{x \to \infty} f(x) = 0$ . Given $g(x) \geq f(x)$ , which statement is always TRUE?

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Problem 25659

Find the linear approximation of $f(x)=\ln x$ at $x=1$ to estimate $\ln (1.25)$ . $L(x)=\square$ , $\ln 1.25 \approx$

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Problem 25660

Maurice's roast cools from $165^{\circ} \mathrm{F}$ to $145^{\circ} \mathrm{F}$ in 10 minutes. Find time to reach $120^{\circ} \mathrm{F}$ using $T(t)=I_{A}+(I_{0}-I_{A}) e^{-k t}$ . Round to the nearest minute.

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Problem 25661

Find $f^{\prime}(x), f^{\prime \prime}(x), f^{\prime \prime \prime}(x)$ for $f(z)=-\frac{3}{z}$ and a formula for $f^{(n)}(x)$ .

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Problem 25662

Maurice's roast cools from $165^{\circ} \mathrm{F}$ to $120^{\circ} \mathrm{F}$ . Use Newton's law: $T(t)=T_{A}+(T_{0}-T_{A}) e^{-k t}$ . How long?

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Problem 25663

Find $f^{\prime}(x)$ , $f^{\prime \prime}(x)$ , $f^{\prime \prime \prime}(x)$ , and $f^{(n)}(x)$ for $f(x)=-\frac{3}{x}$ .

See Solution

Problem 25664

For which value(s) of $c$ is the integral $\int_{c}^{0} \frac{3}{x+2} dx$ improper?

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Problem 25665

Find the derivative of $y=\ln (x+\tan (x))$ . What is $y^{\prime}$ ?

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Problem 25666

Determine where the function is decreasing by finding where its derivative is negative. Provide the function's formula.

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Problem 25667

Find the integral: $\int \frac{x+14}{x^{2}-4}+\frac{3}{x+2} \, dx$

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Problem 25668

Evaluate the following integrals using substitution:
1. $\int x \sqrt{x^{2}+1} \, dx$
2. $\int \frac{6x+3}{x^{2}+x+1} \, dx$
3. $\int \cos^{3} x \sin x \, dx$
4. $\int \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx$

See Solution

Problem 25669

How long for an investment of \ $20,000 to reach \$70,000 at 15% interest compounded continuously? About$ \square$ years.

See Solution

Problem 25670

Find the limit: $\lim _{x \rightarrow 1} \frac{\sqrt{5 x-1}-2 x}{x-1}$ .

See Solution

Problem 25671

Estimate $\cos(0.49\pi)$ using linear approximation.
1. Identify the function $y=\square$ .
2. Find $a=$ (near $0.49\pi$ ).
3. Determine $L(x)=m x+b$ .
4. Approximate $\cos(0.49\pi) \approx$ .

See Solution

Problem 25672

Find the derivative $G'(x)$ of $G(x)=\int_{x}^{e^{x}}\left(\frac{\cos t}{t+1}+C\right) dt$ .

See Solution

Problem 25673

Differentiate the function $V(t)=t^{-3/5}+t^{9}$ . Find $V^{\prime}(t)=$ .

See Solution

Problem 25674

Find the absolute extrema of $f(x)=3 x^{2/3}-2 x$ on $[-1,1]$ . Where are the max and min values?

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Problem 25675

Differentiate the function F(t) = (6t - 5)². Find F'(t) = ?

See Solution

Problem 25676

Calculate the following definite integrals:
1. $\int_{0}^{2}(8-x^{2}+4 x^{3}+e^{-x}) dx$
2. $\int_{1}^{5}(1/x) dx$
3. $\int_{0}^{\pi/2}(\sin x+1) dx$
4. $\int_{0}^{\pi/4}(\sec^{2} x) dx$

See Solution

Problem 25677

Which statement is TRUE by the Comparison Test? Consider the integrals involving $\frac{2^{x}}{x}$ and $\frac{\cos x}{x}$ .

See Solution

Problem 25678

Find the indefinite integral: $\int \frac{d}{d x}(x^{7}+9 x^{3}+1) d x = \square$

See Solution

Problem 25679

Find the derivative of the function $f(x)=e^{-x^{4}}$ . What is $f^{\prime}(x)=\square$ ?

See Solution

Problem 25680

Find the derivative of the function $f(x)=\tan^{-1}(\cos(5x))$ , denoted as $f^{\prime}(x)=?$ .

See Solution

Problem 25681

Find the total oil leaked in the first 3 minutes if $r(t)=500-4t$ liters/minute. Provide the answer as a decimal.

See Solution

Problem 25682

Evaluate the Riemann sum of $f(x)$ on $[-2,4]$ using 2 equal-width rectangles and right endpoints. Choices: 12, 14, 16, 18, 24.

See Solution

Problem 25683

Calculate the integral $\int_{2}^{5} f^{\prime}(x) \, dx$ using the given function values for $f$ and $f^{\prime}$ .

See Solution

Problem 25684

A scientist finds a bacteria strain that triples every 6 days. What is its growth rate?

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Problem 25685

Calculate the limit: $\lim _{x \rightarrow 0^{+}}\left(e^{\left(x \ln (x)-\frac{x^{2}}{4}+\ln (4)+\frac{1}{4}\right)}\right)$ .

See Solution

Problem 25686

Find the indefinite integral $\int \frac{\ln (x)}{3 x} d x$ using the substitution $u=\ln (x)$ . Which transformed integral is correct?

See Solution

Problem 25687

Explain why Rolle's Theorem and the Mean Value Theorem apply to any polynomial on any interval $(a, b)$ .

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Problem 25688

Find the function $S(t)$ for monthly sales declining at $S^{\prime}(t)=-25 t^{\frac{2}{3}}-60$ , starting from 2,350.

See Solution

Problem 25689

Find the function $S(t)$ for monthly sales declining at $S^{\prime}(t)=-25 t^{\frac{2}{3}}-60$ from 2,350 to 1,000. Approximate $t$ when $S(t)=1,000$ .

See Solution

Problem 25690

Find the area of each small square, approximate the orange area using rectangles, and compute the definite integral of $g(x)=\sqrt[3]{x}$ .

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Problem 25691

Evaluate the following limits and function value for $g(x)$ defined piecewise:
(i) $\lim _{x \rightarrow 1^{-}} g(x)$ , (ii) $\lim _{x \rightarrow 1} g(x)$ , (iii) $g(1)$ , (iv) $\lim _{x \rightarrow 2^{-}} g(x)$ , (v) $\lim _{x \rightarrow 2^{+}} g(x)$ , (vi) $\lim _{x \rightarrow 2} g(x)$ .

See Solution

Problem 25692

Find $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow -\infty} f(x)$ for $f(x)=\left(\frac{x^{2}}{x+1}-\frac{x^{2}}{x-1}\right)$ . Also, sketch a graph with limits: $\lim _{t \rightarrow \infty} f(t)=2$ , $\lim _{t \rightarrow 0^{-}} f(t)=-c$ , $\lim _{t \rightarrow -\infty} f(t)=0$ , and $\lim _{t \rightarrow 4} f(t)=3$ .

See Solution

Problem 25693

Find these limits: (a) $\lim _{x \rightarrow \infty} \frac{3 x+2 \sqrt{x}}{1-x}$ , (b) $\lim _{x \rightarrow-\infty} \frac{2 x-5}{|3 x+2|}$ , (c) $\lim _{x \rightarrow \infty} \frac{5 x^{2}+\sin x}{3 x^{2}+\cos x}$ .

See Solution

Problem 25694

Find the differentials $d y$ for $y=\sqrt[3]{x^{2}+2}$ and $y=e^{\sqrt{x}}$ .

See Solution

Problem 25695

Use the function $f(x)=x^{3}-9 x^{2}+24 x$ on $[1,6]$ to find extrema, inflection points, and intervals of increase/decrease and concavity.

See Solution

Problem 25696

Evaluate the integral $\int \frac{-\sin (x)}{1+\cos ^{2}(x)} d x$ using substitution $v=\cos x$ . Rewrite it in another form.

See Solution

Problem 25697

Find the second derivative of the function $f(x)=\ln(e^{x})$ . What is $f''(x)$ ?

See Solution

Problem 25698

Use Rolle's Theorem on $f(x)=4x-x^2$ in $[0,4]$ . Find $c$ where $f'(c)=0$ and state your conclusions.

See Solution

Problem 25699

Differentiate the function $V(t)=t^{-3/5}+t^{9}$ to find $V'(t)=\square$ .

See Solution

Problem 25700

Differentiate the function $F(t)=(6t-5)^{2}$ . Find $F^{\prime}(t)=\square$ .

See Solution

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Calculus

Problem 25601

Evaluate the series: S=∑n=1∞(−1)n1n+4S = \sum_{n=1}^{\infty}(-1)^{n} \frac{1}{n+4}S=∑n=1∞​(−1)nn+41​.

Problem 25602

Define α\alphaα, f(x)f(x)f(x), and LLL for the limit: lim⁡x→4(2x2−7)=25\lim_{x \rightarrow 4}(2x^{2}-7)=25limx→4​(2x2−7)=25.

Problem 25603

Problem 25604

Bestimmen Sie die ersten Ableitungen der Funktionen: a) f(x)=x10+10xf(x)=\frac{x}{10}+\frac{10}{x}f(x)=10x​+x10​ und skizzieren Sie die Graphen. b) f(x)=x−2(2sin⁡(x)+cos⁡(x))f(x)=x^{-2}(2 \sin(x)+\cos(x))f(x)=x−2(2sin(x)+cos(x)).

Problem 25605

Problem 25606

Find the number of iPhones, xxx, to minimize average cost given C(x)=200000+180x+0.004x2C(x)=200000+180x+0.004x^{2}C(x)=200000+180x+0.004x2. What is the average cost?

Problem 25607

Problem 25608

Find the derivative dydx\frac{d y}{d x}dxdy​ using implicit differentiation for the equation x2y−y2=36x^{2} y - y^{2} = 36x2y−y2=36.

Problem 25609

Find the derivative dpdq\frac{d p}{d q}dqdp​ using implicit differentiation for the equation: p2−pq=9p2q2p^{2}-p q=9 p^{2} q^{2}p2−pq=9p2q2.

Problem 25610

Bestimmen Sie die Ableitungsfunktion f′(x)f'(x)f′(x) für f(x)=3(x−2)2+xf(x)=3(x-2)^{2}+xf(x)=3(x−2)2+x mit Ableitungsregeln.

Problem 25611

Find the slope and equation of the tangent line to the curve 6x2−y2=xy−66 x^{2}-y^{2}=x y-66x2−y2=xy−6 at the point (−1,4)(-1,4)(−1,4).

Problem 25612

Find the differential of the functions: 11. y=e5xy=e^{5x}y=e5x, 12. y=[incomplete function]y=\text{[incomplete function]}y=[incomplete function].

Problem 25613

Evaluate the integral: ∫6x4+x2dx\int \frac{6 x}{\sqrt{4+x^{2}}} d x∫4+x2​6x​dx

Problem 25614

Hanson's parents deposited \$6,966 at 14% continuous interest. He withdrew \$7,760 for tuition. How long was it in the account?

Problem 25615

Find the tangent line to y=exy=e^{x}y=ex parallel to 2x−y=52x-y=52x−y=5.

Problem 25616

Evaluate the series 30∑k=1∞(−i)k3k+130 \sum_{k=1}^{\infty} \frac{(-i)^{k}}{3^{k+1}}30∑k=1∞​3k+1(−i)k​.

Problem 25617

Problem 25618

Calculate the sum: 10∑k=0∞(−i)k3k+110 \sum_{k=0}^{\infty} \frac{(-i)^{k}}{3^{k+1}}10∑k=0∞​3k+1(−i)k​.

Problem 25619

Find f(π)f(\pi)f(π) if f(0)=4f(0)=4f(0)=4 and f′(x)=−sin⁡(x)−e−x+3x2f^{\prime}(x)=-\sin (x)-e^{-x}+3 x^{2}f′(x)=−sin(x)−e−x+3x2. Choose from the options.

Problem 25620

Calculate the integral: ∫x2(x+3) dx\int x^{2}(x+3) \, dx∫x2(x+3)dx

Problem 25621

Calculate the integral ∫−10e−x dx\int_{-1}^{0} e^{-x} \, dx∫−10​e−xdx.

Problem 25622

Evaluate the integral ∫−ππf(x)dx\int_{-\pi}^{\pi} f(x) dx∫−ππ​f(x)dx where f(x)=2x4f(x) = 2x^4f(x)=2x4 for −π≤x<0-\pi \leq x < 0−π≤x<0 and f(x)=5sin⁡(x)f(x) = 5 \sin(x)f(x)=5sin(x) for 0≤x≤π0 \leq x \leq \pi0≤x≤π.

Problem 25623

Calculate the integral ∫−1115e12xdx\int_{-1}^{1} \frac{1}{5} e^{\frac{1}{2} x} d x∫−11​51​e21​xdx.

Problem 25624

Calculate the integral: ∫153xdx\int_{1}^{5} \frac{3}{x} d x∫15​x3​dx

Problem 25625

Find the inflection point of the function g(x)=−4x3+5g(x) = -4 x^{3} + 5g(x)=−4x3+5.

Problem 25626

Find ∫47f(x)dx\int_{4}^{7} f(x) d x∫47​f(x)dx given ∫27f(x)dx=12\int_{2}^{7} f(x) d x=12∫27​f(x)dx=12 and ∫24(f(x)+5)dx=8\int_{2}^{4}(f(x)+5) d x=8∫24​(f(x)+5)dx=8.

Problem 25627

Find the inflection point of f(x)f(x)f(x) and its inverse f−1(x)f^{-1}(x)f−1(x) where f(x)=2(x+3)3−1f(x)=2(x+3)^{3}-1f(x)=2(x+3)3−1.

Problem 25628

Problem 25629

Evaluate the integral ∫cos⁡x1+sin⁡2xdx\int \frac{\cos x}{1+\sin ^{2} x} d x∫1+sin2xcosx​dx using u=sin⁡xu=\sin xu=sinx. Rewrite and find the result.

Problem 25630

Find the inflection points of g(x)=−4x3+5g(x) = -4x^3 + 5g(x)=−4x3+5 and g−1(x)g^{-1}(x)g−1(x).

Problem 25631

What is the kinetic energy (KE) of a 1 kg rock just before hitting the bottom of a 120 m canyon?

Problem 25632

Prove that for the pdf f(x)=2xαe−x2/αf(x)=\frac{2 x}{\alpha} e^{-x^{2} / \alpha}f(x)=α2x​e−x2/α (0<X<∞0<X<\infty0<X<∞), i) μ=0.5απ\mu=0.5 \sqrt{\alpha \pi}μ=0.5απ​ ii) σ2=α(1−π4)\sigma^{2}=\alpha(1-\frac{\pi}{4})σ2=α(1−4π​).

Problem 25633

Calculate the indefinite integral: ∫ddx(x5+8x2+3)dx=□\int \frac{d}{d x}(x^{5}+8 x^{2}+3) d x = \square∫dxd​(x5+8x2+3)dx=□

Problem 25634

Find the limit: lim⁡x→0∫x0t2+4dtx\lim _{x \rightarrow 0} \frac{\int_{x}^{0} \sqrt{t^{2}+4} d t}{x}x→0lim​x∫x0​t2+4​dt​. What is its value?

Problem 25635

Find g′(9)g^{\prime}(9)g′(9) for g(x)=∫2xf(t)dtg(x)=\int_{2}^{\sqrt{x}} f(t) dtg(x)=∫2x​​f(t)dt, given f(0)=4,f(1)=0,f(3)=6,f(9)=2f(0)=4, f(1)=0, f(3)=6, f(9)=2f(0)=4,f(1)=0,f(3)=6,f(9)=2.

Problem 25636

Bestimme die Ableitungsfunktionen für: f1(x)=x2+4f_{1}(x)=x^{2}+4f1​(x)=x2+4, f2(x)=x2+4xf_{2}(x)=x^{2}+4 xf2​(x)=x2+4x, f3(x)=−3x2f_{3}(x)=-3 x^{2}f3​(x)=−3x2, f4(x)=x2−2x+1f_{4}(x)=x^{2}-2 x+1f4​(x)=x2−2x+1, f5(x)=(x−2)2f_{5}(x)=(x-2)^{2}f5​(x)=(x−2)2.

Problem 25637

Evaluate the integral: ∫12(1+1x)dx\int_{1}^{2}\left(1+\frac{1}{x}\right) d x∫12​(1+x1​)dx

Problem 25638

Gegeben ist f(x)=12x3+2x2−1f(x)=\frac{1}{2} x^{3}+2 x^{2}-1f(x)=21​x3+2x2−1. Bestimmen Sie das Monotonieverhalten und die Steigung der Tangente in P(1|f(1)).

Problem 25639

Problem 25640

Compute the integral ∫−1πf(x)dx\int_{-1}^{\pi} f(x) dx∫−1π​f(x)dx where f(x)=−x2f(x) = -x^{2}f(x)=−x2 for x≤0x \leq 0x≤0 and f(x)=−sin⁡x+Cf(x) = -\sin x + Cf(x)=−sinx+C for x>0x > 0x>0.

Problem 25641

Find the average rate of change of g(x)=−2x2−5xg(x)=-2x^{2}-5xg(x)=−2x2−5x from x=−2x=-2x=−2 to x=3x=3x=3. Simplify your answer.

Problem 25642

Find g′(x)g'(x)g′(x) for g(x)=∫2x3sin⁡(t)etdtg(x)=\int_{2}^{x^{3}} \sin(t)e^{t} dtg(x)=∫2x3​sin(t)etdt. Options include 3x2sin⁡(x3)ex33x^{2}\sin(x^{3})e^{x^{3}}3x2sin(x3)ex3 and others.

Problem 25643

Evaluate the series: $S = \sum_{n=1}^{\infty}(-1)^{n} \frac{1}{n+4}$ .

Define $\alpha$ , $f(x)$ , and $L$ for the limit: $\lim_{x \rightarrow 4}(2x^{2}-7)=25$ .

Bestimmen Sie die ersten Ableitungen der Funktionen: a) $f(x)=\frac{x}{10}+\frac{10}{x}$ und skizzieren Sie die Graphen. b) $f(x)=x^{-2}(2 \sin(x)+\cos(x))$ .

Find the number of iPhones, $x$ , to minimize average cost given $C(x)=200000+180x+0.004x^{2}$ . What is the average cost?

Find the derivative $\frac{d y}{d x}$ using implicit differentiation for the equation $x^{2} y - y^{2} = 36$ .

Find the derivative $\frac{d p}{d q}$ using implicit differentiation for the equation: $p^{2}-p q=9 p^{2} q^{2}$ .

Bestimmen Sie die Ableitungsfunktion $f'(x)$ für $f(x)=3(x-2)^{2}+x$ mit Ableitungsregeln.

Find the slope and equation of the tangent line to the curve $6 x^{2}-y^{2}=x y-6$ at the point $(-1,4)$ .

Find the differential of the functions: 11. $y=e^{5x}$ , 12. $y=\text{[incomplete function]}$ .

Evaluate the integral: $\int \frac{6 x}{\sqrt{4+x^{2}}} d x$

Find the tangent line to $y=e^{x}$ parallel to $2x-y=5$ .

Evaluate the series $30 \sum_{k=1}^{\infty} \frac{(-i)^{k}}{3^{k+1}}$ .

Calculate the sum: $10 \sum_{k=0}^{\infty} \frac{(-i)^{k}}{3^{k+1}}$ .

Find $f(\pi)$ if $f(0)=4$ and $f^{\prime}(x)=-\sin (x)-e^{-x}+3 x^{2}$ . Choose from the options.

Calculate the integral: $\int x^{2}(x+3) \, dx$

Calculate the integral $\int_{-1}^{0} e^{-x} \, dx$ .

Evaluate the integral $\int_{-\pi}^{\pi} f(x) dx$ where $f(x) = 2x^4$ for $-\pi \leq x < 0$ and $f(x) = 5 \sin(x)$ for $0 \leq x \leq \pi$ .

Calculate the integral $\int_{-1}^{1} \frac{1}{5} e^{\frac{1}{2} x} d x$ .

Calculate the integral: $\int_{1}^{5} \frac{3}{x} d x$

Find the inflection point of the function $g(x) = -4 x^{3} + 5$ .

Find $\int_{4}^{7} f(x) d x$ given $\int_{2}^{7} f(x) d x=12$ and $\int_{2}^{4}(f(x)+5) d x=8$ .

Find the inflection point of $f(x)$ and its inverse $f^{-1}(x)$ where $f(x)=2(x+3)^{3}-1$ .

Evaluate the integral $\int \frac{\cos x}{1+\sin ^{2} x} d x$ using $u=\sin x$ . Rewrite and find the result.

Find the inflection points of $g(x) = -4x^3 + 5$ and $g^{-1}(x)$ .

Prove that for the pdf $f(x)=\frac{2 x}{\alpha} e^{-x^{2} / \alpha}$ ( $0<X<\infty$ ), i) $\mu=0.5 \sqrt{\alpha \pi}$ ii) $\sigma^{2}=\alpha(1-\frac{\pi}{4})$ .

Calculate the indefinite integral: $\int \frac{d}{d x}(x^{5}+8 x^{2}+3) d x = \square$

Find the limit: $\lim _{x \rightarrow 0} \frac{\int_{x}^{0} \sqrt{t^{2}+4} d t}{x}$ . What is its value?

Find $g^{\prime}(9)$ for $g(x)=\int_{2}^{\sqrt{x}} f(t) dt$ , given $f(0)=4, f(1)=0, f(3)=6, f(9)=2$ .

Bestimme die Ableitungsfunktionen für: $f_{1}(x)=x^{2}+4$ , $f_{2}(x)=x^{2}+4 x$ , $f_{3}(x)=-3 x^{2}$ , $f_{4}(x)=x^{2}-2 x+1$ , $f_{5}(x)=(x-2)^{2}$ .

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

Gegeben ist $f(x)=\frac{1}{2} x^{3}+2 x^{2}-1$ . Bestimmen Sie das Monotonieverhalten und die Steigung der Tangente in P(1|f(1)).

Compute the integral $\int_{-1}^{\pi} f(x) dx$ where $f(x) = -x^{2}$ for $x \leq 0$ and $f(x) = -\sin x + C$ for $x > 0$ .

Find the average rate of change of $g(x)=-2x^{2}-5x$ from $x=-2$ to $x=3$ . Simplify your answer.

Find $g'(x)$ for $g(x)=\int_{2}^{x^{3}} \sin(t)e^{t} dt$ . Options include $3x^{2}\sin(x^{3})e^{x^{3}}$ and others.

Finde a für $g_{a}(2)=6$ , a für $g'_{a}(-2)=0$ und erkläre, wie man Extrempunkte von $g_{a}$ findet.

Identify the true identity for $g(x)=\int f(x) d x$ among the following options.

Find the steepest slope of $f(x)=\sqrt{x+2}$ on $[0,1]$ , the secant line equation, and compare slopes.

Find the surface area of the curve $x=y^{2/3}$ from $(0,0)$ to $(1,1)$ when revolved around the $y$ -axis.

Jenna shoots an arrow with an initial velocity of $11 \, \mathrm{m/s}$ from a $2 \, \mathrm{m}$ high platform. Find its velocity and acceleration after $3 \, s$ . $V(t)=h^{\prime \prime}(t)=$

Approximate $\int_{2}^{6} f(x) dx$ using a Riemann sum with 2 rectangles and left endpoints.

Evaluate the integral $\int_{0}^{8} \frac{1}{2\left(x^{1 / 3}\right)} d x$ . Does it converge or diverge?

For the integral $\int_{14}^{\infty} \frac{\sqrt{x}}{x^{p}} d x$ to converge, which condition on $p$ is TRUE?

Find critical points and extrema for: A. $f(x)=x^{3}-12 x$ on $[-1,4]$ , B. $f(x)=x^{3}-6 x^{2}+6 x-4$ on $[-1,4]$ , C. $g(x)=\frac{x}{x^{2}+1}$ on $[1,5]$ .

Find the function $f(x)$ given that $f^{\prime}(x)=x^{2}$ and $f(1)=2$ .

Estimate the integral $\frac{2 \pi}{3} \int_{0}^{1} y^{2/3} \sqrt{9y^{2/3}+4} \, dy$ .

Let $f$ be continuous with $\lim_{x \to 0^+} f(x) = \infty$ and $\lim_{x \to \infty} f(x) = 0$ . Given $g(x) \geq f(x)$ , which statement is always TRUE?

Find the linear approximation of $f(x)=\ln x$ at $x=1$ to estimate $\ln (1.25)$ . $L(x)=\square$ , $\ln 1.25 \approx$

Maurice's roast cools from $165^{\circ} \mathrm{F}$ to $145^{\circ} \mathrm{F}$ in 10 minutes. Find time to reach $120^{\circ} \mathrm{F}$ using $T(t)=I_{A}+(I_{0}-I_{A}) e^{-k t}$ . Round to the nearest minute.

Find $f^{\prime}(x), f^{\prime \prime}(x), f^{\prime \prime \prime}(x)$ for $f(z)=-\frac{3}{z}$ and a formula for $f^{(n)}(x)$ .

Maurice's roast cools from $165^{\circ} \mathrm{F}$ to $120^{\circ} \mathrm{F}$ . Use Newton's law: $T(t)=T_{A}+(T_{0}-T_{A}) e^{-k t}$ . How long?

Find $f^{\prime}(x)$ , $f^{\prime \prime}(x)$ , $f^{\prime \prime \prime}(x)$ , and $f^{(n)}(x)$ for $f(x)=-\frac{3}{x}$ .

For which value(s) of $c$ is the integral $\int_{c}^{0} \frac{3}{x+2} dx$ improper?

Find the derivative of $y=\ln (x+\tan (x))$ . What is $y^{\prime}$ ?

Find the integral: $\int \frac{x+14}{x^{2}-4}+\frac{3}{x+2} \, dx$

How long for an investment of \ $20,000 to reach \$70,000 at 15% interest compounded continuously? About$ \square$ years.

Find the limit: $\lim _{x \rightarrow 1} \frac{\sqrt{5 x-1}-2 x}{x-1}$ .

Estimate $\cos(0.49\pi)$ using linear approximation.
1. Identify the function $y=\square$ .
2. Find $a=$ (near $0.49\pi$ ).
3. Determine $L(x)=m x+b$ .
4. Approximate $\cos(0.49\pi) \approx$ .

Find the derivative $G'(x)$ of $G(x)=\int_{x}^{e^{x}}\left(\frac{\cos t}{t+1}+C\right) dt$ .

Differentiate the function $V(t)=t^{-3/5}+t^{9}$ . Find $V^{\prime}(t)=$ .

Find the absolute extrema of $f(x)=3 x^{2/3}-2 x$ on $[-1,1]$ . Where are the max and min values?