Calculus

Problem 15001

Find the rate of change of $y$ where $y=\log(2x+1)$ with respect to $x$ .

See Solution

Problem 15002

Evaluate the integral from 4 to 5 of $(2+2y)^{2} dy$ . Enter a fraction, integer, or exact decimal.

See Solution

Problem 15003

Find the average rate of change in value per year of the vehicle from years 5 to 10 for $f(x)=4500(0.98)^{x}$ .

See Solution

Problem 15004

Evaluate $\int_{0}^{2} g(t) dt$ and $\int_{1}^{5} g(t) dt$ for the function $g(t)$ with given intercepts.

See Solution

Problem 15005

Analyze the function $f(x)=\sin (x)-\frac{1}{3} \sin (3 x)+\frac{1}{5} \sin (5 x)$ over $[0, \pi]$ .
(a) Find $f'(x)$ and $f''(x)$ .
(b) Determine relative extrema and points of inflection (round to four decimal places).

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Problem 15006

Determine the positivity/negativity of $f(x)=18-2 x^{2}$ on $[0,4]$ ; sketch it and find the Riemann sums with $n=4$ .

See Solution

Problem 15007

Find the length $x$ and width $y$ of a rectangle with area maximum, bounded by $y=\frac{6-x}{2}$ .

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Problem 15008

Evaluate the integral $\int_{0}^{6}(0.6 x^{2}-3) dx$ for $n=4$ ; find $\Delta x$ and grid points.

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Problem 15009

Evaluate the integral $\int_{0}^{6}(0.6 x^{2}-3) d x$ for $n=4$ : graph, find $\Delta x$ , grid points, and Riemann sums.

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Problem 15010

Given the function $f(x)=18-2 x^{2}$ on $[0,4]$ , do the following:
a. Sketch $f(x)$ . b. Find net area using left, right, and midpoint Riemann sums with $n=4$ . c. Identify intervals contributing positively/negatively to net area.
Approximate values: Left Riemann sum = 44, Right = 12, Midpoint = $\square$ .

See Solution

Problem 15011

Find $\frac{d y}{d x}$ if $y=\ln(2 x^{2}-3 y^{2})$ . Choose from the options given.

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Problem 15012

Evaluate the integral $\int_{0}^{6}(0.6 x^{2}-3)dx$ with $n=4$ . Find $\Delta x$ , grid points, and Riemann sums.

See Solution

Problem 15013

Find the local min and max of $f(x)=2x^{3}-24x^{2}+72x+6$ by finding its derivative, setting it to zero, and solving for $x$ . Then, substitute $x$ back into $f(x)$ for the values.

See Solution

Problem 15014

Bestimme die Häufungspunkte der Folge $a_{n}=\frac{(-1)^{n} \cdot n^{2}}{(3 \cdot n+3)^{2}}$ . Anzahl und größter Punkt?

See Solution

Problem 15015

Bestimmen Sie die ersten beiden Ableitungen von $9 x \cdot e^{-x}$ .

See Solution

Problem 15016

Find the absolute max and min of $f(x)=4-6 x^{2}$ for $-5 \leq x \leq 1$ . Max is 4 at $x=$ , min is at $x=$ .

See Solution

Problem 15017

Find the max and min of $f(x)=2x^3+12x^2-72x+1$ on $[-6, 3]$ . Use calculus for derivatives.

See Solution

Problem 15018

Bestimmen Sie die Stammfunktionen für die folgenden Funktionen: a) $f(x)=x^{4}$ , b) $f(x)=x^{-2}$ , c) $f(x)=3 x^{4}-2 x-2$ , d) $f(x)=\frac{2}{5} x^{2}+1,2 \frac{1}{x^{2}}$ , e) $f(x)=12 x^{4}$ , f) $f(x)=\frac{2}{3} x^{-5}$ , g) $f(x)=\frac{1}{4} x^{2}$ , h) $f(x)=3 \sin (x)$ .

See Solution

Problem 15019

Evaluate the integral $\int_{0}^{6}(0.6 x^{2}-3) dx$ with $n=4$ :
a. Sketch the graph.
b. Find $\Delta x$ and grid points $x_{0}, x_{1}, \ldots, x_{n}$ .
c. Calculate left and right Riemann sums.

See Solution

Problem 15020

Evaluate the integral $\int_{0}^{6}(0.6 x^{2}-3) d x$ for $n=4$ . Find $\Delta x$ , grid points, and Riemann sums.

See Solution

Problem 15021

Evaluate the integral $\int_{0}^{6}(0.6 x^{2}-3) dx$ with $n=4$ . Find $\Delta x$ , grid points, and Riemann sums.

See Solution

Problem 15022

Find the absolute extremum of the function $f(x)=4 x^{3}-3$ .

See Solution

Problem 15023

Find the limit: $\lim _{n \rightarrow \infty} \frac{n^{2}+(-1)^{n} \cdot n}{(n+\sqrt{3})^{2}}$

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Problem 15024

Kent deposits \ $20,000 at a continuous interest rate of$ 6.21\%$. Will he afford the car in 6 years?

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Problem 15025

A snowball's radius decreases at $0.3 \mathrm{~cm/min}$ . Find the volume decrease rate when the radius is $17 \mathrm{~cm}$ . $\frac{\mathrm{cm}^{3}}{\min}$

See Solution

Problem 15026

Given $x y=3$ and $\frac{d y}{d t}=2$ , find $\frac{d x}{d t}$ when $x=3$ .

See Solution

Problem 15027

A snowball's radius decreases at $0.4 \mathrm{~cm} / \mathrm{min}$ . Find the volume decrease rate when the radius is $14 \mathrm{~cm}$ . Answer in $\frac{\mathrm{cm}^{3}}{\min}$ .

See Solution

Problem 15028

Calculate the average value of the function $f(x)=-\cos x$ from $x=-\frac{\pi}{2}$ to $x=\frac{\pi}{2}$ .

See Solution

Problem 15029

Find the average value of $f(x)=x(x-1)$ over the interval $[5,8]$ .

See Solution

Problem 15030

Graph the integrand for $\int_{1}^{9}\left(\frac{3}{x}+1\right) d x$ with $n=4$ . Find $\Delta x$ , $x_i$ , and Riemann sums.

See Solution

Problem 15031

Find the correct expression for $\frac{d V}{d t}$ and calculate the volume change rate when $r=2.9$ inches and $\frac{d r}{d t}=0.4$ in/s.

See Solution

Problem 15032

A snowball's radius decreases at $0.1 \mathrm{~cm} / \mathrm{min}$ . Find the volume's decrease rate when the radius is $13 \mathrm{~cm}$ . Answer in $\frac{\mathrm{cm}^{3}}{\min }$ .

See Solution

Problem 15033

Find $\frac{d x}{d t}$ given the equation $x y + y^{3} + x^{4} = 4$ , with $y=-2$ , $\frac{d y}{d t}=5$ , and $x=2$ .

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Problem 15034

Gegeben ist $f_{a}(x)=-a x^{3}+4 a x$ . a) Zeigen Sie Punktsymmetrie. b) Bestimmen Sie Punkte $P(-2,0)$ und $Q(2,0)$ . c) Nachweis von Hoch- und Tiefpunkt. d) Finden Sie die Wendetangente mit Steigung $m=8$ .

See Solution

Problem 15035

Find the rate of volume decrease of a snowball when its radius is 18 cm, given the radius decreases at $0.3 \mathrm{~cm/min}$ . $\frac{\mathrm{cm}^{3}}{\min}$

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Problem 15036

Untersuchen Sie das Steigen und Fallen der Funktionen: a) $f(x)=x^{5}$ , b) $f(x)=x^{2}-4 x$ , c) $f(x)=1-x^{3}$ , d) $f(x)=x^{3}-4 x$ , e) $f(x)=1$ , f) $f(x)=\frac{1}{x}$ .

See Solution

Problem 15037

Find $\frac{d x}{d t}$ given the equation $-x y+y^{2}+x^{4}=99$ with $\frac{d y}{d t}=4$ , $y=-3$ , and $x=3$ .

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Problem 15038

An oil spill is a cylinder with height 4 in and radius $r(t)$ . Find $\frac{d V}{d t}$ and volume change at $r=2.4$ in, rate $0.2$ in/s.

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Problem 15039

Evaluate $\frac{d}{d x} \int_{a}^{x} f(t) d t$ and $\frac{d}{d x} \int_{a}^{b} f(t) d t$ . Simplify your answer.

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Problem 15040

Find the antiderivative $F(x)$ for: (a) $f(x)=\sqrt{x}-x^{4}$ , (b) $f(x)=\frac{1}{x}-\frac{1}{x^{2}}$ with $F(-e)=1$ , (c) $f(x)=4 e^{3 x+1}+2$ . Simplify $F(x)$ .

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Problem 15041

Evaluate the area functions:
a. $A(-2)$ , b. $F(8)$ , c. $A(4)$ , d. $F(4)$ , e. $A(8)$ for the function $f$ .

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Problem 15042

Analyze the function $f(x)=2 x-e^{x}$ : (a) Determine where $f$ is increasing or decreasing. (b) Find concavity. (c) Sketch $f$ with local extrema. (d) Count $x$ -intercepts and explain.

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Problem 15043

Find the horizontal asymptote of the function $f(x)=2^{x-1}+3$ .

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Problem 15044

Tomatenpflanze: Gegeben ist $v(t)=-0,1 t^{3}+t^{2}$ . Bestimme $h(t)$ und beantworte: a) Wachstumsdauer? b) Maximalhöhe? c) Höhe bei maximalem Wachstum?

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Problem 15045

Find the second derivative of $f(x) = -x e^{\frac{x}{4}}$ .

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Problem 15046

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

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Problem 15047

Determine where the function $f(x) = x^{4} - 72x^{2}$ is increasing and decreasing. Sketch the graph with horizontal tangents.

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Problem 15048

Find the second derivative of $f(x)=-x e^{\frac{x}{4}-10}$ .

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Problem 15049

Determine if the function $m(x)=3 x^{3}$ is concave up, down, or neither on the interval $(-\infty, 0)$ .

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Problem 15050

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{3x}{x+2}$ , where $h \neq 0$ .

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Problem 15051

e) Betrachte die Schuldenfunktion $S(t)=-0,08 t^{3}+3,5 t^{2}+10,6 t+237$ .
a) Was beschreibt die Ableitung $S'$ ? b) In welchem Jahr war die Neuverschuldung am höchsten? c) Wann wird Nullverschuldung erreicht? d) Welcher Fehler wurde 2005 in der Meldung über sinkende Staatsschulden gemacht?

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Problem 15052

Find the second derivative, $f^{\prime \prime}(x)$ , for the function $f(x)=4 x^{6}-2 x^{2}+6 x-9$ .

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Problem 15053

Find the linearization $L(x)$ of the function $f(x)=\ln \left(x^{2}\right)$ at $x=e$ .

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Problem 15054

Find the second derivative $h^{\prime \prime}(x)$ for $h(x)=9 x^{-5}-8 x^{-7}$ .

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Problem 15055

Find the second derivative $f^{\prime \prime}(x)$ for the function $f(x)=(x^{2}+1)^{7}$ .

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Problem 15056

Zeige, dass die Funktion $f(x)=(x-a)^{3}$ bei $x=a$ eine Wendestelle hat. (3Pkt)

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Problem 15057

Find the inflection points for the function $f(x)=x^{3}+33 x^{2}$ . What are the coordinates? A. $\square$ B. No inflection points.

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Problem 15058

Find the drug concentration $C(20)$ after 20 minutes using $C(t)=0.08(1-e^{-0.2 t})$ . Round to three decimal places.

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Problem 15059

Determine where the graph of $f(x)=x^{20}+8x^{2}$ is concave up, concave down, and find the inflection points.

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Problem 15060

Find the marginal daily production of labor given by $q(n)=12(0.3 n+0.02 n^{2})^{1.6}$ . Calculate $\frac{d q}{d n}$ .

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Problem 15061

A car wash's daily production is $q(n)=12\left(0.3 n+0.02 n^{2}\right)^{1 / 2}$ . Find $\frac{d q}{d n}$ and estimate it for $n=5$ .

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Problem 15062

A computer company found the price-demand relationship $p=1512-0.14 x^{2}$ and revenue $R(x)=1512 x-0.14 x^{3}$ . Find local maxima for $R(x)$ .

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Problem 15063

A computer company found the price-demand relationship as $p=1512-0.14 x^{2}$ for $0<x<90$ . What are the concavity intervals for $R(x)=1512x-0.14x^{3}$ ?

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Problem 15064

A car wash's daily production is $q(n)=12(0.3 n+0.02 n^{2})^{1.6}$ . Find $\frac{d q}{d n}$ and estimate it for $n=5$ .

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Problem 15065

Differentiate the function $\frac{3 x^{2}-1}{2 x+x^{3}}$ with respect to $x$ .

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Problem 15066

Find the derivative of $(3 x^{3}+2)^{5}$ with respect to $x$ .

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Problem 15067

Calculate the derivative $\left.\frac{d y}{d x}\right|_{x=\frac{\pi}{2}}$ for $y=\frac{x}{\cot x}$ .

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Problem 15068

Find the second derivative of $f(x)=(x^{2}+2)^{9}$ . What is $f^{\prime \prime}(x)$ ?

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Problem 15069

Find the derivative $\frac{d y}{d x}$ at $x=\frac{\pi}{4}$ for $y=3 \sin x \cos x + 2 \tan x$ .

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Problem 15070

Find the consumers' surplus when the demand function is $p(x)=\left(-\frac{1}{400}\right) x+12$ and price is 8.

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Problem 15071

Calculate the half-life of I-10 with a decay rate of $8.912\%$ per day.

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Problem 15072

Find the derivative $\frac{dy}{dx}$ using implicit differentiation for $8xy - \frac{y}{8} = \frac{7}{x}$ .

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Problem 15073

Calculate the doubling time for a city with an annual growth rate of $7.5\%$ .

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Problem 15074

Find the derivative using implicit differentiation for $(x y)^{2}+y^{2}=9$ ; compute $\frac{d x}{d y}$ .

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Problem 15075

Find the derivative $\frac{d y}{d x}$ using implicit differentiation for $\ln (x y)-x \ln (y)=6 y$ .

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Problem 15076

Find the limit as $n$ approaches infinity: $\operatorname{Lim}_{n \rightarrow \infty}\left(1-\frac{1}{n^{2}}\right)^{n+1}$ .

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Problem 15077

Find the slope of the tangent line using implicit differentiation for $4 x^{2}+2 y^{2}=12$ at the point (1,-2). $\frac{d y}{d x} =$

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Problem 15078

Find the consumers' surplus when the price is set at 7000 for the demand function $p(x)=10000-0.048 x^{2}$ .

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Problem 15079

Find the linear approximation $L(s)$ for $f=\frac{p(1+s)}{1+s p}$ , where $p$ is constant and $s$ is small.

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Problem 15080

Approximate the gene frequency $f$ in the next generation using a linear approximation $L(s)$ for small $s$ :
$L(s)=$

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Problem 15081

Find the limit of $a_{n}=\left(\frac{n-1}{n+3}\right)^{2 n+1}$ as $n$ approaches infinity.

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Problem 15082

Find $\frac{d y}{d x}$ for $P=x^{0.6} y^{0.4}$ at $x=100$ , $y=500,000$ . Result: $\$ \square$ per worker.

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Problem 15083

What is the temperature of a drink after 50 minutes if it starts at $5^{\circ} \mathrm{C}$ and reaches $10^{\circ} \mathrm{C}$ in 25 minutes?

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Problem 15084

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

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Problem 15085

Find the limit: $\lim _{x \rightarrow \infty}-\frac{3 x^{2}}{4 x+4}$ .

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Problem 15086

Find the limit: $\lim _{x \rightarrow-\infty} \frac{3 x^{3}}{3 x^{2}-2}$ .

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Problem 15087

Show that for any real numbers $a$ and $b$ , $|\sin b - \sin a| \leq |b - a|$ using the Mean Value theorem.

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Problem 15088

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

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Problem 15089

Find the slope of the tangent line for $h(x)=\frac{(3x-4)^{2}}{x}$ at $x=-2$ .

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Problem 15090

Given $z=\frac{x y}{2}$ , with $\frac{d z}{d t}=-12$ , $\frac{d x}{d t}=3$ , $z=4$ , and $y=6$ , find $\frac{d y}{d t}$ .

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Problem 15091

Find the arc length differential for the curve $C$ given by $\vec{r}(t)=\langle e^{t^{2}}, \ln(t+1), 2-t^{3}\rangle$ .

See Solution

Problem 15092

Find $c$ if the rate of change of $f(x)=\sqrt{2x}$ at $x=c$ is 4 times that at $x=1$ .

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Problem 15093

Find the positive value of $a$ where the slope of the tangent to $f(x)=5 e^{3 x^{3}}$ at $(a, f(a))$ equals 6.

See Solution

Problem 15094

Find the $x$ where the tangent line to $y=\frac{\csc ^{2}\left(\frac{x}{2}\right)}{3 x+4}$ is horizontal for $0 \leq x \leq 2 \pi$ .

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Problem 15095

Find the value of $x$ where the tangents of $f(x)=2 e^{3 x}$ and $g(x)=5 x^{3}$ are parallel.

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Problem 15096

Calculate the line integral of $f(x, y, z) = xy^2$ along the path from $(1, 1, 1)$ to $(2, 2, 2)$ to $(-9, 6, 4)$ .

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Problem 15097

Differentiate the function $y=11 \ln \left(x^{23} \sqrt{4 x+5}\right)$ after simplifying using logarithm properties.

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Problem 15098

Find $f'(3)$ for $f(x)=x \cdot g(h(x))$ given $g(4)=2$ , $g'(4)=3$ , $h(3)=4$ , and $h'(3)=-2$ .

See Solution

Problem 15099

Find the critical numbers of $f(x)=x+\sin(2x)$ for $0 \leq x \leq \pi$ .

See Solution

Problem 15100

Find the positive value of $a$ where the slope of the tangent to $f(x)=5 e^{3 x^{3}}$ at $(a, f(a))$ equals 6.

See Solution

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Calculus

Problem 15001

Find the rate of change of yyy where y=log⁡(2x+1)y=\log(2x+1)y=log(2x+1) with respect to xxx.

Problem 15002

Evaluate the integral from 4 to 5 of (2+2y)2dy(2+2y)^{2} dy(2+2y)2dy. Enter a fraction, integer, or exact decimal.

Problem 15003

Find the average rate of change in value per year of the vehicle from years 5 to 10 for f(x)=4500(0.98)xf(x)=4500(0.98)^{x}f(x)=4500(0.98)x.

Problem 15004

Evaluate ∫02g(t)dt\int_{0}^{2} g(t) dt∫02​g(t)dt and ∫15g(t)dt\int_{1}^{5} g(t) dt∫15​g(t)dt for the function g(t)g(t)g(t) with given intercepts.

Problem 15005

Problem 15006

Determine the positivity/negativity of f(x)=18−2x2f(x)=18-2 x^{2}f(x)=18−2x2 on [0,4][0,4][0,4]; sketch it and find the Riemann sums with n=4n=4n=4.

Problem 15007

Find the length xxx and width yyy of a rectangle with area maximum, bounded by y=6−x2y=\frac{6-x}{2}y=26−x​.

Problem 15008

Evaluate the integral ∫06(0.6x2−3)dx\int_{0}^{6}(0.6 x^{2}-3) dx∫06​(0.6x2−3)dx for n=4n=4n=4; find Δx\Delta xΔx and grid points.

Problem 15009

Evaluate the integral ∫06(0.6x2−3)dx\int_{0}^{6}(0.6 x^{2}-3) d x∫06​(0.6x2−3)dx for n=4n=4n=4: graph, find Δx\Delta xΔx, grid points, and Riemann sums.

Problem 15010

Problem 15011

Find dydx\frac{d y}{d x}dxdy​ if y=ln⁡(2x2−3y2)y=\ln(2 x^{2}-3 y^{2})y=ln(2x2−3y2). Choose from the options given.

Problem 15012

Evaluate the integral ∫06(0.6x2−3)dx\int_{0}^{6}(0.6 x^{2}-3)dx∫06​(0.6x2−3)dx with n=4n=4n=4. Find Δx\Delta xΔx, grid points, and Riemann sums.

Problem 15013

Find the local min and max of f(x)=2x3−24x2+72x+6f(x)=2x^{3}-24x^{2}+72x+6f(x)=2x3−24x2+72x+6 by finding its derivative, setting it to zero, and solving for xxx. Then, substitute xxx back into f(x)f(x)f(x) for the values.

Problem 15014

Bestimme die Häufungspunkte der Folge an=(−1)n⋅n2(3⋅n+3)2a_{n}=\frac{(-1)^{n} \cdot n^{2}}{(3 \cdot n+3)^{2}}an​=(3⋅n+3)2(−1)n⋅n2​. Anzahl und größter Punkt?

Problem 15015

Bestimmen Sie die ersten beiden Ableitungen von 9x⋅e−x9 x \cdot e^{-x}9x⋅e−x.

Problem 15016

Find the absolute max and min of f(x)=4−6x2f(x)=4-6 x^{2}f(x)=4−6x2 for −5≤x≤1-5 \leq x \leq 1−5≤x≤1. Max is 4 at x=x=x=, min is at x=x=x=.

Problem 15017

Find the max and min of f(x)=2x3+12x2−72x+1f(x)=2x^3+12x^2-72x+1f(x)=2x3+12x2−72x+1 on [−6,3][-6, 3][−6,3]. Use calculus for derivatives.

Problem 15018

Problem 15019

Evaluate the integral ∫06(0.6x2−3)dx\int_{0}^{6}(0.6 x^{2}-3) dx∫06​(0.6x2−3)dx with n=4n=4n=4: a. Sketch the graph. b. Find Δx\Delta xΔx and grid points x0,x1,…,xnx_{0}, x_{1}, \ldots, x_{n}x0​,x1​,…,xn​. c. Calculate left and right Riemann sums.

Problem 15020

Evaluate the integral ∫06(0.6x2−3)dx\int_{0}^{6}(0.6 x^{2}-3) d x∫06​(0.6x2−3)dx for n=4n=4n=4. Find Δx\Delta xΔx, grid points, and Riemann sums.

Problem 15021

Evaluate the integral ∫06(0.6x2−3)dx\int_{0}^{6}(0.6 x^{2}-3) dx∫06​(0.6x2−3)dx with n=4n=4n=4. Find Δx\Delta xΔx, grid points, and Riemann sums.

Problem 15022

Find the absolute extremum of the function f(x)=4x3−3f(x)=4 x^{3}-3f(x)=4x3−3.

Problem 15023

Find the limit: lim⁡n→∞n2+(−1)n⋅n(n+3)2\lim _{n \rightarrow \infty} \frac{n^{2}+(-1)^{n} \cdot n}{(n+\sqrt{3})^{2}}limn→∞​(n+3​)2n2+(−1)n⋅n​

Problem 15024

Kent deposits \20,000atacontinuousinterestrateof20,000 at a continuous interest rate of 20,000atacontinuousinterestrateof6.21\%$. Will he afford the car in 6 years?

Problem 15025

A snowball's radius decreases at 0.3 cm/min0.3 \mathrm{~cm/min}0.3 cm/min. Find the volume decrease rate when the radius is 17 cm17 \mathrm{~cm}17 cm. cm3min⁡\frac{\mathrm{cm}^{3}}{\min}mincm3​

Problem 15026

Given xy=3x y=3xy=3 and dydt=2\frac{d y}{d t}=2dtdy​=2, find dxdt\frac{d x}{d t}dtdx​ when x=3x=3x=3.

Problem 15027

A snowball's radius decreases at 0.4 cm/min0.4 \mathrm{~cm} / \mathrm{min}0.4 cm/min. Find the volume decrease rate when the radius is 14 cm14 \mathrm{~cm}14 cm. Answer in cm3min⁡\frac{\mathrm{cm}^{3}}{\min}mincm3​.

Problem 15028

Calculate the average value of the function f(x)=−cos⁡xf(x)=-\cos xf(x)=−cosx from x=−π2x=-\frac{\pi}{2}x=−2π​ to x=π2x=\frac{\pi}{2}x=2π​.

Problem 15029

Find the average value of f(x)=x(x−1)f(x)=x(x-1)f(x)=x(x−1) over the interval [5,8][5,8][5,8].

Problem 15030

Graph the integrand for ∫19(3x+1)dx\int_{1}^{9}\left(\frac{3}{x}+1\right) d x∫19​(x3​+1)dx with n=4n=4n=4. Find Δx\Delta xΔx, xix_ixi​, and Riemann sums.

Problem 15031

Find the correct expression for dVdt\frac{d V}{d t}dtdV​ and calculate the volume change rate when r=2.9r=2.9r=2.9 inches and drdt=0.4\frac{d r}{d t}=0.4dtdr​=0.4 in/s.

Problem 15032

A snowball's radius decreases at 0.1 cm/min0.1 \mathrm{~cm} / \mathrm{min}0.1 cm/min. Find the volume's decrease rate when the radius is 13 cm13 \mathrm{~cm}13 cm. Answer in cm3min⁡\frac{\mathrm{cm}^{3}}{\min }mincm3​.

Problem 15033

Find dxdt\frac{d x}{d t}dtdx​ given the equation xy+y3+x4=4x y + y^{3} + x^{4} = 4xy+y3+x4=4, with y=−2y=-2y=−2, dydt=5\frac{d y}{d t}=5dtdy​=5, and x=2x=2x=2.

Problem 15034

Gegeben ist fa(x)=−ax3+4axf_{a}(x)=-a x^{3}+4 a xfa​(x)=−ax3+4ax. a) Zeigen Sie Punktsymmetrie. b) Bestimmen Sie Punkte P(−2,0)P(-2,0)P(−2,0) und Q(2,0)Q(2,0)Q(2,0). c) Nachweis von Hoch- und Tiefpunkt. d) Finden Sie die Wendetangente mit Steigung m=8m=8m=8.

Problem 15035

Find the rate of volume decrease of a snowball when its radius is 18 cm, given the radius decreases at 0.3 cm/min0.3 \mathrm{~cm/min}0.3 cm/min. cm3min⁡\frac{\mathrm{cm}^{3}}{\min}mincm3​

Problem 15036

Untersuchen Sie das Steigen und Fallen der Funktionen: a) f(x)=x5f(x)=x^{5}f(x)=x5, b) f(x)=x2−4xf(x)=x^{2}-4 xf(x)=x2−4x, c) f(x)=1−x3f(x)=1-x^{3}f(x)=1−x3, d) f(x)=x3−4xf(x)=x^{3}-4 xf(x)=x3−4x, e) f(x)=1f(x)=1f(x)=1, f) f(x)=1xf(x)=\frac{1}{x}f(x)=x1​.

Problem 15037

Find dxdt\frac{d x}{d t}dtdx​ given the equation −xy+y2+x4=99-x y+y^{2}+x^{4}=99−xy+y2+x4=99 with dydt=4\frac{d y}{d t}=4dtdy​=4, y=−3y=-3y=−3, and x=3x=3x=3.

Problem 15038

An oil spill is a cylinder with height 4 in and radius r(t)r(t)r(t). Find dVdt\frac{d V}{d t}dtdV​ and volume change at r=2.4r=2.4r=2.4 in, rate 0.20.20.2 in/s.

Problem 15039

Evaluate ddx∫axf(t)dt\frac{d}{d x} \int_{a}^{x} f(t) d tdxd​∫ax​f(t)dt and ddx∫abf(t)dt\frac{d}{d x} \int_{a}^{b} f(t) d tdxd​∫ab​f(t)dt. Simplify your answer.

Problem 15040

Find the antiderivative F(x)F(x)F(x) for: (a) f(x)=x−x4f(x)=\sqrt{x}-x^{4}f(x)=x​−x4, (b) f(x)=1x−1x2f(x)=\frac{1}{x}-\frac{1}{x^{2}}f(x)=x1​−x21​ with F(−e)=1F(-e)=1F(−e)=1, (c) f(x)=4e3x+1+2f(x)=4 e^{3 x+1}+2f(x)=4e3x+1+2. Simplify F(x)F(x)F(x).

Problem 15041

Evaluate the area functions: a. A(−2)A(-2)A(−2), b. F(8)F(8)F(8), c. A(4)A(4)A(4), d. F(4)F(4)F(4), e. A(8)A(8)A(8) for the function fff.

Find the rate of change of $y$ where $y=\log(2x+1)$ with respect to $x$ .

Evaluate the integral from 4 to 5 of $(2+2y)^{2} dy$ . Enter a fraction, integer, or exact decimal.

Find the average rate of change in value per year of the vehicle from years 5 to 10 for $f(x)=4500(0.98)^{x}$ .

Evaluate $\int_{0}^{2} g(t) dt$ and $\int_{1}^{5} g(t) dt$ for the function $g(t)$ with given intercepts.

Determine the positivity/negativity of $f(x)=18-2 x^{2}$ on $[0,4]$ ; sketch it and find the Riemann sums with $n=4$ .

Find the length $x$ and width $y$ of a rectangle with area maximum, bounded by $y=\frac{6-x}{2}$ .

Evaluate the integral $\int_{0}^{6}(0.6 x^{2}-3) dx$ for $n=4$ ; find $\Delta x$ and grid points.

Evaluate the integral $\int_{0}^{6}(0.6 x^{2}-3) d x$ for $n=4$ : graph, find $\Delta x$ , grid points, and Riemann sums.

Find $\frac{d y}{d x}$ if $y=\ln(2 x^{2}-3 y^{2})$ . Choose from the options given.

Evaluate the integral $\int_{0}^{6}(0.6 x^{2}-3)dx$ with $n=4$ . Find $\Delta x$ , grid points, and Riemann sums.

Find the local min and max of $f(x)=2x^{3}-24x^{2}+72x+6$ by finding its derivative, setting it to zero, and solving for $x$ . Then, substitute $x$ back into $f(x)$ for the values.

Bestimme die Häufungspunkte der Folge $a_{n}=\frac{(-1)^{n} \cdot n^{2}}{(3 \cdot n+3)^{2}}$ . Anzahl und größter Punkt?

Bestimmen Sie die ersten beiden Ableitungen von $9 x \cdot e^{-x}$ .

Find the absolute max and min of $f(x)=4-6 x^{2}$ for $-5 \leq x \leq 1$ . Max is 4 at $x=$ , min is at $x=$ .

Find the max and min of $f(x)=2x^3+12x^2-72x+1$ on $[-6, 3]$ . Use calculus for derivatives.

Evaluate the integral $\int_{0}^{6}(0.6 x^{2}-3) dx$ with $n=4$ :
a. Sketch the graph.
b. Find $\Delta x$ and grid points $x_{0}, x_{1}, \ldots, x_{n}$ .
c. Calculate left and right Riemann sums.

Evaluate the integral $\int_{0}^{6}(0.6 x^{2}-3) d x$ for $n=4$ . Find $\Delta x$ , grid points, and Riemann sums.

Evaluate the integral $\int_{0}^{6}(0.6 x^{2}-3) dx$ with $n=4$ . Find $\Delta x$ , grid points, and Riemann sums.

Find the absolute extremum of the function $f(x)=4 x^{3}-3$ .

Find the limit: $\lim _{n \rightarrow \infty} \frac{n^{2}+(-1)^{n} \cdot n}{(n+\sqrt{3})^{2}}$

Kent deposits \ $20,000 at a continuous interest rate of$ 6.21\%$. Will he afford the car in 6 years?

A snowball's radius decreases at $0.3 \mathrm{~cm/min}$ . Find the volume decrease rate when the radius is $17 \mathrm{~cm}$ . $\frac{\mathrm{cm}^{3}}{\min}$

Given $x y=3$ and $\frac{d y}{d t}=2$ , find $\frac{d x}{d t}$ when $x=3$ .

A snowball's radius decreases at $0.4 \mathrm{~cm} / \mathrm{min}$ . Find the volume decrease rate when the radius is $14 \mathrm{~cm}$ . Answer in $\frac{\mathrm{cm}^{3}}{\min}$ .

Calculate the average value of the function $f(x)=-\cos x$ from $x=-\frac{\pi}{2}$ to $x=\frac{\pi}{2}$ .

Find the average value of $f(x)=x(x-1)$ over the interval $[5,8]$ .

Graph the integrand for $\int_{1}^{9}\left(\frac{3}{x}+1\right) d x$ with $n=4$ . Find $\Delta x$ , $x_i$ , and Riemann sums.

Find the correct expression for $\frac{d V}{d t}$ and calculate the volume change rate when $r=2.9$ inches and $\frac{d r}{d t}=0.4$ in/s.

A snowball's radius decreases at $0.1 \mathrm{~cm} / \mathrm{min}$ . Find the volume's decrease rate when the radius is $13 \mathrm{~cm}$ . Answer in $\frac{\mathrm{cm}^{3}}{\min }$ .

Find $\frac{d x}{d t}$ given the equation $x y + y^{3} + x^{4} = 4$ , with $y=-2$ , $\frac{d y}{d t}=5$ , and $x=2$ .

Gegeben ist $f_{a}(x)=-a x^{3}+4 a x$ . a) Zeigen Sie Punktsymmetrie. b) Bestimmen Sie Punkte $P(-2,0)$ und $Q(2,0)$ . c) Nachweis von Hoch- und Tiefpunkt. d) Finden Sie die Wendetangente mit Steigung $m=8$ .

Find the rate of volume decrease of a snowball when its radius is 18 cm, given the radius decreases at $0.3 \mathrm{~cm/min}$ . $\frac{\mathrm{cm}^{3}}{\min}$

Untersuchen Sie das Steigen und Fallen der Funktionen: a) $f(x)=x^{5}$ , b) $f(x)=x^{2}-4 x$ , c) $f(x)=1-x^{3}$ , d) $f(x)=x^{3}-4 x$ , e) $f(x)=1$ , f) $f(x)=\frac{1}{x}$ .

Find $\frac{d x}{d t}$ given the equation $-x y+y^{2}+x^{4}=99$ with $\frac{d y}{d t}=4$ , $y=-3$ , and $x=3$ .

An oil spill is a cylinder with height 4 in and radius $r(t)$ . Find $\frac{d V}{d t}$ and volume change at $r=2.4$ in, rate $0.2$ in/s.

Evaluate $\frac{d}{d x} \int_{a}^{x} f(t) d t$ and $\frac{d}{d x} \int_{a}^{b} f(t) d t$ . Simplify your answer.

Find the antiderivative $F(x)$ for: (a) $f(x)=\sqrt{x}-x^{4}$ , (b) $f(x)=\frac{1}{x}-\frac{1}{x^{2}}$ with $F(-e)=1$ , (c) $f(x)=4 e^{3 x+1}+2$ . Simplify $F(x)$ .

Evaluate the area functions:
a. $A(-2)$ , b. $F(8)$ , c. $A(4)$ , d. $F(4)$ , e. $A(8)$ for the function $f$ .

Analyze the function $f(x)=2 x-e^{x}$ : (a) Determine where $f$ is increasing or decreasing. (b) Find concavity. (c) Sketch $f$ with local extrema. (d) Count $x$ -intercepts and explain.

Find the horizontal asymptote of the function $f(x)=2^{x-1}+3$ .

Tomatenpflanze: Gegeben ist $v(t)=-0,1 t^{3}+t^{2}$ . Bestimme $h(t)$ und beantworte: a) Wachstumsdauer? b) Maximalhöhe? c) Höhe bei maximalem Wachstum?

Find the second derivative of $f(x) = -x e^{\frac{x}{4}}$ .

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

Determine where the function $f(x) = x^{4} - 72x^{2}$ is increasing and decreasing. Sketch the graph with horizontal tangents.

Find the second derivative of $f(x)=-x e^{\frac{x}{4}-10}$ .

Determine if the function $m(x)=3 x^{3}$ is concave up, down, or neither on the interval $(-\infty, 0)$ .

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{3x}{x+2}$ , where $h \neq 0$ .

Find the second derivative, $f^{\prime \prime}(x)$ , for the function $f(x)=4 x^{6}-2 x^{2}+6 x-9$ .

Find the linearization $L(x)$ of the function $f(x)=\ln \left(x^{2}\right)$ at $x=e$ .

Find the second derivative $h^{\prime \prime}(x)$ for $h(x)=9 x^{-5}-8 x^{-7}$ .

Find the second derivative $f^{\prime \prime}(x)$ for the function $f(x)=(x^{2}+1)^{7}$ .

Zeige, dass die Funktion $f(x)=(x-a)^{3}$ bei $x=a$ eine Wendestelle hat. (3Pkt)

Find the inflection points for the function $f(x)=x^{3}+33 x^{2}$ . What are the coordinates? A. $\square$ B. No inflection points.

Find the drug concentration $C(20)$ after 20 minutes using $C(t)=0.08(1-e^{-0.2 t})$ . Round to three decimal places.

Determine where the graph of $f(x)=x^{20}+8x^{2}$ is concave up, concave down, and find the inflection points.

Find the marginal daily production of labor given by $q(n)=12(0.3 n+0.02 n^{2})^{1.6}$ . Calculate $\frac{d q}{d n}$ .

A car wash's daily production is $q(n)=12\left(0.3 n+0.02 n^{2}\right)^{1 / 2}$ . Find $\frac{d q}{d n}$ and estimate it for $n=5$ .

A computer company found the price-demand relationship $p=1512-0.14 x^{2}$ and revenue $R(x)=1512 x-0.14 x^{3}$ . Find local maxima for $R(x)$ .

A computer company found the price-demand relationship as $p=1512-0.14 x^{2}$ for $0<x<90$ . What are the concavity intervals for $R(x)=1512x-0.14x^{3}$ ?

A car wash's daily production is $q(n)=12(0.3 n+0.02 n^{2})^{1.6}$ . Find $\frac{d q}{d n}$ and estimate it for $n=5$ .

Differentiate the function $\frac{3 x^{2}-1}{2 x+x^{3}}$ with respect to $x$ .

Find the derivative of $(3 x^{3}+2)^{5}$ with respect to $x$ .

Calculate the derivative $\left.\frac{d y}{d x}\right|_{x=\frac{\pi}{2}}$ for $y=\frac{x}{\cot x}$ .

Find the second derivative of $f(x)=(x^{2}+2)^{9}$ . What is $f^{\prime \prime}(x)$ ?

Find the derivative $\frac{d y}{d x}$ at $x=\frac{\pi}{4}$ for $y=3 \sin x \cos x + 2 \tan x$ .

Find the consumers' surplus when the demand function is $p(x)=\left(-\frac{1}{400}\right) x+12$ and price is 8.

Calculate the half-life of I-10 with a decay rate of $8.912\%$ per day.

Find the derivative $\frac{dy}{dx}$ using implicit differentiation for $8xy - \frac{y}{8} = \frac{7}{x}$ .

Calculate the doubling time for a city with an annual growth rate of $7.5\%$ .

Find the derivative using implicit differentiation for $(x y)^{2}+y^{2}=9$ ; compute $\frac{d x}{d y}$ .