Calculus

Problem 20901

Find the limits of $f(x)=\frac{x^{2}+20}{x^{3}+17}$ as $x \rightarrow \infty$ and $x \rightarrow -\infty$ .

See Solution

Problem 20902

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $2 y^{2}=\frac{5 x-3}{5 x+3}$ .

See Solution

Problem 20903

Describe the long-term behavior of the solution to $\frac{d P}{d t}=\frac{1}{2} P(3-P)$ if $P(0)$ is positive.

See Solution

Problem 20904

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $x=\sin y$ . $\frac{d y}{d x}=\square$

See Solution

Problem 20905

Prove that $f(x)$ is continuous at $x=-4$ , where $f(x)=\frac{3 x^{2}+11 x-4}{x+4}$ for $x \neq -4$ and $f(-4)=-13$ .

See Solution

Problem 20906

Find $\frac{d r}{d \theta}$ using implicit differentiation for $\tan(r \theta^{4})=\frac{1}{4}$ .

See Solution

Problem 20907

Find the limit: $\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+7 x}-\sqrt{x^{2}-3 x}\right)$ . Simplify your answer.

See Solution

Problem 20908

Prove that the function $f(x)=\left\{\begin{array}{ll} \frac{2 x^{2}+8 x-10}{x+5} & x \neq-5 \\ -12 & x=-5 \end{array}\right.$ is continuous at $x=-5$ .

See Solution

Problem 20909

Find the slope of the curve $5 y^{9}+7 x^{4}=2 y+10 x$ at the point (1,1). What is it?

See Solution

Problem 20910

Evaluate the integral: $\int\left(x^{2}+x+9\right) \sqrt{x+8} \, dx$ and use $C$ for the constant of integration.

See Solution

Problem 20911

Find the slope of the curve $y^{4}=y^{2}-x^{2}$ at the point $\left(\frac{\sqrt{3}}{4}, \frac{\sqrt{3}}{2}\right)$ .

See Solution

Problem 20912

How long to triple an investment with continuous compounding at $5\%$ ?

See Solution

Problem 20913

Evaluate the integral $\int x^{3} e^{2 x} d x$ using the reduction formula for $a \neq 0$ .

See Solution

Problem 20914

Show that the reproduction rate $r(C)$ approaches zero as sugar level $C$ approaches zero: $\lim_{C \rightarrow 0} r(C)=0$ .

See Solution

Problem 20915

Rewrite the logistic equation problem for a flu outbreak on a college campus with 1000 students.
1. Plot the slope field for $\frac{d y}{d x}=0.9906 y(1000-y), \quad y(0)=1$ .
2. Find and plot the solution with the slope field.
3. Calculate infected students after 6 days.
4. Will all students get infected or recover over time?
5. Estimate days until all are infected or recovered.

See Solution

Problem 20916

What is the predicted population in 4 years using the model $P(t)=221,000 e^{-0.021 t}$ ? Round to the nearest person.

See Solution

Problem 20917

Calculate the work (in joules) to pump water from a cone tank with height 13 m, base 26 m, and spout 3 m. Use density 1000 kg/m³ and g = 9.8 m/s². Answer in scientific notation to three significant figures.

See Solution

Problem 20918

Find the general antiderivative of $f(x)=4 \sqrt{x}-3 \sqrt[3]{x}$ and check by differentiating. Use $C$ .

See Solution

Problem 20919

Evaluate the integral: $\int 8 \cos^{3} 2x \, dx = \square$

See Solution

Problem 20920

Find the general antiderivative of $f(t)=\frac{2 t-4+6 \sqrt{t}}{\sqrt{t}}$ and verify by differentiation. Use $C$ for the constant.

See Solution

Problem 20921

Find the general antiderivative of $g(v)=3 \cos (v)-\frac{4}{\sqrt{1-v^{2}}$ and check by differentiation. Use $C$.

See Solution

Problem 20922

Find the derivative $f^{\prime}(x)$ for the function $f(x)=(x^{2}+7)^{5}$ .

See Solution

Problem 20923

Find the general antiderivative of $f(x)=6^{x}+3 \sinh (x)$ and verify by differentiation. Use $C$ for the constant.

See Solution

Problem 20924

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\sin \left(x^{2}\right)$ .

See Solution

Problem 20925

Find the number of units $x$ for max profit from $R(x)=132x-0.0135x^2$ and $C(x)=9000+66x-0.027x^2+0.00001x^3$ .

See Solution

Problem 20926

Calculate the work in MJ to pump water from a trough with dimensions $a=9 \mathrm{~m}, b=14 \mathrm{~m}, c=13 \mathrm{~m}, h=4 \mathrm{~m}$ . Use $\rho=1000 \mathrm{~kg/m}^{3}$ and $g=9.8 \mathrm{~m/s}^{2}$ . Round to two decimal places.

See Solution

Problem 20927

Find the antiderivative $F$ of $f(x)=2 e^{x}-8 x$ with $F(0)=5$ and verify by comparing $f$ and $F$ graphs.

See Solution

Problem 20928

A town's population dropped from 210,000 to 207,000 in 3 years. Find the population after 6 more years. It will be about $\square$ .

See Solution

Problem 20929

Calculate the work in MJ to pump water from a trough with dimensions $a=9 \mathrm{~m}, b=14 \mathrm{~m}, c=13 \mathrm{~m}, h=4 \mathrm{~m}$ . Density is $1000 \mathrm{~kg/m}^{3}$ , $g=9.8 \mathrm{~m/s}^{2}$ . Answer to two decimal places.

See Solution

Problem 20930

Evaluate the integral: $\int 7 \sin ^{2} 5 x \, dx = \square$

See Solution

Problem 20931

Find the function $f$ such that $f(x)=f'(x)=\frac{x+1}{\sqrt{x}}$ and $f(1)=3$ .

See Solution

Problem 20932

A town's population dropped from 220,000 to 217,000 in 4 years. What's the population in 6 more years? About $\square$ .

See Solution

Problem 20933

Find the function $f$ given $f''(x)=-2+12x-12x^2$ , $f(0)=6$ , and $f'(0)=14$ .

See Solution

Problem 20934

Calculate the work in MJ to pump water from a trough with dimensions $a=9 \mathrm{~m}, b=14 \mathrm{~m}, c=13 \mathrm{~m}, h=4 \mathrm{~m}$ . Use $\rho=1000 \mathrm{~kg/m^{3}}$ and $g=9.8 \mathrm{~m/s^{2}}$ . Round to two decimal places.

See Solution

Problem 20935

Find the function $f$ such that $f^{\prime \prime}(\theta)=\sin (\theta)+\cos (\theta)$ with $f(0)=5$ and $f^{\prime}(0)=1$ .

See Solution

Problem 20936

Find the function $f$ given that $f''(x)=4+6x+24x^2$ , $f(0)=3$ , and $f(1)=13$ .

See Solution

Problem 20937

Calculate the average value of the function $f(x)=9$ over the interval $[10,90]$ .

See Solution

Problem 20938

Find the slope of $x^{3}+y^{3}-65xy=0$ at $(4,16)$ and $(16,4)$ , and points with horizontal/vertical tangents.

See Solution

Problem 20939

Find the average sales from day 0 to day 2 for the function $S(x) = 200x + 9x^2$ .

See Solution

Problem 20940

Find the average temperature from time $t=0$ to $t=10$ for $T(t)=-0.3 t^{2}+4 t+70$ .

See Solution

Problem 20941

Find a function $f$ where $f'(x)=3x^3$ and the line $81x+y=0$ is tangent to its graph. $f(x)=$

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Problem 20942

Find the position of a particle with $v(t)=5 \cos (t)+3 \sin (t)$ and $s(0)=9$ . What is $s(t)$ ?

See Solution

Problem 20943

Is the series $\frac{1}{2}+\frac{1}{5}+\frac{1}{8}+\frac{1}{11}+\frac{1}{14}+\cdots$ convergent or divergent?

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Problem 20944

Evaluate the integral: $\int 9 \sin ^{2} 5 x \, d x = \square$

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Problem 20945

Evaluate the integral from 1 to infinity: $\int_{1}^{\infty} x^{4} e^{-x^{5}} d x$ .

See Solution

Problem 20946

Evaluate the integral of $\sin^{5}(8x)$ with respect to $x$ : $\int \sin^{5}(8x) \, dx = \square$

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Problem 20947

Find the particle's position $s(t)$ given $a(t)=2 t+9$ , $s(0)=5$ , and $v(0)=-4$ .

See Solution

Problem 20948

Find $\frac{d x}{d t}$ , $\frac{d y}{d t}$ , and $\frac{d y}{d x}$ for $x=5 t^{3}+6 t$ , $y=3 t-4 t^{2}$ .

See Solution

Problem 20949

Find the limit: $\lim _{x \rightarrow 2} \frac{\sin \left(\frac{1}{x-2}\right)}{(x-2)}$ .

See Solution

Problem 20950

Bestimmen Sie die positive Extremstelle $t_{1}$ von $B(t)=b \cdot e^{k \cdot t-\frac{c}{2} \cdot t^{2}}$ und den Einfluss von $c$ auf $t_{1}$ .

See Solution

Problem 20951

A stone drops from a 100 m tower. Find: (a) height $s(t)$ , (b) time to ground, (c) impact velocity, (d) time if thrown down at 5 m/s.

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Problem 20952

Find $\frac{d f}{d x}$ at $x=\frac{1}{8}$ and $\frac{d f^{-1}}{d x}$ at $x=f\left(\frac{1}{8}\right)$ for $f(x)=9-8x$ . Show $\left.\frac{d f^{-1}}{d x}\right|_{x=f\left(\frac{1}{8}\right)}=\frac{1}{\left.\frac{d f}{d x}\right|_{x=\frac{1}{8}}}$ .

See Solution

Problem 20953

Approximate the area under $f(x)=2x$ from $a=3$ to $b=4$ using 5 rectangles. Then find the exact area using a definite integral.

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Problem 20954

Evaluate the integral from 2 to infinity: $\int_{2}^{\infty} \frac{x^{2}}{x^{3}+1} d x$ .

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Problem 20955

A bacteria culture grows proportionally. It had 3,000 at 9 AM and 3,700 at noon. How many at midnight? There will be about $\square$ bacteria at midnight.

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Problem 20956

Determine the values of $x$ for the convergence of the series $\sum_{n=0}^{\infty} e^{n x}$ using interval notation.

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Problem 20957

Evaluate the integral from 1 to infinity: $\int_{1}^{\infty} x^{-5} d x$ .

See Solution

Problem 20958

Find the antiderivative of $\frac{d x}{d t}=5 e^{t}-6$ with the condition $x(0)=6$ .

See Solution

Problem 20959

Evaluate the integral $\int \sin^{5}(6x) \, dx$ .

See Solution

Problem 20960

Find the general antiderivative of $f(x)=9 x^{2/7}+8 x^{-6/7}$ and check by differentiating. Use $C$ for the constant.

See Solution

Problem 20961

Find the derivative $\frac{d y}{d x}$ for $y=(\csc x+\cot x)(\csc x-\cot x)$ .

See Solution

Problem 20962

Find the general antiderivative of $f(x)=\frac{2 x^{4}+2 x^{3}-x}{x^{3}}$ for $x>0$ . Use $C$ for the constant.

See Solution

Problem 20963

How long to triple an investment compounded continuously at $9\%$ ? It takes about $\square$ years.

See Solution

Problem 20964

Determine if the polynomial $f(x)=6 x^{4}-2 x^{3}-5 x-6$ has a real zero between -1 and 0 using the Intermediate Value Theorem.

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Problem 20965

Find the function $f$ given that $f^{\prime \prime}(x)=28 x^{3}-18 x^{2}+8 x$ . Use constants $C$ and $D$ .

See Solution

Problem 20966

Find the function $f$ given that $f'(x)=1+3\sqrt{x}$ and $f(4)=29$ . What is $f(x)$ ?

See Solution

Problem 20967

Find the general antiderivative of $g(t)=\frac{8+t+t^{2}}{\sqrt{t}}$ and verify by differentiating. Use constant $C$ .

See Solution

Problem 20968

Find the derivative of $F(x)=\int_{x}^{6x} \cos(t^{5}) dt$ . What is $F'(x)$ ?

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Problem 20969

Find the antiderivative $F$ of $f(x)=5 x^{4}-8 x^{5}$ with $F(0)=2$ . Verify by comparing graphs of $f$ and $F$ .

See Solution

Problem 20970

Given the function $y=f(x)$ with an inverse, passing through $(5,6)$ and slope $\frac{1}{5}$ , find $\frac{d f^{-1}}{d x}$ at $x=6$ .

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Problem 20971

Find the derivative of $y=\ln(6x)+3x$ with respect to $x$ . What is $\frac{dy}{dx}$ ?

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Problem 20972

A simple pendulum with a length of $50 \mathrm{~cm}$ and max displacement of $8 \mathrm{~cm}$ : a) Find the period. b) Find the horizontal position function. c) Find the velocity function. d) Find the acceleration function.

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Problem 20973

Find the position of a particle given its velocity $v(t)=\sin(t)-\cos(t)$ and initial position $s(0)=4$ .

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Problem 20974

Find the derivative of $y=5 \ln \left(\frac{7}{x}\right)$ . What is $\frac{d y}{d x}=\square$ ?

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Problem 20975

Verify if the function $f(x)=x^{3}+2$ has an inverse and find $\left(f^{-1}\right)^{\prime}(-1)$ .

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Problem 20976

Find the derivative of $y=\left(x^{4} \ln x\right)^{6}$ with respect to $x$ . What is $\frac{d y}{d x}=?$

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Problem 20977

Calculate the extra earnings from a \ $10,000 investment at$ 5\%$ compounded continuously vs. semi-annually for 4 years.

See Solution

Problem 20978

Calculate the integral: $\int\left(\frac{6}{x}+\frac{x}{6}\right) d x=\square$

See Solution

Problem 20979

Find the derivative of $y$ where $y=\frac{\ln x}{4+\ln x}$ . What is $\frac{d y}{d x}=\square$ ?

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Problem 20980

What is the value today of a \$400 investment at a 3.5% continuous interest rate after 100 years? It will be worth \$\square.

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Problem 20981

Calculate the integral: $\int\left(x-8 x^{4}+\frac{1}{6 x}\right) d x$

See Solution

Problem 20982

Find the derivative of $y$ with respect to $x$ for $y=\frac{\ln x}{4+\ln x}$ . What is $\frac{\mathrm{dy}}{\mathrm{dx}}$ ?

See Solution

Problem 20983

Find the integral of $-3(e^{5x}+1)$ with respect to $x$ .

See Solution

Problem 20984

Find all functions $f(x)$ such that $f'(x)=x^{3}$ and $f(0)=9$ . What is $f(x)$ ?

See Solution

Problem 20985

Evaluate the integral: $\int \sin^{3} 5\theta \cos^{-2} 5\theta d\theta = \square$

See Solution

Problem 20986

Find $k$ in the equation $\int(4 x-3)^{3}dx=k(4 x-3)^{4}+C$ .

See Solution

Problem 20987

Evaluate the integral: $\int \cos^{3} 2\theta \sin^{-2} 2\theta d\theta = \square$

See Solution

Problem 20988

Find the derivative of $\ln \left(\frac{3 x+1}{3 x-1}\right)$ . What is it equal to?

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Problem 20989

Find a parametric equation for the rim of a cylindrical can with $x^{2}+y^{2}=1$ and height 3, then compute the surface integral.

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Problem 20990

Calculate the integral: $\int 4 e^{-5 x} \mathrm{dx}=\square$

See Solution

Problem 20991

Find the constant $k$ in the integral $\int 2 e^{-3 t} \mathrm{dt}=k e^{-3 t}+\mathrm{C}$ .

See Solution

Problem 20992

Find $\frac{dy}{d\theta}$ for $y=16\theta(\sin(\ln(16\theta))+\cos(\ln(16\theta)))$ .

See Solution

Problem 20993

Find the derivative of $y = \ln \sqrt{\frac{(x+9)^{15}}{(x-2)^{60}}}$ . What is $\frac{d y}{d x} = \square$ ?

See Solution

Problem 20994

Find the tangent line equation for $y=\ln(2+x)+\frac{2}{2+x}$ at the point (-1,2).

See Solution

Problem 20995

Determine the values of $x$ for which the series $\sum_{n=1}^{\infty}(-6)^{n} x^{n}$ converges and find its sum.

See Solution

Problem 20996

Differentiate $y=\sqrt{(x^{2}+1)(x-1)^{2}}$ using logarithmic differentiation. Find $\frac{\mathrm{dy}}{\mathrm{dx}}=\square$ .

See Solution

Problem 20997

Evaluate the integral: $\int \cos ^{3} 8 \theta \sin ^{-2} 8 \theta d \theta$ .

See Solution

Problem 20998

Find the derivative of $y$ using logarithmic differentiation, where $y=\sqrt[6]{\frac{t}{3t+1}}$ . $\frac{\mathrm{dy}}{\mathrm{dt}}=\square$

See Solution

Problem 20999

Determine the values of $x$ for which the series $\sum_{n=1}^{\infty}(x+9)^{n}$ converges and find its sum.

See Solution

Problem 21000

Gegeben ist eine Funktion $f$ mit einem Wendepunkt bei $x = 1$ , abnehmender Steigung, $f'$ schneidet die x-Achse bei $x = 2$ .

See Solution

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Calculus

Problem 20901

Find the limits of f(x)=x2+20x3+17f(x)=\frac{x^{2}+20}{x^{3}+17}f(x)=x3+17x2+20​ as x→∞x \rightarrow \inftyx→∞ and x→−∞x \rightarrow -\inftyx→−∞.

Problem 20902

Find dydx\frac{d y}{d x}dxdy​ using implicit differentiation for the equation 2y2=5x−35x+32 y^{2}=\frac{5 x-3}{5 x+3}2y2=5x+35x−3​.

Problem 20903

Describe the long-term behavior of the solution to dPdt=12P(3−P)\frac{d P}{d t}=\frac{1}{2} P(3-P)dtdP​=21​P(3−P) if P(0)P(0)P(0) is positive.

Problem 20904

Find dydx\frac{d y}{d x}dxdy​ using implicit differentiation for the equation x=sin⁡yx=\sin yx=siny. dydx=□\frac{d y}{d x}=\squaredxdy​=□

Problem 20905

Prove that f(x)f(x)f(x) is continuous at x=−4x=-4x=−4, where f(x)=3x2+11x−4x+4f(x)=\frac{3 x^{2}+11 x-4}{x+4}f(x)=x+43x2+11x−4​ for x≠−4x \neq -4x=−4 and f(−4)=−13f(-4)=-13f(−4)=−13.

Problem 20906

Find drdθ\frac{d r}{d \theta}dθdr​ using implicit differentiation for tan⁡(rθ4)=14\tan(r \theta^{4})=\frac{1}{4}tan(rθ4)=41​.

Problem 20907

Find the limit: lim⁡x→∞(x2+7x−x2−3x)\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+7 x}-\sqrt{x^{2}-3 x}\right)x→∞lim​(x2+7x​−x2−3x​). Simplify your answer.

Problem 20908

Prove that the function f(x)={2x2+8x−10x+5x≠−5−12x=−5f(x)=\left\{\begin{array}{ll} \frac{2 x^{2}+8 x-10}{x+5} & x \neq-5 \\ -12 & x=-5 \end{array}\right.f(x)={x+52x2+8x−10​−12​x=−5x=−5​ is continuous at x=−5x=-5x=−5.

Problem 20909

Find the slope of the curve 5y9+7x4=2y+10x5 y^{9}+7 x^{4}=2 y+10 x5y9+7x4=2y+10x at the point (1,1). What is it?

Problem 20910

Evaluate the integral: ∫(x2+x+9)x+8 dx\int\left(x^{2}+x+9\right) \sqrt{x+8} \, dx∫(x2+x+9)x+8​dx and use CCC for the constant of integration.

Problem 20911

Find the slope of the curve y4=y2−x2y^{4}=y^{2}-x^{2}y4=y2−x2 at the point (34,32)\left(\frac{\sqrt{3}}{4}, \frac{\sqrt{3}}{2}\right)(43​​,23​​).

Problem 20912

How long to triple an investment with continuous compounding at 5%5\%5%?

Problem 20913

Evaluate the integral ∫x3e2xdx\int x^{3} e^{2 x} d x∫x3e2xdx using the reduction formula for a≠0a \neq 0a=0.

Problem 20914

Show that the reproduction rate r(C)r(C)r(C) approaches zero as sugar level CCC approaches zero: lim⁡C→0r(C)=0\lim_{C \rightarrow 0} r(C)=0limC→0​r(C)=0.

Problem 20915

Problem 20916

What is the predicted population in 4 years using the model P(t)=221,000e−0.021tP(t)=221,000 e^{-0.021 t}P(t)=221,000e−0.021t? Round to the nearest person.

Problem 20917

Calculate the work (in joules) to pump water from a cone tank with height 13 m, base 26 m, and spout 3 m. Use density 1000 kg/m³ and g = 9.8 m/s². Answer in scientific notation to three significant figures.

Problem 20918

Find the general antiderivative of f(x)=4x−3x3f(x)=4 \sqrt{x}-3 \sqrt[3]{x}f(x)=4x​−33x​ and check by differentiating. Use CCC.

Problem 20919

Evaluate the integral: ∫8cos⁡32x dx=□\int 8 \cos^{3} 2x \, dx = \square∫8cos32xdx=□

Problem 20920

Find the general antiderivative of f(t)=2t−4+6ttf(t)=\frac{2 t-4+6 \sqrt{t}}{\sqrt{t}}f(t)=t​2t−4+6t​​ and verify by differentiation. Use CCC for the constant.

Problem 20921

Find the general antiderivative of $g(v)=3 \cos (v)-\frac{4}{\sqrt{1-v^{2}}$ and check by differentiation. Use $C$.

Problem 20922

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=(x2+7)5f(x)=(x^{2}+7)^{5}f(x)=(x2+7)5.

Problem 20923

Find the general antiderivative of f(x)=6x+3sinh⁡(x)f(x)=6^{x}+3 \sinh (x)f(x)=6x+3sinh(x) and verify by differentiation. Use CCC for the constant.

Problem 20924

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=sin⁡(x2)f(x)=\sin \left(x^{2}\right)f(x)=sin(x2).

Problem 20925

Find the number of units xxx for max profit from R(x)=132x−0.0135x2R(x)=132x-0.0135x^2R(x)=132x−0.0135x2 and C(x)=9000+66x−0.027x2+0.00001x3C(x)=9000+66x-0.027x^2+0.00001x^3C(x)=9000+66x−0.027x2+0.00001x3.

Problem 20926

Problem 20927

Find the antiderivative FFF of f(x)=2ex−8xf(x)=2 e^{x}-8 xf(x)=2ex−8x with F(0)=5F(0)=5F(0)=5 and verify by comparing fff and FFF graphs.

Problem 20928

A town's population dropped from 210,000 to 207,000 in 3 years. Find the population after 6 more years. It will be about □\square□.

Problem 20929

Problem 20930

Evaluate the integral: ∫7sin⁡25x dx=□\int 7 \sin ^{2} 5 x \, dx = \square∫7sin25xdx=□

Problem 20931

Find the function fff such that f(x)=f′(x)=x+1xf(x)=f'(x)=\frac{x+1}{\sqrt{x}}f(x)=f′(x)=x​x+1​ and f(1)=3f(1)=3f(1)=3.

Problem 20932

A town's population dropped from 220,000 to 217,000 in 4 years. What's the population in 6 more years? About □\square□.

Problem 20933

Find the function fff given f′′(x)=−2+12x−12x2f''(x)=-2+12x-12x^2f′′(x)=−2+12x−12x2, f(0)=6f(0)=6f(0)=6, and f′(0)=14f'(0)=14f′(0)=14.

Problem 20934

Problem 20935

Find the function fff such that f′′(θ)=sin⁡(θ)+cos⁡(θ)f^{\prime \prime}(\theta)=\sin (\theta)+\cos (\theta)f′′(θ)=sin(θ)+cos(θ) with f(0)=5f(0)=5f(0)=5 and f′(0)=1f^{\prime}(0)=1f′(0)=1.

Problem 20936

Find the function fff given that f′′(x)=4+6x+24x2f''(x)=4+6x+24x^2f′′(x)=4+6x+24x2, f(0)=3f(0)=3f(0)=3, and f(1)=13f(1)=13f(1)=13.

Problem 20937

Calculate the average value of the function f(x)=9f(x)=9f(x)=9 over the interval [10,90][10,90][10,90].

Problem 20938

Find the slope of x3+y3−65xy=0x^{3}+y^{3}-65xy=0x3+y3−65xy=0 at (4,16)(4,16)(4,16) and (16,4)(16,4)(16,4), and points with horizontal/vertical tangents.

Problem 20939

Find the average sales from day 0 to day 2 for the function S(x)=200x+9x2S(x) = 200x + 9x^2S(x)=200x+9x2.

Problem 20940

Find the average temperature from time t=0t=0t=0 to t=10t=10t=10 for T(t)=−0.3t2+4t+70T(t)=-0.3 t^{2}+4 t+70T(t)=−0.3t2+4t+70.

Problem 20941

Find a function fff where f′(x)=3x3f'(x)=3x^3f′(x)=3x3 and the line 81x+y=081x+y=081x+y=0 is tangent to its graph. f(x)=f(x)=f(x)=

Problem 20942

Find the limits of $f(x)=\frac{x^{2}+20}{x^{3}+17}$ as $x \rightarrow \infty$ and $x \rightarrow -\infty$ .

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $2 y^{2}=\frac{5 x-3}{5 x+3}$ .

Describe the long-term behavior of the solution to $\frac{d P}{d t}=\frac{1}{2} P(3-P)$ if $P(0)$ is positive.

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $x=\sin y$ . $\frac{d y}{d x}=\square$

Prove that $f(x)$ is continuous at $x=-4$ , where $f(x)=\frac{3 x^{2}+11 x-4}{x+4}$ for $x \neq -4$ and $f(-4)=-13$ .

Find $\frac{d r}{d \theta}$ using implicit differentiation for $\tan(r \theta^{4})=\frac{1}{4}$ .

Find the limit: $\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+7 x}-\sqrt{x^{2}-3 x}\right)$ . Simplify your answer.

Prove that the function $f(x)=\left\{\begin{array}{ll} \frac{2 x^{2}+8 x-10}{x+5} & x \neq-5 \\ -12 & x=-5 \end{array}\right.$ is continuous at $x=-5$ .

Find the slope of the curve $5 y^{9}+7 x^{4}=2 y+10 x$ at the point (1,1). What is it?

Evaluate the integral: $\int\left(x^{2}+x+9\right) \sqrt{x+8} \, dx$ and use $C$ for the constant of integration.

Find the slope of the curve $y^{4}=y^{2}-x^{2}$ at the point $\left(\frac{\sqrt{3}}{4}, \frac{\sqrt{3}}{2}\right)$ .

How long to triple an investment with continuous compounding at $5\%$ ?

Evaluate the integral $\int x^{3} e^{2 x} d x$ using the reduction formula for $a \neq 0$ .

Show that the reproduction rate $r(C)$ approaches zero as sugar level $C$ approaches zero: $\lim_{C \rightarrow 0} r(C)=0$ .

What is the predicted population in 4 years using the model $P(t)=221,000 e^{-0.021 t}$ ? Round to the nearest person.

Find the general antiderivative of $f(x)=4 \sqrt{x}-3 \sqrt[3]{x}$ and check by differentiating. Use $C$ .

Evaluate the integral: $\int 8 \cos^{3} 2x \, dx = \square$

Find the general antiderivative of $f(t)=\frac{2 t-4+6 \sqrt{t}}{\sqrt{t}}$ and verify by differentiation. Use $C$ for the constant.

Find the derivative $f^{\prime}(x)$ for the function $f(x)=(x^{2}+7)^{5}$ .

Find the general antiderivative of $f(x)=6^{x}+3 \sinh (x)$ and verify by differentiation. Use $C$ for the constant.

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\sin \left(x^{2}\right)$ .

Find the number of units $x$ for max profit from $R(x)=132x-0.0135x^2$ and $C(x)=9000+66x-0.027x^2+0.00001x^3$ .

Find the antiderivative $F$ of $f(x)=2 e^{x}-8 x$ with $F(0)=5$ and verify by comparing $f$ and $F$ graphs.

A town's population dropped from 210,000 to 207,000 in 3 years. Find the population after 6 more years. It will be about $\square$ .

Evaluate the integral: $\int 7 \sin ^{2} 5 x \, dx = \square$

Find the function $f$ such that $f(x)=f'(x)=\frac{x+1}{\sqrt{x}}$ and $f(1)=3$ .

A town's population dropped from 220,000 to 217,000 in 4 years. What's the population in 6 more years? About $\square$ .

Find the function $f$ given $f''(x)=-2+12x-12x^2$ , $f(0)=6$ , and $f'(0)=14$ .

Find the function $f$ such that $f^{\prime \prime}(\theta)=\sin (\theta)+\cos (\theta)$ with $f(0)=5$ and $f^{\prime}(0)=1$ .

Find the function $f$ given that $f''(x)=4+6x+24x^2$ , $f(0)=3$ , and $f(1)=13$ .

Calculate the average value of the function $f(x)=9$ over the interval $[10,90]$ .

Find the slope of $x^{3}+y^{3}-65xy=0$ at $(4,16)$ and $(16,4)$ , and points with horizontal/vertical tangents.

Find the average sales from day 0 to day 2 for the function $S(x) = 200x + 9x^2$ .

Find the average temperature from time $t=0$ to $t=10$ for $T(t)=-0.3 t^{2}+4 t+70$ .

Find a function $f$ where $f'(x)=3x^3$ and the line $81x+y=0$ is tangent to its graph. $f(x)=$

Find the position of a particle with $v(t)=5 \cos (t)+3 \sin (t)$ and $s(0)=9$ . What is $s(t)$ ?

Is the series $\frac{1}{2}+\frac{1}{5}+\frac{1}{8}+\frac{1}{11}+\frac{1}{14}+\cdots$ convergent or divergent?

Evaluate the integral: $\int 9 \sin ^{2} 5 x \, d x = \square$

Evaluate the integral from 1 to infinity: $\int_{1}^{\infty} x^{4} e^{-x^{5}} d x$ .

Evaluate the integral of $\sin^{5}(8x)$ with respect to $x$ : $\int \sin^{5}(8x) \, dx = \square$

Find the particle's position $s(t)$ given $a(t)=2 t+9$ , $s(0)=5$ , and $v(0)=-4$ .

Find $\frac{d x}{d t}$ , $\frac{d y}{d t}$ , and $\frac{d y}{d x}$ for $x=5 t^{3}+6 t$ , $y=3 t-4 t^{2}$ .

Find the limit: $\lim _{x \rightarrow 2} \frac{\sin \left(\frac{1}{x-2}\right)}{(x-2)}$ .

Bestimmen Sie die positive Extremstelle $t_{1}$ von $B(t)=b \cdot e^{k \cdot t-\frac{c}{2} \cdot t^{2}}$ und den Einfluss von $c$ auf $t_{1}$ .

A stone drops from a 100 m tower. Find: (a) height $s(t)$ , (b) time to ground, (c) impact velocity, (d) time if thrown down at 5 m/s.

Approximate the area under $f(x)=2x$ from $a=3$ to $b=4$ using 5 rectangles. Then find the exact area using a definite integral.

Evaluate the integral from 2 to infinity: $\int_{2}^{\infty} \frac{x^{2}}{x^{3}+1} d x$ .

A bacteria culture grows proportionally. It had 3,000 at 9 AM and 3,700 at noon. How many at midnight? There will be about $\square$ bacteria at midnight.

Determine the values of $x$ for the convergence of the series $\sum_{n=0}^{\infty} e^{n x}$ using interval notation.

Evaluate the integral from 1 to infinity: $\int_{1}^{\infty} x^{-5} d x$ .

Find the antiderivative of $\frac{d x}{d t}=5 e^{t}-6$ with the condition $x(0)=6$ .

Evaluate the integral $\int \sin^{5}(6x) \, dx$ .

Find the general antiderivative of $f(x)=9 x^{2/7}+8 x^{-6/7}$ and check by differentiating. Use $C$ for the constant.

Find the derivative $\frac{d y}{d x}$ for $y=(\csc x+\cot x)(\csc x-\cot x)$ .

Find the general antiderivative of $f(x)=\frac{2 x^{4}+2 x^{3}-x}{x^{3}}$ for $x>0$ . Use $C$ for the constant.

How long to triple an investment compounded continuously at $9\%$ ? It takes about $\square$ years.

Determine if the polynomial $f(x)=6 x^{4}-2 x^{3}-5 x-6$ has a real zero between -1 and 0 using the Intermediate Value Theorem.

Find the function $f$ given that $f^{\prime \prime}(x)=28 x^{3}-18 x^{2}+8 x$ . Use constants $C$ and $D$ .

Find the function $f$ given that $f'(x)=1+3\sqrt{x}$ and $f(4)=29$ . What is $f(x)$ ?

Find the general antiderivative of $g(t)=\frac{8+t+t^{2}}{\sqrt{t}}$ and verify by differentiating. Use constant $C$ .

Find the derivative of $F(x)=\int_{x}^{6x} \cos(t^{5}) dt$ . What is $F'(x)$ ?

Find the antiderivative $F$ of $f(x)=5 x^{4}-8 x^{5}$ with $F(0)=2$ . Verify by comparing graphs of $f$ and $F$ .

Given the function $y=f(x)$ with an inverse, passing through $(5,6)$ and slope $\frac{1}{5}$ , find $\frac{d f^{-1}}{d x}$ at $x=6$ .

Find the derivative of $y=\ln(6x)+3x$ with respect to $x$ . What is $\frac{dy}{dx}$ ?

A simple pendulum with a length of $50 \mathrm{~cm}$ and max displacement of $8 \mathrm{~cm}$ : a) Find the period. b) Find the horizontal position function. c) Find the velocity function. d) Find the acceleration function.

Find the position of a particle given its velocity $v(t)=\sin(t)-\cos(t)$ and initial position $s(0)=4$ .

Find the derivative of $y=5 \ln \left(\frac{7}{x}\right)$ . What is $\frac{d y}{d x}=\square$ ?