Calculus

Problem 20801

Evaluate the integral: $\int \frac{5 x^{2}+2 x-5}{x^{3}-x} d x$

See Solution

Problem 20802

Use substitution to simplify $\int (\sec^2 x) \ln(\tan x + 8) \, dx$ to $\int \ln u \, du$ and evaluate it.

See Solution

Problem 20803

Berechnen Sie die mittlere Steigung von $f$ in den Intervallen: a) $f(x)=\frac{1}{2} x^{2}+1, I=[1 ; 3]$ , b) $f(x)=\frac{4}{x}, I=[2 ; 8]$ , c) $f(x)=2x, I=[0 ; 1]$ , d) $f(x)=2^{x}, I=[1 ; 3]$ .

See Solution

Problem 20804

Use substitution to simplify $\int(\cos x) \ln (\sin x+2) d x$ to $\int \ln u \, du$ and evaluate it.

See Solution

Problem 20805

Gegeben sind $f(x)=2^{x}$ und $g(x)=-2x+4$ . Skizzieren Sie die Graphen und zeigen Sie, dass sie bei $S(1|2)$ schneiden. Berechnen Sie die Fläche zwischen den Graphen und den Achsen.

See Solution

Problem 20806

Calculate the value of $\int_{2}^{3} \frac{x+x^{2}}{5} dx - \int_{2}^{3} \frac{x-x^{2}}{5} dx + 2 \cdot \int_{2}^{3} \frac{x^{2}}{5} dx$ .

See Solution

Problem 20807

Bestimmen Sie die durchschnittliche Steigung der Funktion $f(x)=2x$ im Intervall $I=[0 ; 1]$ .

See Solution

Problem 20808

Bestimme die Ableitung von $f(x)=3 \cdot 2^{x}$ und schreibe sie als allgemeine Exponentialfunktion.

See Solution

Problem 20809

Bestimmen Sie die Ableitung von $f(x)=3 \cdot 2^{x}$ und formulieren Sie das Ergebnis als allgemeine Exponentialfunktion.

See Solution

Problem 20810

Bestimmen Sie die Tangentengleichung von $f(x)=\frac{1}{2} \cdot e^{2 x}$ bei $x=0$ .

See Solution

Problem 20811

Berechnen Sie die mittlere Steigung von $f(x)=\frac{4}{x}$ im Intervall $[2 ; 8]$ und von $f(x)=2^{x}$ im Intervall $[1 ; 3]$ .

See Solution

Problem 20812

Gegeben ist die Funktion $f_{r}(x)=\frac{1}{4} x^{4}-\frac{9}{2} r x^{2}+r$ . Bestimmen Sie a) $f_{-2}$ und $f_{\frac{2}{3}}$ , b) die Extremstellen und deren Minima, c) die $y$ -Werte der Extrempunkte und Bedingungen für die $y$ -Achse.

See Solution

Problem 20813

Find the area between the revenue $R(t)=100+10t$ and costs $C(t)=66+5t$ from $t=0$ to $t=5$ .

See Solution

Problem 20814

Gegeben ist die Funktion $f(x)=-x^{3}+3 x^{2}$ . Zeichne die Tangente bei $P(1 \mid 2)$ ein und bestimme die Steigung $m$ durch $f'(1)$ .

See Solution

Problem 20815

Evaluate these limits carefully: i. $\lim _{x \rightarrow \infty} \frac{x^{2}}{2 x^{2}+e^{x}}$ ii. $\lim _{x \rightarrow-\infty} \frac{x^{2}}{2 x^{2}+e^{x}}$ iii. $\lim _{x \rightarrow \infty} \frac{3 e^{2 x}-x-3}{e^{2 x}+1}$ iv. $\lim _{x \rightarrow-\infty} \frac{3 e^{2 x}-x-3}{e^{2 x}+1}$

See Solution

Problem 20816

Construct and simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=9-8 x^{3}$ .

See Solution

Problem 20817

Evaluate the integral using integration by parts: $\int(6-x) 6^{x} d x$ (Include constant $C$ ).

See Solution

Problem 20818

Zeigen Sie, dass die Tangentengleichung an $P\left(x_{0} \mid f\left(x_{0}\right)\right)$ ist $y=f^{\prime}\left(x_{0}\right) \cdot x+f\left(x_{0}\right)-f^{\prime}\left(x_{0}\right) \cdot x_{0}$ . Welche Bedingung muss $f$ bei $x_{0}$ erfüllen?

See Solution

Problem 20819

Find $\lim _{x \rightarrow 1}(x+1) f(x)$ given $f(x)=f(3 x+1)$ and $\lim _{x \rightarrow 4} \frac{f(x)-x^{2}+13}{x-4}=12$ . Answer: 6.

See Solution

Problem 20820

Find the derivative of the function $f(x)=3x^{2}+3x-2$ , denoted as $f^{\prime}(x)$ .

See Solution

Problem 20821

Evaluate the integral using integration by parts: $\int x(x+4)^{6} dx$ . Use $C$ for the constant of integration.

See Solution

Problem 20822

Find the limit of the population function $P(t)=\frac{300 t}{2 t^{2}+8}$ as $t \rightarrow \infty$ and graph it.

See Solution

Problem 20823

Evaluate the integral using integration by parts: $\int x^{5} \ln (x) d x$ (include constant $C$ ).

See Solution

Problem 20824

Find the limit: $\lim _{x \rightarrow 1}(x+1) f(x)$ given $f(x)=f(3x+1)$ and $\lim _{x \rightarrow 4} \frac{f(x)-x^{2}+13}{x-4}=12$ .

See Solution

Problem 20825

Find the area between the marginal cost $C(x)=0.01 x^{2}-3 x+229$ and marginal revenue $R(x)=429-2 x$ curves from $x=0$ .

See Solution

Problem 20826

Find the series expansion for $F(x)=-5+\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n+2}}{(2 n+1) !(2 n+2)}$ .

See Solution

Problem 20827

Determine where the function $f(x)=(x^{2}-4)^{2}$ is decreasing.

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Problem 20828

Calculate the area between the curve $y=3 x \ln (x)$ , the $x$ -axis, $x=1$ , and $x=e$ .

See Solution

Problem 20829

Finde die Ableitung von $f(x) = 3x^2 + 3x - 2$ .

See Solution

Problem 20830

Evaluate these limits and justify your answers: (a) $\lim _{x \rightarrow \infty} \frac{\ln x+\sqrt[10]{x}}{x^{0.1}}$ (b) $\lim _{x \rightarrow \infty} \frac{\ln x+0.99^{x}}{\sqrt[100]{x}}$ (c) $\lim _{x \rightarrow \infty} \frac{\ln x+1.01^{x}}{x^{5}}$

See Solution

Problem 20831

Invest \ $13,484 at 6.1% interest compounded continuously. Find the function$ P(t)$, balances for 1, 2, 5, 10 years, and doubling time.

See Solution

Problem 20832

Determine where the function $f(x)=5 x^{3}+9 x^{2}-7 x+3$ is increasing.

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Problem 20833

Prove that $\frac{d}{d x}(\csc x)=-\csc x \cot x$ using definitions: $\cot x=1 / \tan x$ , $\csc x=1 / \sin x$ , $\sec x=1 / \cos x$ .

See Solution

Problem 20834

Bestimme die Tangentengleichung $\mathrm{t}$ an den Graphen von $\mathrm{f}$ im Punkt $P$ und wo sie die $x$ -Achse schneidet.

See Solution

Problem 20835

Berechne die Ableitung von $f(x) = \frac{x}{2} + 2$ .

See Solution

Problem 20836

Determine if $f(x)=\frac{x^{4}}{e^{x}}$ has a global max or min on $(-\infty, \infty)$ . Justify your answer.

See Solution

Problem 20837

Differentiate $\cos^{5}(x^{2}-3x)$ .

See Solution

Problem 20838

Umformen und Ableiten der Funktionen: a) $f(x)=x \cdot (x^{2}+2x-3)$ , $f^{\prime}(x)=$ b) $f(x)=(x+1) \cdot (x-2)$ , $f^{\prime}(x)=$ c) $f(x)=(2x+0.5)^{2}$ , $f^{\prime}(x)=$

See Solution

Problem 20839

Given $f(-3) = 0$ , $f'(-3) = 3$ , $f''(-3) = -6$ , $f'''(-3) = 9$ , $f^{(4)}(-3) = -12$ , $f^{(5)}(-3) = 15$ , $f^{(6)}(-3) = -18$ , find $f(x)$ .

See Solution

Problem 20840

Bestimmen Sie die Ableitung $f^{\prime}(x)$ für die Funktionen: a) $f(x)=a x^{2}$ , b) $f(x)=x^{2}+a$ , c) $f(x)=\frac{x^{2}}{a}$ , d) $f(x)=a^{2} x$ , e) $f(x)=a x^{2}+b x$ , f) $f(x)=a x^{2}+b$ , g) $f(x)=b x^{2}+a$ , h) $f(x)=a x^{3}+a^{3} x$ .

See Solution

Problem 20841

Find the difference quotient for $f(x)=x^{2}+4 x-6$ . Choose from: 1) $2 x+h+4$ , 2) $8 x-12$ , 3) $10 x+h-12$ , 4) $\frac{2 h x+h^{2}+8 x-12}{h}$ .

See Solution

Problem 20842

Find the derivatives: $f(x) = 3x^{2} + 4x - 3$ , $f(x) = 2x - 1$ , $f(x) = 8x + 2$ .

See Solution

Problem 20843

An object at $130^{\circ} \mathrm{F}$ cools in water at $40^{\circ} \mathrm{F}$ . Find $F(t)$ with cooling constant $k=0.2$ . $F(t)=$

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Problem 20844

Skizziere den Graphen der Funktion $f(x)=1-x^{2}$ im Intervall $[0,1]$ und teile es in 4 gleich große Teile. Berechne $U_{4}$ und $O_{4}$ .

See Solution

Problem 20845

How long to double \$1000 at 8.5% interest, compounded continuously? Answer: 8.15 years.

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Problem 20846

Find the horizontal asymptote of $y(t)=\frac{0.8 t+1000}{5 t+4}, t \geq 15$ and interpret its meaning.

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Problem 20847

Find the derivative of $\ln \left[(4 x^{2})(3 x^{4}-6)\right]$ with respect to $x$ .

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Problem 20848

Find intervals for the function $y=\cos (x-\pi)$ on $[0,4 \pi]$ where the average rate of change is negative and when the instantaneous rate is zero.

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Problem 20849

Find the integral of $\frac{1}{(4x^{2}+9)^{2}}$ .

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Problem 20850

A lake has 400 fish. After 5 months, find $P(t)$ and state the biologist's concern about reaching 2500 fish.

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Problem 20851

Find the derivative of the function $f(z)=7 e^{z-5}$ . What is $\frac{d f}{d z}=?$

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Problem 20852

A student invests \$ 2500 at a 1.3% interest rate compounded continuously. How long to reach \$ 3300?

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Problem 20853

Find a closed form for the derivatives $f^{(n)}(4)$ given: $f(4) = -4$ , $f'(4) = 16$ , $f''(4) = -64$ , $f'''(4) = 256$ , $f^{(4)}(4) = -1024$ .

See Solution

Problem 20854

Given $y=\cos (x-\pi)$ for $x \in[0,4 \pi]$ , find: a) Interval with negative average rate of change (sketch needed) b) $\mathrm{x}$ -value(s) where instantaneous rate of change is zero (use sketch)

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Problem 20855

Berechne die Fläche $F$ zwischen der Funktion $f(x)=-0,25 x^{2}+1,5 x-3$ und der $x$ -Achse im Intervall $I[-1,6]$ .

See Solution

Problem 20856

Find where the function $f(x)=-x^{3}-2 e^{x-1}+0.8(x+3)^{2}$ is decreasing. Round to two decimal places.

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Problem 20857

Find where the function $f(x)=\frac{1}{3} x^{3}+x^{2}-x+3$ is increasing. Round to four decimal places.

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Problem 20858

Find the relative minimum values of $f(x)=0.02 x^{4}-0.05 x^{3}-0.75 x^{2}+x+2$ . Round to four decimal places. Options: 1) $-5.6830,-5.4732$ 2) $-3.8300,5.1064$ 3) $-3.8457,5.0810$ 4) $-5.7194,-5.5102$ 5) None.

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Problem 20859

Rewrite $y=\sec(\tan x)$ as $y=f(u)$ and $u=g(x)$ , then find $\frac{dy}{dx}$ . What are $f(u)$ and $g(x)$ ?

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Problem 20860

Express $y=e^{-6 x}$ as $y=f(u)$ and $u=g(x)$ , then find $\frac{d y}{d x}$ . Which option is correct? A. $y=6 u, u=e^{-x}$ B. $y=-6 u, u=e^{x}$ C. $y=e^{u}, u=-6 x$ D. $y=-e^{u}, u=6 x$

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Problem 20861

Find the derivative of $y=(\csc x-\cot x)^{-1}$ . What is $\frac{\mathrm{dy}}{\mathrm{dx}}$ ?

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Problem 20862

Find the derivative of $y=\sqrt{-8+3x}$ . What is $\frac{d y}{d x}$ ?

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Problem 20863

Deposit \$2200 yearly at 8% interest compounded continuously. Find future value after 10 years and total interest earned.

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Problem 20864

Fox population grows at $9\%$ yearly from $15,000$ in 2013.
(a) Model: $n(t)=15,000 e^{0.09 t}$ . (b) Estimate population in 2020. (c) Years to reach $22,000$ ?

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Problem 20865

What does $100+\int_{0}^{15} n^{\prime}(t) d t$ represent for a bee population starting with 100 bees?

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Problem 20866

Find the derivative of $f(x)=\sqrt{9+x \tan x}$ . What is $\frac{d}{d x} \sqrt{9+x \tan x}=\square$ ?

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Problem 20867

Find the value(s) of $c$ from the Mean Value Theorem for Integrals for these functions on their intervals: 43. $f(x)=x-2\sqrt{x}$ , $[0,2]$ ; 44. $f(x)=\frac{9}{x^{3}}$ , $[1,3]$ .

See Solution

Problem 20868

Find the derivative of $y=3 x e^{-6 x}-3 e^{x^{2}}$ . What is $\frac{\mathrm{dy}}{\mathrm{dx}}$ ?

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Problem 20869

Find the bacteria population after 24 hours with growth rates of $15\%$ (no antibiotic) and $8\%$ (with antibiotic) from 22 initial bacteria. Round to whole numbers.

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Problem 20870

Find the time $t$ in years for \ $5500 at 10\% interest to accumulate to \$11000 using$ A=Pe^{rt}$.

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Problem 20871

An object with velocity $v(t)=t(9-t)$ starts at -2 units. Find its position 18 seconds later.

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Problem 20872

Find the average value of $f(x)=4-x^{2}$ over $[-2,2]$ and where it equals this average. Then do the same for $f(x)=\frac{4\left(x^{2}+1\right)}{x^{2}}$ over $[1,3]$ .

See Solution

Problem 20873

Find the derivative of $q=\cos \left(\frac{t}{\sqrt{t+5}}\right)$ and express it as $\frac{d q}{d t}=\square$ .

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Problem 20874

Sketch the graph of $R$ . For the fish population $P(t)=\frac{2600}{1+12 e^{-\frac{t}{5}}$: a. Initial fish count? b. Population limit as $t \rightarrow \infty$? c. Years to reach 1000 fish?

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Problem 20875

Find the initial amount and half-life of the isotope given $A(t)=700 e^{-0.02827 t}$ . Initial amount is 700g; half-life is $\square$ years.

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Problem 20876

A bowl of soup cools according to $T(t)=51+137 e^{-0.05 t}$ . Find the initial temp, temp after 10 min, and time to reach $100^{\circ} F$ .

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Problem 20877

Find $f(x)$ given $f(1)=1$ and the tangent line at $(x, f(x))$ has slope $\frac{5}{x}$ . What is $f(x)$ ?

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Problem 20878

Radium-221 has a half-life of $30 \mathrm{sec}$ . How long for $92\%$ of a sample to decay? Round to nearest whole number. $\mathrm{sec}$

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Problem 20879

Check if $f(x)=\cos(3x)$ meets Rolle's Theorem conditions on $[\frac{\pi}{12}, \frac{7\pi}{12}]$ and find $c$ .

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Problem 20880

Given the formula $A(t)=700 e^{-0.02838 t}$ , find the initial amount and half-life of the isotope.

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Problem 20881

Find $V(t)$ , the total hours of video uploaded by time $t$ , given $v(t)=1.1 t^{2}-2.6 t+2.3$ .

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Problem 20882

Given the function $f(x)=x^{3}-4 x^{2}-16 x+2$ on $[-4,4]$ , is $f$ continuous? Find $f^{\prime}(x)$ , $f(-4)$ , and $f(4)$ . Can Rolle's theorem apply? If yes, find $c$ where $f^{\prime}(c)=0$ .

See Solution

Problem 20883

Find the derivative of $y=\cos \left(e^{-4 \theta^{3}}\right)$ . What is $\frac{d y}{d \theta}$ ?

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Problem 20884

Find the function $f(x)$ where $f(1)=1$ and the tangent line at $(x, f(x))$ has slope $\frac{7}{x}$ .

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Problem 20885

Given $f(x, y)=\frac{x+1}{y+1}$ , find and simplify: a. $f_{x}$ , b. $f_{y}$ , c. $f_{x x}$ , d. $f_{y y}$ , e. $f_{x y}$ .

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Problem 20886

Find $(f \circ g)^{\prime}(1)$ where $f(u)=u^{5}+9$ , $u=g(x)=\sqrt{x}$ , and $x=1$ .

See Solution

Problem 20887

Find the initial amount and half-life of a radioactive isotope given by $A(t)=500 e^{-0.02839 t}$ .

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Problem 20888

Estimate the integral $\int_{0}^{50} f(t) \, dt$ using the emissions data for nitrogen oxides from 1940 to 1990.

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Problem 20889

Find the tangent line to $y=\sqrt{x^{2}-x+10}$ at $x=3$ . What is the equation of the tangent line?

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Problem 20890

Find the 9th-degree Taylor polynomial $p_{9}(x)$ at 0 for the solution of $y'' - x^{2} y = 0$ with $y(0)=-1, y'(0)=1$ .

See Solution

Problem 20891

Evaluate the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}+5}}{\sqrt{5 x^{2}+1}}$

See Solution

Problem 20892

Estimate the slope of the tangent line to $f(x)=\frac{x}{x-2}$ at $x=3$ .

See Solution

Problem 20893

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $8 x^{2} y + 3 x y^{2} = -4$ .

See Solution

Problem 20894

Find the min and max values of $f(x)=x^{3}-6x^{2}+9x+7$ on the interval $[-1,9]$ .

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Problem 20895

An athlete needs 20 calories/pound/day. If intake is $H$ , find the differential equation for weight $W(t)$ and its limit as $t \to \infty$ for $W(0)=170$ , $H=3700$ .

See Solution

Problem 20896

Find the intervals for population $P$ where it is increasing ( $<P<$ ) and decreasing ( $P>$ ) in the logistic model $\frac{d P}{d t}=\frac{8}{500} P(5-P)$ .

See Solution

Problem 20897

Find the average value of $f(x)$ over the interval and where $f(x)$ equals that average for: 47. $f(x)=4-x^{2}, \quad[-2,2]$ 48. $f(x)=\frac{4\left(x^{2}+1\right)}{x^{2}},[1,3]$ 49. $f(x)=\sin x,[0, \pi]$ 50. $f(x)=\cos x,[0, \pi / 2]$

See Solution

Problem 20898

Dead leaves accumulate at 4 g/cm²/yr and decompose at 75%/yr.
A. Write the differential equation for $Q$ : $\frac{d Q}{d t}=$
B. Sketch the solution showing equilibrium.
Initial leaves: $Q(0)=$ Equilibrium level: $Q_{eq}=$ Does equilibrium depend on initial condition? A. yes B. no

See Solution

Problem 20899

Find $\frac{\mathrm{dy}}{\mathrm{dx}}$ using implicit differentiation for the equation $x^{4}-16 x y+y^{4}=1$ .

See Solution

Problem 20900

Find equilibrium solutions for $\frac{d P}{d t}=\frac{1}{2} P$ and $\frac{d P}{d t}=\frac{1}{2} P(3-P)$ . Determine stability.

See Solution

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Calculus

Problem 20801

Evaluate the integral: ∫5x2+2x−5x3−xdx\int \frac{5 x^{2}+2 x-5}{x^{3}-x} d x∫x3−x5x2+2x−5​dx

Problem 20802

Use substitution to simplify ∫(sec⁡2x)ln⁡(tan⁡x+8) dx\int (\sec^2 x) \ln(\tan x + 8) \, dx∫(sec2x)ln(tanx+8)dx to ∫ln⁡u du\int \ln u \, du∫lnudu and evaluate it.

Problem 20803

Problem 20804

Use substitution to simplify ∫(cos⁡x)ln⁡(sin⁡x+2)dx\int(\cos x) \ln (\sin x+2) d x∫(cosx)ln(sinx+2)dx to ∫ln⁡u du\int \ln u \, du∫lnudu and evaluate it.

Problem 20805

Gegeben sind f(x)=2xf(x)=2^{x}f(x)=2x und g(x)=−2x+4g(x)=-2x+4g(x)=−2x+4. Skizzieren Sie die Graphen und zeigen Sie, dass sie bei S(1∣2)S(1|2)S(1∣2) schneiden. Berechnen Sie die Fläche zwischen den Graphen und den Achsen.

Problem 20806

Calculate the value of ∫23x+x25dx−∫23x−x25dx+2⋅∫23x25dx\int_{2}^{3} \frac{x+x^{2}}{5} dx - \int_{2}^{3} \frac{x-x^{2}}{5} dx + 2 \cdot \int_{2}^{3} \frac{x^{2}}{5} dx∫23​5x+x2​dx−∫23​5x−x2​dx+2⋅∫23​5x2​dx.

Problem 20807

Bestimmen Sie die durchschnittliche Steigung der Funktion f(x)=2xf(x)=2xf(x)=2x im Intervall I=[0;1]I=[0 ; 1]I=[0;1].

Problem 20808

Bestimme die Ableitung von f(x)=3⋅2xf(x)=3 \cdot 2^{x}f(x)=3⋅2x und schreibe sie als allgemeine Exponentialfunktion.

Problem 20809

Bestimmen Sie die Ableitung von f(x)=3⋅2xf(x)=3 \cdot 2^{x}f(x)=3⋅2x und formulieren Sie das Ergebnis als allgemeine Exponentialfunktion.

Problem 20810

Bestimmen Sie die Tangentengleichung von f(x)=12⋅e2xf(x)=\frac{1}{2} \cdot e^{2 x}f(x)=21​⋅e2x bei x=0x=0x=0.

Problem 20811

Berechnen Sie die mittlere Steigung von f(x)=4xf(x)=\frac{4}{x}f(x)=x4​ im Intervall [2;8][2 ; 8][2;8] und von f(x)=2xf(x)=2^{x}f(x)=2x im Intervall [1;3][1 ; 3][1;3].

Problem 20812

Problem 20813

Find the area between the revenue R(t)=100+10tR(t)=100+10tR(t)=100+10t and costs C(t)=66+5tC(t)=66+5tC(t)=66+5t from t=0t=0t=0 to t=5t=5t=5.

Problem 20814

Gegeben ist die Funktion f(x)=−x3+3x2f(x)=-x^{3}+3 x^{2}f(x)=−x3+3x2. Zeichne die Tangente bei P(1∣2)P(1 \mid 2)P(1∣2) ein und bestimme die Steigung mmm durch f′(1)f'(1)f′(1).

Problem 20815

Problem 20816

Construct and simplify the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=9−8x3f(x)=9-8 x^{3}f(x)=9−8x3.

Problem 20817

Evaluate the integral using integration by parts: ∫(6−x)6xdx\int(6-x) 6^{x} d x∫(6−x)6xdx (Include constant CCC).

Problem 20818

Problem 20819

Find lim⁡x→1(x+1)f(x)\lim _{x \rightarrow 1}(x+1) f(x)limx→1​(x+1)f(x) given f(x)=f(3x+1)f(x)=f(3 x+1)f(x)=f(3x+1) and lim⁡x→4f(x)−x2+13x−4=12\lim _{x \rightarrow 4} \frac{f(x)-x^{2}+13}{x-4}=12limx→4​x−4f(x)−x2+13​=12. Answer: 6.

Problem 20820

Find the derivative of the function f(x)=3x2+3x−2f(x)=3x^{2}+3x-2f(x)=3x2+3x−2, denoted as f′(x)f^{\prime}(x)f′(x).

Problem 20821

Evaluate the integral using integration by parts: ∫x(x+4)6dx\int x(x+4)^{6} dx∫x(x+4)6dx. Use CCC for the constant of integration.

Problem 20822

Find the limit of the population function P(t)=300t2t2+8P(t)=\frac{300 t}{2 t^{2}+8}P(t)=2t2+8300t​ as t→∞t \rightarrow \inftyt→∞ and graph it.

Problem 20823

Evaluate the integral using integration by parts: ∫x5ln⁡(x)dx\int x^{5} \ln (x) d x∫x5ln(x)dx (include constant CCC).

Problem 20824

Find the limit: lim⁡x→1(x+1)f(x)\lim _{x \rightarrow 1}(x+1) f(x)x→1lim​(x+1)f(x) given f(x)=f(3x+1)f(x)=f(3x+1)f(x)=f(3x+1) and lim⁡x→4f(x)−x2+13x−4=12\lim _{x \rightarrow 4} \frac{f(x)-x^{2}+13}{x-4}=12limx→4​x−4f(x)−x2+13​=12.

Problem 20825

Find the area between the marginal cost C(x)=0.01x2−3x+229C(x)=0.01 x^{2}-3 x+229C(x)=0.01x2−3x+229 and marginal revenue R(x)=429−2xR(x)=429-2 xR(x)=429−2x curves from x=0x=0x=0.

Problem 20826

Find the series expansion for F(x)=−5+∑n=0∞(−1)nx2n+2(2n+1)!(2n+2)F(x)=-5+\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n+2}}{(2 n+1) !(2 n+2)}F(x)=−5+∑n=0∞​(2n+1)!(2n+2)(−1)nx2n+2​.

Problem 20827

Determine where the function f(x)=(x2−4)2f(x)=(x^{2}-4)^{2}f(x)=(x2−4)2 is decreasing.

Problem 20828

Calculate the area between the curve y=3xln⁡(x)y=3 x \ln (x)y=3xln(x), the xxx-axis, x=1x=1x=1, and x=ex=ex=e.

Problem 20829

Finde die Ableitung von f(x)=3x2+3x−2f(x) = 3x^2 + 3x - 2f(x)=3x2+3x−2.

Problem 20830

Problem 20831

Invest \13,484at6.113,484 at 6.1% interest compounded continuously. Find the function 13,484at6.1P(t)$, balances for 1, 2, 5, 10 years, and doubling time.

Problem 20832

Determine where the function f(x)=5x3+9x2−7x+3f(x)=5 x^{3}+9 x^{2}-7 x+3f(x)=5x3+9x2−7x+3 is increasing.

Problem 20833

Prove that ddx(csc⁡x)=−csc⁡xcot⁡x\frac{d}{d x}(\csc x)=-\csc x \cot xdxd​(cscx)=−cscxcotx using definitions: cot⁡x=1/tan⁡x\cot x=1 / \tan xcotx=1/tanx, csc⁡x=1/sin⁡x\csc x=1 / \sin xcscx=1/sinx, sec⁡x=1/cos⁡x\sec x=1 / \cos xsecx=1/cosx.

Problem 20834

Bestimme die Tangentengleichung t\mathrm{t}t an den Graphen von f\mathrm{f}f im Punkt PPP und wo sie die xxx-Achse schneidet.

Problem 20835

Berechne die Ableitung von f(x)=x2+2 f(x) = \frac{x}{2} + 2 f(x)=2x​+2.

Problem 20836

Determine if f(x)=x4exf(x)=\frac{x^{4}}{e^{x}}f(x)=exx4​ has a global max or min on (−∞,∞)(-\infty, \infty)(−∞,∞). Justify your answer.

Problem 20837

Differentiate cos⁡5(x2−3x)\cos^{5}(x^{2}-3x)cos5(x2−3x).

Problem 20838

Problem 20839

Problem 20840

Problem 20841

Find the difference quotient for f(x)=x2+4x−6f(x)=x^{2}+4 x-6f(x)=x2+4x−6. Choose from: 1) 2x+h+42 x+h+42x+h+4, 2) 8x−128 x-128x−12, 3) 10x+h−1210 x+h-1210x+h−12, 4) 2hx+h2+8x−12h\frac{2 h x+h^{2}+8 x-12}{h}h2hx+h2+8x−12​.

Problem 20842

Find the derivatives: f(x)=3x2+4x−3f(x) = 3x^{2} + 4x - 3f(x)=3x2+4x−3, f(x)=2x−1f(x) = 2x - 1f(x)=2x−1, f(x)=8x+2f(x) = 8x + 2f(x)=8x+2.

Problem 20843

An object at 130∘F130^{\circ} \mathrm{F}130∘F cools in water at 40∘F40^{\circ} \mathrm{F}40∘F. Find F(t)F(t)F(t) with cooling constant k=0.2k=0.2k=0.2. F(t)= F(t)= F(t)=

Problem 20844

Evaluate the integral: $\int \frac{5 x^{2}+2 x-5}{x^{3}-x} d x$

Use substitution to simplify $\int (\sec^2 x) \ln(\tan x + 8) \, dx$ to $\int \ln u \, du$ and evaluate it.

Use substitution to simplify $\int(\cos x) \ln (\sin x+2) d x$ to $\int \ln u \, du$ and evaluate it.

Gegeben sind $f(x)=2^{x}$ und $g(x)=-2x+4$ . Skizzieren Sie die Graphen und zeigen Sie, dass sie bei $S(1|2)$ schneiden. Berechnen Sie die Fläche zwischen den Graphen und den Achsen.

Calculate the value of $\int_{2}^{3} \frac{x+x^{2}}{5} dx - \int_{2}^{3} \frac{x-x^{2}}{5} dx + 2 \cdot \int_{2}^{3} \frac{x^{2}}{5} dx$ .

Bestimmen Sie die durchschnittliche Steigung der Funktion $f(x)=2x$ im Intervall $I=[0 ; 1]$ .

Bestimme die Ableitung von $f(x)=3 \cdot 2^{x}$ und schreibe sie als allgemeine Exponentialfunktion.

Bestimmen Sie die Ableitung von $f(x)=3 \cdot 2^{x}$ und formulieren Sie das Ergebnis als allgemeine Exponentialfunktion.

Bestimmen Sie die Tangentengleichung von $f(x)=\frac{1}{2} \cdot e^{2 x}$ bei $x=0$ .

Berechnen Sie die mittlere Steigung von $f(x)=\frac{4}{x}$ im Intervall $[2 ; 8]$ und von $f(x)=2^{x}$ im Intervall $[1 ; 3]$ .

Find the area between the revenue $R(t)=100+10t$ and costs $C(t)=66+5t$ from $t=0$ to $t=5$ .

Gegeben ist die Funktion $f(x)=-x^{3}+3 x^{2}$ . Zeichne die Tangente bei $P(1 \mid 2)$ ein und bestimme die Steigung $m$ durch $f'(1)$ .

Construct and simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=9-8 x^{3}$ .

Evaluate the integral using integration by parts: $\int(6-x) 6^{x} d x$ (Include constant $C$ ).

Find $\lim _{x \rightarrow 1}(x+1) f(x)$ given $f(x)=f(3 x+1)$ and $\lim _{x \rightarrow 4} \frac{f(x)-x^{2}+13}{x-4}=12$ . Answer: 6.

Find the derivative of the function $f(x)=3x^{2}+3x-2$ , denoted as $f^{\prime}(x)$ .

Evaluate the integral using integration by parts: $\int x(x+4)^{6} dx$ . Use $C$ for the constant of integration.

Find the limit of the population function $P(t)=\frac{300 t}{2 t^{2}+8}$ as $t \rightarrow \infty$ and graph it.

Evaluate the integral using integration by parts: $\int x^{5} \ln (x) d x$ (include constant $C$ ).

Find the limit: $\lim _{x \rightarrow 1}(x+1) f(x)$ given $f(x)=f(3x+1)$ and $\lim _{x \rightarrow 4} \frac{f(x)-x^{2}+13}{x-4}=12$ .

Find the area between the marginal cost $C(x)=0.01 x^{2}-3 x+229$ and marginal revenue $R(x)=429-2 x$ curves from $x=0$ .

Find the series expansion for $F(x)=-5+\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n+2}}{(2 n+1) !(2 n+2)}$ .

Determine where the function $f(x)=(x^{2}-4)^{2}$ is decreasing.

Calculate the area between the curve $y=3 x \ln (x)$ , the $x$ -axis, $x=1$ , and $x=e$ .

Finde die Ableitung von $f(x) = 3x^2 + 3x - 2$ .

Invest \ $13,484 at 6.1% interest compounded continuously. Find the function$ P(t)$, balances for 1, 2, 5, 10 years, and doubling time.

Determine where the function $f(x)=5 x^{3}+9 x^{2}-7 x+3$ is increasing.

Prove that $\frac{d}{d x}(\csc x)=-\csc x \cot x$ using definitions: $\cot x=1 / \tan x$ , $\csc x=1 / \sin x$ , $\sec x=1 / \cos x$ .

Bestimme die Tangentengleichung $\mathrm{t}$ an den Graphen von $\mathrm{f}$ im Punkt $P$ und wo sie die $x$ -Achse schneidet.

Berechne die Ableitung von $f(x) = \frac{x}{2} + 2$ .

Determine if $f(x)=\frac{x^{4}}{e^{x}}$ has a global max or min on $(-\infty, \infty)$ . Justify your answer.

Differentiate $\cos^{5}(x^{2}-3x)$ .

Find the difference quotient for $f(x)=x^{2}+4 x-6$ . Choose from: 1) $2 x+h+4$ , 2) $8 x-12$ , 3) $10 x+h-12$ , 4) $\frac{2 h x+h^{2}+8 x-12}{h}$ .

Find the derivatives: $f(x) = 3x^{2} + 4x - 3$ , $f(x) = 2x - 1$ , $f(x) = 8x + 2$ .

An object at $130^{\circ} \mathrm{F}$ cools in water at $40^{\circ} \mathrm{F}$ . Find $F(t)$ with cooling constant $k=0.2$ . $F(t)=$

Skizziere den Graphen der Funktion $f(x)=1-x^{2}$ im Intervall $[0,1]$ und teile es in 4 gleich große Teile. Berechne $U_{4}$ und $O_{4}$ .

Find the horizontal asymptote of $y(t)=\frac{0.8 t+1000}{5 t+4}, t \geq 15$ and interpret its meaning.

Find the derivative of $\ln \left[(4 x^{2})(3 x^{4}-6)\right]$ with respect to $x$ .

Find intervals for the function $y=\cos (x-\pi)$ on $[0,4 \pi]$ where the average rate of change is negative and when the instantaneous rate is zero.

Find the integral of $\frac{1}{(4x^{2}+9)^{2}}$ .

A lake has 400 fish. After 5 months, find $P(t)$ and state the biologist's concern about reaching 2500 fish.

Find the derivative of the function $f(z)=7 e^{z-5}$ . What is $\frac{d f}{d z}=?$

Find a closed form for the derivatives $f^{(n)}(4)$ given: $f(4) = -4$ , $f'(4) = 16$ , $f''(4) = -64$ , $f'''(4) = 256$ , $f^{(4)}(4) = -1024$ .

Given $y=\cos (x-\pi)$ for $x \in[0,4 \pi]$ , find: a) Interval with negative average rate of change (sketch needed) b) $\mathrm{x}$ -value(s) where instantaneous rate of change is zero (use sketch)

Berechne die Fläche $F$ zwischen der Funktion $f(x)=-0,25 x^{2}+1,5 x-3$ und der $x$ -Achse im Intervall $I[-1,6]$ .

Find where the function $f(x)=-x^{3}-2 e^{x-1}+0.8(x+3)^{2}$ is decreasing. Round to two decimal places.

Find where the function $f(x)=\frac{1}{3} x^{3}+x^{2}-x+3$ is increasing. Round to four decimal places.

Rewrite $y=\sec(\tan x)$ as $y=f(u)$ and $u=g(x)$ , then find $\frac{dy}{dx}$ . What are $f(u)$ and $g(x)$ ?

Find the derivative of $y=(\csc x-\cot x)^{-1}$ . What is $\frac{\mathrm{dy}}{\mathrm{dx}}$ ?

Find the derivative of $y=\sqrt{-8+3x}$ . What is $\frac{d y}{d x}$ ?

Fox population grows at $9\%$ yearly from $15,000$ in 2013.
(a) Model: $n(t)=15,000 e^{0.09 t}$ . (b) Estimate population in 2020. (c) Years to reach $22,000$ ?

What does $100+\int_{0}^{15} n^{\prime}(t) d t$ represent for a bee population starting with 100 bees?

Find the derivative of $f(x)=\sqrt{9+x \tan x}$ . What is $\frac{d}{d x} \sqrt{9+x \tan x}=\square$ ?

Find the value(s) of $c$ from the Mean Value Theorem for Integrals for these functions on their intervals: 43. $f(x)=x-2\sqrt{x}$ , $[0,2]$ ; 44. $f(x)=\frac{9}{x^{3}}$ , $[1,3]$ .

Find the derivative of $y=3 x e^{-6 x}-3 e^{x^{2}}$ . What is $\frac{\mathrm{dy}}{\mathrm{dx}}$ ?

Find the bacteria population after 24 hours with growth rates of $15\%$ (no antibiotic) and $8\%$ (with antibiotic) from 22 initial bacteria. Round to whole numbers.

Find the time $t$ in years for \ $5500 at 10\% interest to accumulate to \$11000 using$ A=Pe^{rt}$.

An object with velocity $v(t)=t(9-t)$ starts at -2 units. Find its position 18 seconds later.

Find the average value of $f(x)=4-x^{2}$ over $[-2,2]$ and where it equals this average. Then do the same for $f(x)=\frac{4\left(x^{2}+1\right)}{x^{2}}$ over $[1,3]$ .

Find the derivative of $q=\cos \left(\frac{t}{\sqrt{t+5}}\right)$ and express it as $\frac{d q}{d t}=\square$ .

Sketch the graph of $R$ . For the fish population $P(t)=\frac{2600}{1+12 e^{-\frac{t}{5}}$: a. Initial fish count? b. Population limit as $t \rightarrow \infty$? c. Years to reach 1000 fish?