Calculus

Problem 23701

Evaluate the integral $\int \frac{z^{8}}{z^{9}+2} d z$ and include the constant of integration $C$ .

See Solution

Problem 23702

Berechne den Flächeninhalt zwischen den Graphen $g(x)=x^{2}$ und $h(x)=2x+4$ im Intervall $[1, 4]$ .

See Solution

Problem 23703

Approximate the area under $f(x)=\frac{1}{2} x^{2}$ from 0 to 2 using 10 left rectangles. Then find the exact area using integrals.

See Solution

Problem 23704

Bestimme die allgemeine Stammfunktion von $f(x)=-\frac{9 x^{7}}{x^{15}}$ .

See Solution

Problem 23705

Find the value of the function $f(x)=\frac{7(\sin(x))^2}{(1-(\cos(x))^2) \cdot \sqrt{x}}$ .

See Solution

Problem 23706

Evaluate the integral $\int_{0}^{\sqrt{\pi}} x \cos \left(x^{2}\right) d x$ .

See Solution

Problem 23707

Find the average distance from the parabola $y=10 x(16-x)$ to the $x$ -axis over the interval $[0,16]$ .

See Solution

Problem 23708

Bestimme die allgemeine Stammfunktion von $f(x)=\frac{9(\sin (x))^{3}}{1-(\cos (x))^{2}}$ .

See Solution

Problem 23709

Bestimme die allgemeine Stammfunktion von $f(x)=\frac{x^{2}+18 x+81}{x+9}$ .

See Solution

Problem 23710

Bestimme die allgemeine Stammfunktion von $f(x)=\frac{7\left(x^{2}-1\right)}{x-1}$ .

See Solution

Problem 23711

Find the speed $v$ that maximizes fuel efficiency $E(v)=\frac{1600 v}{v^{2}+6400}$ for limits 100 km/h and 50 km/h. Identify intervals for increasing and decreasing efficiency from 0 to 100 km/h.

See Solution

Problem 23712

Solve the initial value problem $\frac{d y}{d x}=5 x-2$ , $y=-1$ at $x=-2$ . Find $y=f(x)$ .

See Solution

Problem 23713

Approximate the area under $f(x)=\frac{1}{2} x^{2}$ from 0 to 2 using 10 left rectangles. Then find the exact area.

See Solution

Problem 23714

Solve the initial value problem $\frac{dy}{dx}=\frac{4}{x}-\frac{1}{x^{2}}$ , $y=-1$ when $x=1$ .

See Solution

Problem 23715

Find the integral and verify by differentiation: $\int e^{-0.01 t}\left(e^{-0.14 t}+7\right) d t$ .

See Solution

Problem 23716

Find the definite integral equal to $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{10 k}{n}\left(\sqrt{1+\frac{5 k}{n}}\right)\left(\frac{5}{n}\right)$ .

See Solution

Problem 23717

Find the definite integral equivalent to $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{10 k}{n}\left(\sqrt{1+\frac{5 k}{n}}\right)$ .

See Solution

Problem 23718

Find the radius of convergence, $R$ , and interval, $I$ , for the series $\sum_{n=1}^{\infty} n !(8 x-1)^{n}$ .

See Solution

Problem 23719

Find the midpoint Riemann sum for $\int_{4}^{6} \sqrt{x^{3}+1} \, dx$ using 4 equal subintervals.

See Solution

Problem 23720

Find the left Riemann sum for $\int_{2}^{8} \cos \left(x^{2}\right) d x$ using $n$ equal subintervals.

See Solution

Problem 23721

Calculate the integral: $\int\left(x^{2}+1\right)^{5} x d x$ using $u=x^{2}+1$ and $\int u^{n} du=\frac{1}{n+1} u^{n+1}+C$ .

See Solution

Problem 23722

Find the function $f(x)$ given its slope $f^{\prime}(x)=-9 x^{2}-4 x+5$ and the point $(0,1)$ .

See Solution

Problem 23723

Find the function $f(x)$ given its slope $f^{\prime}(x)=e^{-x}+x^{7}$ and the point $(0,-9)$ .

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Problem 23724

Given values of $R(t)$ and that $R$ is increasing and concave down, which statement about the trapezoidal sum is true?

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Problem 23725

Which definite integral matches $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{10 k}{n}\left(\sqrt{1+\frac{5 k}{n}}\right)\left(\frac{5}{n}\right)$ ? (A) $\int_{1}^{6} 10 \sqrt{x} d x$ (B) $\int_{1}^{6} 2 x \sqrt{x} d x$ (C) $\int_{0}^{5} 10 \sqrt{1+x} d x$ (D) $\int_{0}^{5} 2 x \sqrt{1-x} d x$

See Solution

Problem 23726

Find the arc length of the curve $y=31-10\left(e^{x / 20}+e^{-x / 20}\right)$ from $x=-20$ to $x=20$ .

See Solution

Problem 23727

Which option is a left Riemann sum for $\int_{2}^{8} \cos(x^2) \, dx$ with $n$ equal subintervals? (A) $\sum_{k=1}^{n}\left(\cos\left(2+\frac{k-1}{n}\right)^{2}\right)\frac{1}{n}$ (B) $\sum_{k=1}^{n}\left(\cos\left(\frac{6k}{n}\right)^{2}\right)\frac{6}{n}$ (C) $\sum_{k=1}^{n}\left(\cos\left(2+\frac{6(k-1)}{n}\right)^{2}\right)\frac{6}{n}$ (D) $\sum_{k=1}^{n}\left(\cos\left(2+\frac{6k}{n}\right)^{2}\right)\frac{6}{n}$

See Solution

Problem 23728

Find the arc length of the curve $y = 31 - 10\left(\sqrt{x}/20 + e^{-x/20}\right)$ from $x = -20$ to $x = 20$ .

See Solution

Problem 23729

Determine if the max/min algorithm can find extrema for these functions: a. $y=x^{3}-5 x^{2}+10, -5 \leq x \leq 5$ b. $y=\frac{3 x}{x-2}, -1 \leq x \leq 3$ c. $y=\frac{x}{x^{2}-4}, x \in[0,5]$

See Solution

Problem 23730

Find the midpoint Riemann sum for $\int_{4}^{6} \sqrt{x^{3}+1} \, dx$ using 4 equal subintervals.

See Solution

Problem 23731

Find the limit: $\lim _{x \rightarrow 10} \frac{x^{2}-99}{x-10}$ .

See Solution

Problem 23732

Find the limit: $\lim _{x \rightarrow 16} \frac{x^{2}-99}{x-10}$ .

See Solution

Problem 23733

Find the limit using l'Hospital's Rule: $\lim _{x \rightarrow 1} \frac{3 \ln x}{8 \sin \pi x}$

See Solution

Problem 23734

Find the integral using substitution or state "IMPOSSIBLE": $\int\left(x^{8}-4\right)^{9} x^{7} d x$ . Use $C$ for the constant.

See Solution

Problem 23735

Find the approximate half-life of a drug that decreases from $3,600 \mathrm{mg} / \mathrm{L}$ to $1,160 \mathrm{mg} / \mathrm{L}$ in one day.

See Solution

Problem 23736

Find $H^{\prime}(x)$ where $H(x)=\int_{1}^{\sin (x)} f(t) dt$ . Options: (a) $-f(\sin (x))$ , (b) $f(x)$ , (c) $f(\sin (x)) \cdot \cos (x)$ , (d) $f(\sin (x))-f(1)$ , (e) $f^{\prime}(\sin (x)) \cdot \cos ^{2}(x)-f(\sin (x)) \cdot \sin (x)$ .

See Solution

Problem 23737

Pipeline: $f$ gibt die Ölmenge seit 6 Uhr an. Erklären Sie $f(7)=60$ , $\frac{f(7)-f(3)}{4}=11$ , $f^{\prime}(5)=12$ . Berechnen Sie die Ölmenge zwischen 11 und 11:15 Uhr mit $f^{\prime}(5)=12$ .

See Solution

Problem 23738

Calculate the power wasted by a waterfall flowing at $1.3 \times 10^{6} \mathrm{~kg/s}$ and falling $55.1 \mathrm{~m}$ . Use $g = 9.81 \mathrm{~m/s^{2}}$ . Answer in W.

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Problem 23739

Bestimmen Sie $x$ , so dass die Ableitung $f'(x) = -\frac{1}{x^2}$ gleich $m = -\frac{1}{9}$ ist.

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Problem 23740

Find the derivative of $h(t)=\ln (\pi) \cdot \sin (t)$ . What is $h^{\prime}(t)$ ? Options: A, B, C, D.

See Solution

Problem 23741

Given $s(t)=5 t^{2}+20 t$ , find $v(t)$ , $a(t)$ , and their values at $t=2$ sec. Show your work.

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Problem 23742

Find $\lim f(x)$ as $x \rightarrow 10$ for $f(x)=x^{2}$ (if $x \neq 10$ , $f(10)=9$ ).

See Solution

Problem 23743

Find $\lim _{x \rightarrow 10} f(x)$ for $f(x)=x^{2}$ when $x \neq 10$ and $f(10)=99$ .

See Solution

Problem 23744

Find the limit: $\lim _{x \rightarrow 10} \sqrt{-x^{2}+20 x-100}$ .

See Solution

Problem 23745

Find the limit: $\lim _{x \rightarrow \infty} \frac{x^{2}}{e^{4 x}-1-4 x}$ .

See Solution

Problem 23746

Find the limit: $\lim _{x \rightarrow-4} \frac{x^{2}-16}{x+4} \ln |x|$

See Solution

Problem 23747

Find the indefinite integral $\int A \cot (x) d x$ . Which option is correct? (Hint: Use $\cot (x) = \frac{\cos(x)}{\sin(x)}$ )

See Solution

Problem 23748

Find the limit: $\lim _{x \rightarrow-\infty} \frac{3 x^{6}-7 x^{5}+x}{5 x^{6}+4 x^{5}-3}$ .

See Solution

Problem 23749

Fetal head circumference $H$ depends on age $t$ weeks: $H=-30.04+1.802 t^{2}-0.9032 t^{2} \log t$ .
(a) Find $\frac{\mathrm{dH}}{\mathrm{dt}}$ . (b) Is it larger at $t=8$ weeks or $t=36$ weeks? (c) Compare $\frac{1}{H} \frac{\mathrm{dH}}{\mathrm{dt}}$ at both times.

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Problem 23750

In the lagrangian $L=U(x, y)+\lambda\left(I-P_{x} x-P_{y} y\right)$ , which statement about utility increase is correct?

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Problem 23751

Bestimmen Sie den Differenzenquotienten der Funktion $f(x)=-2x^2+4$ für die Intervalle I: a) I=[-1 ; 3], b) I=[-2 ; 2].

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Problem 23752

How long to double \ $3250 at a continuous compound rate of$ 6.5\%$ to reach \$ 6500?

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Problem 23753

If $m$ decreases by 1 and $\lambda=-5$ , how does $L$ change? Options: Rise by 5, No change, Fall by 5, None.

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Problem 23754

Differentiate $y=(5 x)^{\ln 5 x}$ using logarithmic differentiation to find $y^{\prime}=\square$ .

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Problem 23755

Find the limit $\lim _{x \rightarrow 0^{+}} \frac{F(x)}{x^{2}}$ where $F(x)=\int_{0}^{x} e^{t}-1 dt$ . Use L'Hôpital's Rule.

See Solution

Problem 23756

Express the limit of the left-endpoint Riemann sum $\mathrm{L}_{\mathrm{n}} = \frac{1}{n} \sum_{i=1}^{n}\left(3+1 \frac{i-1}{n}\right)$ as a definite integral.

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Problem 23757

Maximize $z=f(x, y)$ with constraints $y-g(x)=0$ and $y-h(x)=0$ . What is the Lagrangian function?

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Problem 23758

Two paint buckets (12 kg at 4 m, 4 kg at 2 m) are connected. Find speed of the 12 kg bucket when it hits the ground using energy conservation. Also, analyze forces, draw diagrams, find acceleration, and calculate its final velocity.

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Problem 23759

Find where $f(x)=8-4x$ equals its average value on $[0,2]$ .

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Problem 23760

Find the derivative of $f(t)=\sqrt{\frac{t}{t^{2}+4}}$ .

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Problem 23761

Find the period of a satellite orbiting at a distance of $6.7 \times 10^{3} \mathrm{~km}$ from Earth's center.

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Problem 23762

Maximize $z=x^{2}+y^{2}-2 x-6 y+14$ with $x^{2}+y^{2}=16$ . Which set gives the first order conditions?

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Problem 23763

Evaluate these limits: 1. $\lim _{k \rightarrow-2} \frac{2-k}{k^{2}+8}$ , 2. $\lim _{x \rightarrow-7} \frac{2 x^{2}+13 x-7}{x^{2}-49}$ , 3. $\lim _{h \rightarrow 9} \frac{3-\sqrt{h}}{9 h-h^{2}}$ , 4. $\lim _{x \rightarrow \infty} \frac{x^{2}-5 x+4}{2 x^{2}+4 x+7}$ .

See Solution

Problem 23764

Maximize $f(x_{1}, x_{2}) = -2 x_{1}^{2} - (x_{2}-x_{1})(x_{1}+x_{2}) + 5 x_{2}$ subject to $2 x_{1}+x_{2} \geq 5$ and $x_{1}+3 x_{2} \geq 8$ . Find the Kuhn-Tucker condition and $f$ at the max.

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Problem 23765

Find the growth rate of fungus size $L(t)=2.6t+1.2\cos\left(\frac{2\pi t}{24}\right)$ .
(a) Calculate $\frac{dL}{dt}$ . (b) Determine the maximum and minimum growth rates.

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Problem 23766

Find points where $f(x)=-\frac{\pi}{6} \sin x$ equals its average value on the interval $[-\pi, 0]$ .

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Problem 23767

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin (3 x)}{3 x}$ .

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Problem 23768

Prove if $\int_{-a}^{a} p(q(x)) d x$ is even or odd, then find its value or simplify it.

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Problem 23769

Find the average value of $f(x)=a x(13-x)$ on $[6,7]$ as a function of $a$ .

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Problem 23770

Analyze the function $f(x)=\frac{x+8}{x^{2}+10 x+16}$ . What is its behavior near the vertical asymptote?

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Problem 23771

Evaluate the following integrals using the Fundamental Theorem of Calculus and the Substitution Rule:
1. $\int_{0}^{1}(v+3)(v-4) d v$
2. $\int_{1}^{4} \frac{6+x^{2}}{x} d x$
3. $\int \cos (4+7 t) d t$

See Solution

Problem 23772

Given the function $f(x)=x^{2}+6$ , find the derivative at $x=2$ , $f(2)$ , and the tangent line's equation.

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Problem 23773

Find the first and second derivatives, $y^{\prime}$ and $y^{\prime \prime}$ , for $y=\cos(x^{2})$ .

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Problem 23774

Evaluate the integral using symmetry: $\int_{-2}^{2}\left(17-|x|^{3}\right) d x$

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Problem 23775

Approximate the area under $f(x)=\frac{1}{2} x^{2}$ from 0 to 2 using 10 left rectangles. Then find the exact area using a definite integral.

See Solution

Problem 23776

A rock is thrown down at $25 \mathrm{~m/s}$ from a 50m cliff. What is its speed just before hitting the ground? A. $33 \mathrm{~m/s}$ B. $40 \mathrm{~m/s}$ C. $56 \mathrm{~m/s}$ D. $75 \mathrm{~m/s}$

See Solution

Problem 23777

Calculate the integral $\int_{-2}^{2}\left(17-|x|^{3}\right) d x$ .

See Solution

Problem 23778

Calculate the integral from -2 to 2 of the function $8 x^{4}-9$ .

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Problem 23779

Find the difference quotient $\frac{f(x+h)-f(x)}{h}, h \neq 0$ , for $f(x)=4 x^{2}+5 x+8$ .

See Solution

Problem 23780

Find the derivatives using the Fundamental Theorem of Calculus for these integrals:
1. $\int_{x}^{0} \sin (2 t) d t$
2. $\int_{1}^{x^{3}} \ln (t^{2}-3 t+2) d t$
3. $\int_{x}^{4 x} e^{t} d t$

See Solution

Problem 23781

What are the new limits of integration for the integral $\int_{3}^{9} f(x) d x$ with the change of variables $u=x^{2}+4$ ?

See Solution

Problem 23782

If $y=x^{2} e^{4 x}$ , find $\frac{d y}{d x}$ . Is it $2 x e^{4 x}+4 x^{2} e^{4 x}$ ? True or False?

See Solution

Problem 23783

Find the new limits of integration when using $u=x^{2}+4$ for $\int_{3}^{9} f(x) d x$ . Lower limit is 13, upper limit is $\square$ .

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Problem 23784

Evaluate the integral $\int 2 x (x^{2}+5)^{3} \, dx$ using the substitution $u=x^{2}+5$ .

See Solution

Problem 23785

Evaluate the integral $\int 2 x\left(x^{2}+1\right)^{4} d x$ .

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Problem 23786

Prüfen Sie, ob $F$ eine Stammfunktion von $f$ ist für: a) $f(x)=x+5, F(x)=\frac{1}{2} x^{2}+5 x+c$ ; c) $f(x)=0,2 x^{3}, F(x)=0,08 x^{4}+7$ .

See Solution

Problem 23787

Evaluate the integral $\int 2 x (x^{2}+1)^{4} d x$ using the substitution $u=x^{2}+1$ . Rewrite it in terms of $u$ .

See Solution

Problem 23788

Find the velocity of the particle at the first time it is at the origin, given $x(t)=\cos \sqrt{t}$ . Round to three decimals.

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Problem 23789

Find $\frac{d y}{d x}$ for $4 x^{3}-3 x y^{2}+y^{3}=28$ at the point $(3,4)$ . Use a calculator and give a decimal.

See Solution

Problem 23790

Evaluate the integral: $\int 2x(x^{2}+1)^{4} dx = \int u^{4} du$ .

See Solution

Problem 23791

Find the derivative of $y=\cos^{3}(4x-1)$ . It's given that $\frac{dy}{dx}=-12 \cos^{2}(4x-1) \sin(4x-1)$ .

See Solution

Problem 23792

If $y=\sqrt{3x-4}$ , is it true that $\frac{dy}{dx}=\frac{2}{2\sqrt{3x-4}}$ ?

See Solution

Problem 23793

Determine if the following statements are True or False:
1. If $x>0$ and $a>1$ , then $\frac{\ln (x)}{\ln (a)}=\ln \left(\frac{x}{a}\right)$
2. If $\lim _{x \rightarrow 2} f(x)=0$ and $\lim _{x \rightarrow 2} g(x)=0$ , then $\lim _{x \rightarrow 2}\left[\frac{f(x)}{g(x)}\right]$ does not exist
3. If $f$ and $g$ are differentiable, then $\frac{d}{d x}[f(x)+g(x)]=f^{\prime}(x)+g^{\prime}(x)$
4. If $f^{\prime}(x)>0$ for $2<x<9$ , then $f$ is increasing on $(2,9)$
5. $\int_{0}^{4} e^{x^{2}} d x=\int_{0}^{7} e^{x^{2}} d x+\int_{7}^{4} e^{x^{2}} d x$

See Solution

Problem 23794

Determine if the statements are True or False. Circle your answer clearly.
1. If $x>0$ and $a>1$ , then $\frac{\ln (x)}{\ln (a)}=\ln \left(\frac{x}{a}\right)$ True False
2. If $\lim _{x \rightarrow 2} f(x)=0$ and $\lim _{x \rightarrow 2} g(x)=0$ , then $\lim _{x \rightarrow 2}\left[\frac{f(x)}{g(x)}\right]$ does not exist True False
3. If $f$ and $g$ are differentiable, then $\frac{d}{d x}[f(x)+g(x)]=f^{\prime}(x)+g^{\prime}(x)$ True False
4. If $f^{\prime}(x)>0$ for $2<x<9$ , then $f$ is increasing on $(2,9)$ True False
5. $\int_{0}^{4} e^{x^{2}} d x=\int_{0}^{7} e^{x^{2}} d x+\int_{7}^{4} e^{x^{2}} d x$ True False

See Solution

Problem 23795

If $y=\sin^{-1}(6x)$ , is $\frac{dy}{dx}=\frac{1}{\sqrt{1-x^{2}}}$ true or false?

See Solution

Problem 23796

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{e^{4 x}-1}{2 x^{2}+6 x}$ using l'Hôpital's Rule when applicable.

See Solution

Problem 23797

Given $4 x^{2}+9 y^{2}=100$ , find $\frac{d x}{d t}$ and $\frac{d y}{d t}$ under specified conditions.

See Solution

Problem 23798

If $\sin y = x + y$ , is it true that $\frac{d^{2} y}{d x^{2}} = \frac{\sin y}{(\cos y - 1)^{3}}$ ? True or False.

See Solution

Problem 23799

Find the first derivative $f'(x)$ and second derivative $f''(x)$ of $f(x) = 4x^4 - x^3 - 2x + 6$ to analyze critical and inflection points.

See Solution

Problem 23800

Find the value of $g'(2)$ where $g(x)=f^{-1}(x)$ and $f(x)=3x^2-2x+1$ .

See Solution

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Calculus

Problem 23701

Evaluate the integral ∫z8z9+2dz\int \frac{z^{8}}{z^{9}+2} d z∫z9+2z8​dz and include the constant of integration CCC.

Problem 23702

Berechne den Flächeninhalt zwischen den Graphen g(x)=x2g(x)=x^{2}g(x)=x2 und h(x)=2x+4h(x)=2x+4h(x)=2x+4 im Intervall [1,4][1, 4][1,4].

Problem 23703

Approximate the area under f(x)=12x2f(x)=\frac{1}{2} x^{2}f(x)=21​x2 from 0 to 2 using 10 left rectangles. Then find the exact area using integrals.

Problem 23704

Bestimme die allgemeine Stammfunktion von f(x)=−9x7x15f(x)=-\frac{9 x^{7}}{x^{15}}f(x)=−x159x7​.

Problem 23705

Find the value of the function f(x)=7(sin⁡(x))2(1−(cos⁡(x))2)⋅xf(x)=\frac{7(\sin(x))^2}{(1-(\cos(x))^2) \cdot \sqrt{x}}f(x)=(1−(cos(x))2)⋅x​7(sin(x))2​.

Problem 23706

Evaluate the integral ∫0πxcos⁡(x2)dx\int_{0}^{\sqrt{\pi}} x \cos \left(x^{2}\right) d x∫0π​​xcos(x2)dx.

Problem 23707

Find the average distance from the parabola y=10x(16−x)y=10 x(16-x)y=10x(16−x) to the xxx-axis over the interval [0,16][0,16][0,16].

Problem 23708

Bestimme die allgemeine Stammfunktion von f(x)=9(sin⁡(x))31−(cos⁡(x))2f(x)=\frac{9(\sin (x))^{3}}{1-(\cos (x))^{2}}f(x)=1−(cos(x))29(sin(x))3​.

Problem 23709

Bestimme die allgemeine Stammfunktion von f(x)=x2+18x+81x+9f(x)=\frac{x^{2}+18 x+81}{x+9}f(x)=x+9x2+18x+81​.

Problem 23710

Bestimme die allgemeine Stammfunktion von f(x)=7(x2−1)x−1f(x)=\frac{7\left(x^{2}-1\right)}{x-1}f(x)=x−17(x2−1)​.

Problem 23711

Find the speed vvv that maximizes fuel efficiency E(v)=1600vv2+6400E(v)=\frac{1600 v}{v^{2}+6400}E(v)=v2+64001600v​ for limits 100 km/h and 50 km/h. Identify intervals for increasing and decreasing efficiency from 0 to 100 km/h.

Problem 23712

Solve the initial value problem dydx=5x−2\frac{d y}{d x}=5 x-2dxdy​=5x−2, y=−1y=-1y=−1 at x=−2x=-2x=−2. Find y=f(x)y=f(x)y=f(x).

Problem 23713

Approximate the area under f(x)=12x2f(x)=\frac{1}{2} x^{2}f(x)=21​x2 from 0 to 2 using 10 left rectangles. Then find the exact area.

Problem 23714

Solve the initial value problem dydx=4x−1x2\frac{dy}{dx}=\frac{4}{x}-\frac{1}{x^{2}}dxdy​=x4​−x21​, y=−1y=-1y=−1 when x=1x=1x=1.

Problem 23715

Find the integral and verify by differentiation: ∫e−0.01t(e−0.14t+7)dt\int e^{-0.01 t}\left(e^{-0.14 t}+7\right) d t∫e−0.01t(e−0.14t+7)dt.

Problem 23716

Find the definite integral equal to lim⁡n→∞∑k=1n10kn(1+5kn)(5n)\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{10 k}{n}\left(\sqrt{1+\frac{5 k}{n}}\right)\left(\frac{5}{n}\right)limn→∞​∑k=1n​n10k​(1+n5k​​)(n5​).

Problem 23717

Find the definite integral equivalent to lim⁡n→∞∑k=1n10kn(1+5kn)\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{10 k}{n}\left(\sqrt{1+\frac{5 k}{n}}\right)limn→∞​∑k=1n​n10k​(1+n5k​​).

Problem 23718

Find the radius of convergence, RRR, and interval, III, for the series ∑n=1∞n!(8x−1)n\sum_{n=1}^{\infty} n !(8 x-1)^{n}∑n=1∞​n!(8x−1)n.

Problem 23719

Find the midpoint Riemann sum for ∫46x3+1 dx\int_{4}^{6} \sqrt{x^{3}+1} \, dx∫46​x3+1​dx using 4 equal subintervals.

Problem 23720

Find the left Riemann sum for ∫28cos⁡(x2)dx\int_{2}^{8} \cos \left(x^{2}\right) d x∫28​cos(x2)dx using nnn equal subintervals.

Problem 23721

Calculate the integral: ∫(x2+1)5xdx\int\left(x^{2}+1\right)^{5} x d x∫(x2+1)5xdx using u=x2+1u=x^{2}+1u=x2+1 and ∫undu=1n+1un+1+C\int u^{n} du=\frac{1}{n+1} u^{n+1}+C∫undu=n+11​un+1+C.

Problem 23722

Find the function f(x)f(x)f(x) given its slope f′(x)=−9x2−4x+5f^{\prime}(x)=-9 x^{2}-4 x+5f′(x)=−9x2−4x+5 and the point (0,1)(0,1)(0,1).

Problem 23723

Find the function f(x)f(x)f(x) given its slope f′(x)=e−x+x7f^{\prime}(x)=e^{-x}+x^{7}f′(x)=e−x+x7 and the point (0,−9)(0,-9)(0,−9).

Problem 23724

Given values of R(t)R(t)R(t) and that RRR is increasing and concave down, which statement about the trapezoidal sum is true?

Problem 23725

Problem 23726

Find the arc length of the curve y=31−10(ex/20+e−x/20)y=31-10\left(e^{x / 20}+e^{-x / 20}\right)y=31−10(ex/20+e−x/20) from x=−20x=-20x=−20 to x=20x=20x=20.

Problem 23727

Problem 23728

Find the arc length of the curve y=31−10(x/20+e−x/20)y = 31 - 10\left(\sqrt{x}/20 + e^{-x/20}\right)y=31−10(x​/20+e−x/20) from x=−20x = -20x=−20 to x=20x = 20x=20.

Problem 23729

Problem 23730

Find the midpoint Riemann sum for ∫46x3+1 dx\int_{4}^{6} \sqrt{x^{3}+1} \, dx∫46​x3+1​dx using 4 equal subintervals.

Problem 23731

Find the limit: lim⁡x→10x2−99x−10\lim _{x \rightarrow 10} \frac{x^{2}-99}{x-10}limx→10​x−10x2−99​.

Problem 23732

Find the limit: lim⁡x→16x2−99x−10\lim _{x \rightarrow 16} \frac{x^{2}-99}{x-10}limx→16​x−10x2−99​.

Problem 23733

Find the limit using l'Hospital's Rule: lim⁡x→13ln⁡x8sin⁡πx \lim _{x \rightarrow 1} \frac{3 \ln x}{8 \sin \pi x} x→1lim​8sinπx3lnx​

Problem 23734

Find the integral using substitution or state "IMPOSSIBLE": ∫(x8−4)9x7dx\int\left(x^{8}-4\right)^{9} x^{7} d x∫(x8−4)9x7dx. Use CCC for the constant.

Problem 23735

Find the approximate half-life of a drug that decreases from 3,600mg/L3,600 \mathrm{mg} / \mathrm{L}3,600mg/L to 1,160mg/L1,160 \mathrm{mg} / \mathrm{L}1,160mg/L in one day.

Problem 23736

Problem 23737

Pipeline: fff gibt die Ölmenge seit 6 Uhr an. Erklären Sie f(7)=60f(7)=60f(7)=60, f(7)−f(3)4=11\frac{f(7)-f(3)}{4}=114f(7)−f(3)​=11, f′(5)=12f^{\prime}(5)=12f′(5)=12. Berechnen Sie die Ölmenge zwischen 11 und 11:15 Uhr mit f′(5)=12f^{\prime}(5)=12f′(5)=12.

Problem 23738

Calculate the power wasted by a waterfall flowing at 1.3×106 kg/s1.3 \times 10^{6} \mathrm{~kg/s}1.3×106 kg/s and falling 55.1 m55.1 \mathrm{~m}55.1 m. Use g=9.81 m/s2g = 9.81 \mathrm{~m/s^{2}}g=9.81 m/s2. Answer in W.

Problem 23739

Bestimmen Sie xxx, so dass die Ableitung f′(x)=−1x2f'(x) = -\frac{1}{x^2}f′(x)=−x21​ gleich m=−19m = -\frac{1}{9}m=−91​ ist.

Problem 23740

Find the derivative of h(t)=ln⁡(π)⋅sin⁡(t)h(t)=\ln (\pi) \cdot \sin (t)h(t)=ln(π)⋅sin(t). What is h′(t)h^{\prime}(t)h′(t)? Options: A, B, C, D.

Problem 23741

Given s(t)=5t2+20ts(t)=5 t^{2}+20 ts(t)=5t2+20t, find v(t)v(t)v(t), a(t)a(t)a(t), and their values at t=2t=2t=2 sec. Show your work.

Problem 23742

Evaluate the integral $\int \frac{z^{8}}{z^{9}+2} d z$ and include the constant of integration $C$ .

Berechne den Flächeninhalt zwischen den Graphen $g(x)=x^{2}$ und $h(x)=2x+4$ im Intervall $[1, 4]$ .

Approximate the area under $f(x)=\frac{1}{2} x^{2}$ from 0 to 2 using 10 left rectangles. Then find the exact area using integrals.

Bestimme die allgemeine Stammfunktion von $f(x)=-\frac{9 x^{7}}{x^{15}}$ .

Find the value of the function $f(x)=\frac{7(\sin(x))^2}{(1-(\cos(x))^2) \cdot \sqrt{x}}$ .

Evaluate the integral $\int_{0}^{\sqrt{\pi}} x \cos \left(x^{2}\right) d x$ .

Find the average distance from the parabola $y=10 x(16-x)$ to the $x$ -axis over the interval $[0,16]$ .

Bestimme die allgemeine Stammfunktion von $f(x)=\frac{9(\sin (x))^{3}}{1-(\cos (x))^{2}}$ .

Bestimme die allgemeine Stammfunktion von $f(x)=\frac{x^{2}+18 x+81}{x+9}$ .

Bestimme die allgemeine Stammfunktion von $f(x)=\frac{7\left(x^{2}-1\right)}{x-1}$ .

Find the speed $v$ that maximizes fuel efficiency $E(v)=\frac{1600 v}{v^{2}+6400}$ for limits 100 km/h and 50 km/h. Identify intervals for increasing and decreasing efficiency from 0 to 100 km/h.

Solve the initial value problem $\frac{d y}{d x}=5 x-2$ , $y=-1$ at $x=-2$ . Find $y=f(x)$ .

Approximate the area under $f(x)=\frac{1}{2} x^{2}$ from 0 to 2 using 10 left rectangles. Then find the exact area.

Solve the initial value problem $\frac{dy}{dx}=\frac{4}{x}-\frac{1}{x^{2}}$ , $y=-1$ when $x=1$ .

Find the integral and verify by differentiation: $\int e^{-0.01 t}\left(e^{-0.14 t}+7\right) d t$ .

Find the definite integral equal to $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{10 k}{n}\left(\sqrt{1+\frac{5 k}{n}}\right)\left(\frac{5}{n}\right)$ .

Find the definite integral equivalent to $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{10 k}{n}\left(\sqrt{1+\frac{5 k}{n}}\right)$ .

Find the radius of convergence, $R$ , and interval, $I$ , for the series $\sum_{n=1}^{\infty} n !(8 x-1)^{n}$ .

Find the midpoint Riemann sum for $\int_{4}^{6} \sqrt{x^{3}+1} \, dx$ using 4 equal subintervals.

Find the left Riemann sum for $\int_{2}^{8} \cos \left(x^{2}\right) d x$ using $n$ equal subintervals.

Calculate the integral: $\int\left(x^{2}+1\right)^{5} x d x$ using $u=x^{2}+1$ and $\int u^{n} du=\frac{1}{n+1} u^{n+1}+C$ .

Find the function $f(x)$ given its slope $f^{\prime}(x)=-9 x^{2}-4 x+5$ and the point $(0,1)$ .

Find the function $f(x)$ given its slope $f^{\prime}(x)=e^{-x}+x^{7}$ and the point $(0,-9)$ .

Given values of $R(t)$ and that $R$ is increasing and concave down, which statement about the trapezoidal sum is true?

Find the arc length of the curve $y=31-10\left(e^{x / 20}+e^{-x / 20}\right)$ from $x=-20$ to $x=20$ .

Find the arc length of the curve $y = 31 - 10\left(\sqrt{x}/20 + e^{-x/20}\right)$ from $x = -20$ to $x = 20$ .

Find the midpoint Riemann sum for $\int_{4}^{6} \sqrt{x^{3}+1} \, dx$ using 4 equal subintervals.

Find the limit: $\lim _{x \rightarrow 10} \frac{x^{2}-99}{x-10}$ .

Find the limit: $\lim _{x \rightarrow 16} \frac{x^{2}-99}{x-10}$ .

Find the limit using l'Hospital's Rule: $\lim _{x \rightarrow 1} \frac{3 \ln x}{8 \sin \pi x}$

Find the integral using substitution or state "IMPOSSIBLE": $\int\left(x^{8}-4\right)^{9} x^{7} d x$ . Use $C$ for the constant.

Find the approximate half-life of a drug that decreases from $3,600 \mathrm{mg} / \mathrm{L}$ to $1,160 \mathrm{mg} / \mathrm{L}$ in one day.

Pipeline: $f$ gibt die Ölmenge seit 6 Uhr an. Erklären Sie $f(7)=60$ , $\frac{f(7)-f(3)}{4}=11$ , $f^{\prime}(5)=12$ . Berechnen Sie die Ölmenge zwischen 11 und 11:15 Uhr mit $f^{\prime}(5)=12$ .

Calculate the power wasted by a waterfall flowing at $1.3 \times 10^{6} \mathrm{~kg/s}$ and falling $55.1 \mathrm{~m}$ . Use $g = 9.81 \mathrm{~m/s^{2}}$ . Answer in W.

Bestimmen Sie $x$ , so dass die Ableitung $f'(x) = -\frac{1}{x^2}$ gleich $m = -\frac{1}{9}$ ist.

Find the derivative of $h(t)=\ln (\pi) \cdot \sin (t)$ . What is $h^{\prime}(t)$ ? Options: A, B, C, D.

Given $s(t)=5 t^{2}+20 t$ , find $v(t)$ , $a(t)$ , and their values at $t=2$ sec. Show your work.

Find $\lim f(x)$ as $x \rightarrow 10$ for $f(x)=x^{2}$ (if $x \neq 10$ , $f(10)=9$ ).

Find $\lim _{x \rightarrow 10} f(x)$ for $f(x)=x^{2}$ when $x \neq 10$ and $f(10)=99$ .

Find the limit: $\lim _{x \rightarrow 10} \sqrt{-x^{2}+20 x-100}$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{x^{2}}{e^{4 x}-1-4 x}$ .

Find the limit: $\lim _{x \rightarrow-4} \frac{x^{2}-16}{x+4} \ln |x|$

Find the indefinite integral $\int A \cot (x) d x$ . Which option is correct? (Hint: Use $\cot (x) = \frac{\cos(x)}{\sin(x)}$ )

Find the limit: $\lim _{x \rightarrow-\infty} \frac{3 x^{6}-7 x^{5}+x}{5 x^{6}+4 x^{5}-3}$ .

In the lagrangian $L=U(x, y)+\lambda\left(I-P_{x} x-P_{y} y\right)$ , which statement about utility increase is correct?

Bestimmen Sie den Differenzenquotienten der Funktion $f(x)=-2x^2+4$ für die Intervalle I: a) I=[-1 ; 3], b) I=[-2 ; 2].

How long to double \ $3250 at a continuous compound rate of$ 6.5\%$ to reach \$ 6500?

If $m$ decreases by 1 and $\lambda=-5$ , how does $L$ change? Options: Rise by 5, No change, Fall by 5, None.

Differentiate $y=(5 x)^{\ln 5 x}$ using logarithmic differentiation to find $y^{\prime}=\square$ .

Find the limit $\lim _{x \rightarrow 0^{+}} \frac{F(x)}{x^{2}}$ where $F(x)=\int_{0}^{x} e^{t}-1 dt$ . Use L'Hôpital's Rule.

Express the limit of the left-endpoint Riemann sum $\mathrm{L}_{\mathrm{n}} = \frac{1}{n} \sum_{i=1}^{n}\left(3+1 \frac{i-1}{n}\right)$ as a definite integral.

Maximize $z=f(x, y)$ with constraints $y-g(x)=0$ and $y-h(x)=0$ . What is the Lagrangian function?

Find where $f(x)=8-4x$ equals its average value on $[0,2]$ .

Find the derivative of $f(t)=\sqrt{\frac{t}{t^{2}+4}}$ .

Find the period of a satellite orbiting at a distance of $6.7 \times 10^{3} \mathrm{~km}$ from Earth's center.

Maximize $z=x^{2}+y^{2}-2 x-6 y+14$ with $x^{2}+y^{2}=16$ . Which set gives the first order conditions?

Find the growth rate of fungus size $L(t)=2.6t+1.2\cos\left(\frac{2\pi t}{24}\right)$ .
(a) Calculate $\frac{dL}{dt}$ . (b) Determine the maximum and minimum growth rates.

Find points where $f(x)=-\frac{\pi}{6} \sin x$ equals its average value on the interval $[-\pi, 0]$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin (3 x)}{3 x}$ .

Prove if $\int_{-a}^{a} p(q(x)) d x$ is even or odd, then find its value or simplify it.

Find the average value of $f(x)=a x(13-x)$ on $[6,7]$ as a function of $a$ .

Analyze the function $f(x)=\frac{x+8}{x^{2}+10 x+16}$ . What is its behavior near the vertical asymptote?

Evaluate the following integrals using the Fundamental Theorem of Calculus and the Substitution Rule:
1. $\int_{0}^{1}(v+3)(v-4) d v$
2. $\int_{1}^{4} \frac{6+x^{2}}{x} d x$
3. $\int \cos (4+7 t) d t$

Given the function $f(x)=x^{2}+6$ , find the derivative at $x=2$ , $f(2)$ , and the tangent line's equation.

Find the first and second derivatives, $y^{\prime}$ and $y^{\prime \prime}$ , for $y=\cos(x^{2})$ .

Evaluate the integral using symmetry: $\int_{-2}^{2}\left(17-|x|^{3}\right) d x$