Calculus

Problem 24001

Find the marginal revenue and profit functions for UB Spectrum, where revenue per copy is \ $0.70 and cost is$ C(x)=90+0.5x+0.001x^{2}$.

See Solution

Problem 24002

Evaluate the derivative of $(f(x))^{2}$ at $x=5$ , given $f(5)=4$ and $f'(5)=3$ . What is the value?

See Solution

Problem 24003

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{5 x^{2}-x+2}{4 x^{2}+5 x-1}$

See Solution

Problem 24004

Evaluate the limits of $f(x)=\frac{8}{x^{3}-1}$ as $x$ approaches 1 from the left and right: $\lim _{x \rightarrow 1^{-}} f(x)$ and $\lim _{x \rightarrow 1^{+}} f(x)$ .

See Solution

Problem 24005

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{9 x^{6}-x}}{x^{3}+6}$ . If none exists, enter DNE.

See Solution

Problem 24006

Find the area of the shaded region between the curves $x=24y^2-8y^3$ and $x=6y^2-18y$ . Set up the integral(s).

See Solution

Problem 24007

Find the limit: $\lim _{x \rightarrow \infty} \sqrt{x^{2}+7}$ . If none, write DNE.

See Solution

Problem 24008

Cost of airing $x$ Super Bowl commercials is $C(x)=80+3000x+0.05x^{2}$ . Find $\bar{C}(4)$ and interpret the average cost.

See Solution

Problem 24009

Calculate the area between the curves $y=x(2-x)$ and $y=4$ from $x=0$ to $x=2$ . Area = $\square$ square units.

See Solution

Problem 24010

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{2-e^{x}}{2+9 e^{x}}$

See Solution

Problem 24011

Calculate the area between the curves $y=x^{2}$ and $y=3x$ at their intersection points.

See Solution

Problem 24012

Find the limit: $\lim _{t \rightarrow \infty} \frac{\sqrt{t}+t^{2}}{4 t-t^{2}}$ . If none, enter DNE.

See Solution

Problem 24013

Determine the horizontal and vertical asymptotes for the curve $y=\frac{2+x^{4}}{x^{2}-x^{4}}$ .

See Solution

Problem 24014

Find intervals where the function $f$ is increasing given that $f^{\prime}(x)=-x^{4}-12 x^{3}+28 x^{2}$ .

See Solution

Problem 24015

Find the area between $\mathrm{MP}_{1}(x)=-x^{2}+20 x-84$ and $\mathrm{MP}_{2}(x)=-x^{2}+18 x-72$ from $x=6$ to $x=8$ .

See Solution

Problem 24016

Find $G'(a)$ for $G(x)=x^{2}-5x+5$ and use it to get tangent line equations at $(0,5)$ and $(4,1)$ .

See Solution

Problem 24017

Graph $f(x)=x^{3}-13 x^{2}+44 x-27$ and find $c$ for the Mean Value Theorem on $[0, 5]$ .

See Solution

Problem 24018

Determine which statements about the integrable function $f$ are true:
1. $\int_{a}^{b} n d x=(b-a) / n$ for any real number $n$ .
2. $\int_{a}^{b} f(x) d x=\int_{a}^{b} f(t) d t$
3. If $f(x) \leq g(x)$ then $\int_{a}^{b} f(x) d x \leq \int_{a}^{b} g(x) d x$
4. $\int_{a}^{b} f(x) d x=f\left(x_{1}\right) \Delta x+\cdots+f\left(x_{n}\right) \Delta x$ where $\Delta x=(b-a) / n$ .
5. $\int_{a}^{b} f(x) d x=\int_{a}^{c} f(x) d x-\int_{b}^{c} f(x) d x$ for any real number $c$ .
6. $\int_{a}^{b} f(x) d x=\int_{b}^{a} f(x) d x$
7. $\int_{a}^{a} f(x) d x=0$
8. $\int_{a}^{b} f(x) d x \geq 0$ measures the area.

See Solution

Problem 24019

Find the minimum value of $f(x)=\frac{x}{4}+\frac{1}{x}$ on the interval $[1,5]$ . What is $x$ ?

See Solution

Problem 24020

Find a function $f$ and a number $a$ such that: $\lim _{h \rightarrow 0} \frac{(1+h)^{7}-1}{h}$ .

See Solution

Problem 24021

Calculate the average value of $f(x)=x^{2}-11$ over the interval from $x=0$ to $x=3$ .

See Solution

Problem 24022

Evaluate these integrals given:
1. $\int_{3}^{7} f(x) d x=16$ , $\int_{3}^{7} g(x) d x=5$ , $\int_{4}^{7} f(x) d x=8$ , $\int_{3}^{4} g(x) d x=2$ .
a. $\int_{3}^{4} 4 f(x) d x=$
b. $\int_{3}^{7}(f(x)-g(x)) d x=$

See Solution

Problem 24023

Find the derivative of $x^{2}+e^{f(x)}+\ln(f(x))$ at $x=2$ given $f(2)=4$ and $f^{\prime}(2)=3$ . Round to three decimal places.

See Solution

Problem 24024

Find the average value of $f(x)=\frac{1}{x}$ from $x=\frac{1}{11}$ to $x=11$ . The average value is $\square$ .

See Solution

Problem 24025

Evaluate these integrals: (i) $\int(4x-3\sqrt{x}+2x^{5})dx$ , (ii) $\int \ln(3x)\frac{1}{x}dx$ , (iii) $\int x^{2}e^{x}dx$ , (iv) $\int \frac{1}{1+x^{2}}+\frac{2}{\sqrt{1-x^{2}}}dx$ .

See Solution

Problem 24026

Plot the line segment for $f(x)=x^{3}+10 x^{2}+17 x-18$ between $x=-4$ and $x=1$ . Find all $c$ for the Mean Value Theorem on $[-4,1]$ .

See Solution

Problem 24027

Given the function $h(x) = \frac{x^{2} + 2x - 3}{x + 2}$ , find its intercepts, asymptotes, and use limits for vertical asymptote behavior.

See Solution

Problem 24028

Evaluate $\int_{3}^{4} 4 f(x) d x$ given $\int_{3}^{7} f(x) dx=16$ and $\int_{4}^{7} f(x) dx=8$ .

See Solution

Problem 24029

Plot the line segment on $f(x)=x^{3}+10x^{2}+17x-18$ between $x=-4$ and $x=1$ . Find values of $c$ for the Mean Value Theorem on $[-4, 1]$ .

See Solution

Problem 24030

Compute the integral $\int_{-1}^{6} f(x) \, \mathrm{d}x$ using the graph of $f$ described by the user.

See Solution

Problem 24031

Plot a line segment on $f(x)=x^{3}-13x^{2}+47x-25$ between $x=0$ and $x=5$ . Find all $c$ for the Mean Value Theorem in $[0, 5]$ .

See Solution

Problem 24032

Calculate the consumer's surplus for the demand curve $p=5-\frac{x}{15}$ at $x=30$ . The result is $\$ \square$ .

See Solution

Problem 24033

Find the average temperature over 15 hours given $T(t)=43+6t-\frac{1}{3}t^{2}$ . Average temp: $\square^{\circ}$ .

See Solution

Problem 24034

Find the value of $\int_{4}^{7} 2 g(x) d x$ given the other integrals.

See Solution

Problem 24035

Evaluate the integrals based on the given functions $f$ and $g$ on $[3,7]$ with provided values.
a. Find $\int_{3}^{4} 4 f(x) d x$ .
b. Find $\int_{3}^{7}(f(x)-g(x)) d x$ .
c. Find $\int_{3}^{4}(f(x)-g(x)) d x$ .
d. Find $\int_{4}^{7}(g(x)-f(x)) d x=\square$ .

See Solution

Problem 24036

Evaluate the integrals based on given functions $f$ and $g$ on $[3,7]$ with specified values.

See Solution

Problem 24037

Find a function $f$ and a number $a$ such that
$\lim _{h \rightarrow 0} \frac{(1+h)^{7}-1}{h}$
represents the derivative of $f$ at $a$ .

See Solution

Problem 24038

Evaluate the integrals based on given functions $f$ and $g$ over $[3,7]$ with specified integral values.

See Solution

Problem 24039

Calculate the volume of the solid formed by revolving $y=\sqrt{25-x^{2}}$ from $x=-5$ to $x=5$ around the $x$ -axis. The volume is $\square$ cubic units.

See Solution

Problem 24040

Find a function $f$ and a number $a$ such that the limit equals the derivative at that point: $\lim _{x \rightarrow \pi / 4} \frac{\tan x-1}{x-\pi / 4}$

See Solution

Problem 24041

Find the volume of the solid formed by revolving the area between $y=2 x^{2}$ and the $x$ -axis from $x=0$ to $x=3$ around the $x$ -axis. The volume is $\square$ cubic units.

See Solution

Problem 24042

Find a function $f$ and a number $a$ such that the limit equals the derivative: $\lim _{h \rightarrow 0} \frac{\cos (\pi+h)+1}{h}$

See Solution

Problem 24043

Evaluate the integrals given $\int_{3}^{7} f(x) dx=16$ , $\int_{3}^{7} g(x) dx=5$ , $\int_{4}^{7} f(x) dx=8$ , $\int_{3}^{4} g(x) dx=2$ .
a. $\int_{3}^{4} 4 f(x) dx$ b. $\int_{3}^{7}(f(x)-g(x)) dx$ c. $\int_{3}^{4}(f(x)-g(x)) dx$ d. $\int_{4}^{7}(g(x)-f(x)) dx$ e. $\int_{4}^{7} 2 g(x) dx$

See Solution

Problem 24044

Given $g^{\prime}(x)=-\frac{x(x-3)}{(x-5)e^{2x}}$ , find if $x=3$ and $x=0$ are local max or min.

See Solution

Problem 24045

Find the derivative $f^{\prime}(a)$ for the function $f(t)=t^{4}-6t$ .

See Solution

Problem 24046

Calculate the volume of the solid formed by revolving $y=\sqrt{x}$ from $x=0$ to $x=4$ around the $x$ -axis. Volume = $\square$ .

See Solution

Problem 24047

Find $f^{\prime}(a)$ for $f(x)=\frac{1}{\sqrt{x+9}}$ when $f^{\prime}(a)=1$ .

See Solution

Problem 24048

Calculate the integral $\int_{0}^{3}\left(4 x-2 x^{3}\right) d x$ given that $I=\frac{45}{4}$ .

See Solution

Problem 24049

Find the derivative of $g(x)=\sqrt{1-x}$ using the definition of derivative: $g^{\prime}(x)=\square$ .

See Solution

Problem 24050

Evaluate the integrals:
a. $\int_{0}^{3}(4x-2x^{3})dx$
b. $\int_{3}^{0}(2x-x^{3})dx$

See Solution

Problem 24051

Evaluate the integral $\int_{0}^{3}(5 x+1) d x$ using right Riemann sums. Simplify your answer.

See Solution

Problem 24052

Find the derivative of $g(x)=x^{2/3}$ at $a \neq 0$ : $g^{\prime}(a)=\ ?$

See Solution

Problem 24053

Calculate the area between the curve $f(x)=e^{2 x}-12$ and the $x$ -axis from $0$ to $\frac{\ln 12}{2}$ .

See Solution

Problem 24054

Differentiate the function $f(t)=4-\frac{2}{3} t$ and find $f^{\prime}(t)=\square$ .

See Solution

Problem 24055

Calculate the area between the curve $f(x)=x(x^{2}-25)$ and the $x$ -axis from $x=-5$ to $x=5$ . Area = $\square$ square units.

See Solution

Problem 24056

Differentiate the function $f(x)=x^{5}-2x+7$ and find $f^{\prime}(x)=\square$ .

See Solution

Problem 24057

Calculate the area between $y=x^{2}-2x$ and the $x$ -axis for these intervals: (a) 0 to 2, (b) 0 to 3, (c) -2 to 2.

See Solution

Problem 24058

Compute $\frac{\partial f}{\partial s}$ for $f(x, y)=x^{4} y^{2}$ where $x=s t, y=s^{2} t$ . Options: A, B, C, D, E.

See Solution

Problem 24059

Calculate the area between $y=x^{2}-2x$ and the $x$ -axis for these intervals: (a) 0 to 2, (b) 0 to 3, (c) -2 to 2.

See Solution

Problem 24060

Find the rate of change of daily oil revenue at the start of 2010, given $q(t)=0.015 t^{2}-0.1 t+5.11$ and oil price decreasing from \$85 at \$19/year.

See Solution

Problem 24061

A company makes alternators with laborers ( $x$ ) and robots ( $y$ ) satisfying $xy=58500$ . With 195 robots increasing by 13/month, how fast are laborers laid off?

See Solution

Problem 24062

Differentiate the function $g(x)=x^{2}(1-2 x)$ .

See Solution

Problem 24063

Differentiate the function $A(s)=-\frac{12}{s^{5}}$ .

See Solution

Problem 24064

Find the intervals where the function is increasing, decreasing, and constant. Given: increasing on $(-4,2)$ . Decreasing: $\square$ .

See Solution

Problem 24065

Find the slope and equation of the tangent line to $16 x^{2}+8 y^{2}=24$ at point $(1,-1)$ .

See Solution

Problem 24066

Evaluate the integral: $\int\left(\sec ^{2} x+7\right) d x$

See Solution

Problem 24067

Find the garden's width that minimizes the area of a $150 \mathrm{~m}^{2}$ garden with a $1 \mathrm{~m}$ and $2 \mathrm{~m}$ border. The height is $5 \sqrt{3} \mathrm{~m}$ ; find the width.

See Solution

Problem 24068

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

See Solution

Problem 24069

Differentiate the function $y=\sqrt{x}(x-10)$ and find $y^{\prime}=\square$ .

See Solution

Problem 24070

Differentiate the function: $y=6 e^{x}+\frac{2}{\sqrt[3]{x}}$ .

See Solution

Problem 24071

Calculate the area between the curve $f(x)=e^{3x}-5$ and the $x$ -axis from $0$ to $\frac{\ln 5}{3}$ . Area is $\square$ square unit(s).

See Solution

Problem 24072

Find the integral of the function: $\int\left(\sec ^{2} x+3\right) d x$ .

See Solution

Problem 24073

A hemispherical tank with radius $9 \mathrm{~m}$ fills at $4 \mathrm{~m}^{3}/\mathrm{min}$ . Find the water level rise rate when $h=7 \mathrm{~m}$ .
Volume relation: $V=\frac{\pi h^{2}(27-h)}{3}$
Differentiate: $\frac{d V}{d t}=\square \frac{d h}{d t}$

See Solution

Problem 24074

Calculate the integral $\int(6 x^{2}-5 x+1)(2 x^{3}-\frac{5}{2} x^{2}+x-2305)^{11} dx$ .

See Solution

Problem 24075

Evaluate the integral: $\int_{-6}^{-3}[-(9-x^{2})] dx + \int_{-3}^{3}[9-x^{2}] dx + \int_{3}^{6}[-(9-x^{2})] dx$ .

See Solution

Problem 24076

Find how fast the daily operating budget $y$ is increasing if $P=5000$ , $x=160$ , and $x$ increases by 10/year.

See Solution

Problem 24077

Calculate the volume of $1.00 \mathrm{~mol} \mathrm{~N}_{2}$ at $10 \mathrm{~atm}$ using van der Waals at critical and Boyle temperatures.

See Solution

Problem 24078

Evaluate the integral $\int_{-\infty}^{\infty} \frac{d x}{x^{2}+25}$ or state if it diverges.

See Solution

Problem 24079

Carbon-14 half-life is 5730 years. A plant had 100 g. Find: a) remaining after 5730 years, b) equation for remaining after $t$ years, c) years for 80 g left, d) rate of change at 100 years.

See Solution

Problem 24080

Find the limit or divergence of the sequence $\left\{\frac{5^{n}}{n !}\right\}$ using the Growth Rates of Sequences Theorem. A. Limit is $\square$ . B. Diverges.

See Solution

Problem 24081

Find the total maintenance cost function $M(x)$ given $M^{\prime}(x)=15x$ and $M(2)=125$ .

See Solution

Problem 24082

Differentiate the function $y=e^{x+6}+4$ and find $y'$ .

See Solution

Problem 24083

Solve the differential equation $\frac{d y}{d x}=4 x+21$ to find the general and particular solutions with $y(0)=-20$ .

See Solution

Problem 24084

Differentiate the function $u=\sqrt[5]{t}+6\sqrt{t^{5}}$ . Find $u'=\square$ .

See Solution

Problem 24085

Differentiate the function $g(u)=\sqrt{3} u+\sqrt{7 u}$ . What is $g^{\prime}(u)=?$

See Solution

Problem 24086

Find the 1st and 2nd derivatives of $f(x)=14 x^{14}+5 x^{5}-x$ .

See Solution

Problem 24087

Find $f^{\prime}\left(\frac{\pi}{4}\right)$ for $f(x)=g(2 \sin (x))$ given $g^{\prime}(\sqrt{2})=\sqrt{2}$ .

See Solution

Problem 24088

Find the first and second derivatives of $G(r)=\sqrt{r}+\sqrt[8]{r}$ . $G'(r)=\square$ , $G''(r)=\square$ .

See Solution

Problem 24089

Find $f^{\prime}\left(\frac{\pi}{4}\right)$ for $f(x)=g(2 \sin (x))$ given $g^{\prime}(\sqrt{2})=\sqrt{2}$ .

See Solution

Problem 24090

Estimate the paint needed for a hemispherical dome (50 m diameter) with a $0.05 \, \mathrm{cm}$ thick coat using differentials.

See Solution

Problem 24091

What does $f^{\prime}(1000)=9$ imply about the cost $C=f(x)$ for producing $x$ yards of fabric?

See Solution

Problem 24092

Evaluate the integral $\int_{1}^{5} \frac{x-1}{x^{2}(x+1)} d x$ .

See Solution

Problem 24093

Find the volume of the solid formed by rotating the area above $y=x^{2}$ , below the $x$ -axis, and right of $x=2$ about the $y$ -axis. $V=\square$

See Solution

Problem 24094

Find the volume $V$ of the solid formed by revolving the region between $y=4$ and $y=4\sqrt{\cos x}$ from $x=-\frac{\pi}{4}$ to $x=\frac{\pi}{4}$ about the x-axis using the washer method. Type an exact answer with $\pi$ .

See Solution

Problem 24095

Find the volume of the solid formed by revolving the area between $x=\sqrt{7 \sin (7 y)}$ , $0 \leq y \leq \frac{\pi}{21}$ , and $x=0$ about the $y$ -axis. The volume is $\square$ cubic units.

See Solution

Problem 24096

Evaluate the integral from -4 to -3: $\int_{-4}^{-3}\left(\frac{y^{6}-7 y^{3}}{y^{5}}\right) d y=\square$

See Solution

Problem 24097

Find the volume of the solid formed by revolving the area between $x=\sqrt{15} y^{2}$ , $x=0$ , $y=-2$ , and $y=2$ around the $y$ -axis. Volume is $\square$ cubic unit(s).

See Solution

Problem 24098

Find the volume of the solid formed by revolving the area between $y=e^{x-1}$ , $y=0$ , $x=1$ , and $x=8$ around the $x$ -axis. $V=\square$ (Use $\pi$ and $e$ as needed.)

See Solution

Problem 24099

Find the volume of the solid formed by revolving the area between $y=\frac{3}{\sqrt{x}}$ , $x=2$ , and $x=5$ around the $x$ -axis. The volume is $\square$ .

See Solution

Problem 24100

Find the volume of the solid formed by revolving the area between $y=2 \sqrt{\sin x}$ , $y=0$ , $x=\frac{\pi}{4}$ , and $x=\frac{3 \pi}{4}$ around the $x$ -axis. The volume is $\square$ cubic units.

See Solution

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Calculus

Problem 24001

Find the marginal revenue and profit functions for UB Spectrum, where revenue per copy is \0.70andcostis0.70 and cost is 0.70andcostisC(x)=90+0.5x+0.001x^{2}$.

Problem 24002

Evaluate the derivative of (f(x))2(f(x))^{2}(f(x))2 at x=5x=5x=5, given f(5)=4f(5)=4f(5)=4 and f′(5)=3f'(5)=3f′(5)=3. What is the value?

Problem 24003

Find the limit as xxx approaches infinity: lim⁡x→∞5x2−x+24x2+5x−1\lim _{x \rightarrow \infty} \frac{5 x^{2}-x+2}{4 x^{2}+5 x-1}x→∞lim​4x2+5x−15x2−x+2​

Problem 24004

Evaluate the limits of f(x)=8x3−1f(x)=\frac{8}{x^{3}-1}f(x)=x3−18​ as xxx approaches 1 from the left and right: lim⁡x→1−f(x)\lim _{x \rightarrow 1^{-}} f(x)limx→1−​f(x) and lim⁡x→1+f(x)\lim _{x \rightarrow 1^{+}} f(x)limx→1+​f(x).

Problem 24005

Find the limit: lim⁡x→∞9x6−xx3+6\lim _{x \rightarrow \infty} \frac{\sqrt{9 x^{6}-x}}{x^{3}+6}x→∞lim​x3+69x6−x​​. If none exists, enter DNE.

Problem 24006

Find the area of the shaded region between the curves x=24y2−8y3x=24y^2-8y^3x=24y2−8y3 and x=6y2−18yx=6y^2-18yx=6y2−18y. Set up the integral(s).

Problem 24007

Find the limit: lim⁡x→∞x2+7\lim _{x \rightarrow \infty} \sqrt{x^{2}+7}x→∞lim​x2+7​. If none, write DNE.

Problem 24008

Cost of airing xxx Super Bowl commercials is C(x)=80+3000x+0.05x2C(x)=80+3000x+0.05x^{2}C(x)=80+3000x+0.05x2. Find Cˉ(4)\bar{C}(4)Cˉ(4) and interpret the average cost.

Problem 24009

Calculate the area between the curves y=x(2−x)y=x(2-x)y=x(2−x) and y=4y=4y=4 from x=0x=0x=0 to x=2x=2x=2. Area = □\square□ square units.

Problem 24010

Find the limit as x x x approaches infinity: lim⁡x→∞2−ex2+9ex \lim _{x \rightarrow \infty} \frac{2-e^{x}}{2+9 e^{x}} limx→∞​2+9ex2−ex​

Problem 24011

Calculate the area between the curves y=x2y=x^{2}y=x2 and y=3xy=3xy=3x at their intersection points.

Problem 24012

Find the limit: lim⁡t→∞t+t24t−t2\lim _{t \rightarrow \infty} \frac{\sqrt{t}+t^{2}}{4 t-t^{2}}t→∞lim​4t−t2t​+t2​. If none, enter DNE.

Problem 24013

Determine the horizontal and vertical asymptotes for the curve y=2+x4x2−x4y=\frac{2+x^{4}}{x^{2}-x^{4}}y=x2−x42+x4​.

Problem 24014

Find intervals where the function fff is increasing given that f′(x)=−x4−12x3+28x2f^{\prime}(x)=-x^{4}-12 x^{3}+28 x^{2}f′(x)=−x4−12x3+28x2.

Problem 24015

Find the area between MP1(x)=−x2+20x−84\mathrm{MP}_{1}(x)=-x^{2}+20 x-84MP1​(x)=−x2+20x−84 and MP2(x)=−x2+18x−72\mathrm{MP}_{2}(x)=-x^{2}+18 x-72MP2​(x)=−x2+18x−72 from x=6x=6x=6 to x=8x=8x=8.

Problem 24016

Find G′(a)G'(a)G′(a) for G(x)=x2−5x+5G(x)=x^{2}-5x+5G(x)=x2−5x+5 and use it to get tangent line equations at (0,5)(0,5)(0,5) and (4,1)(4,1)(4,1).

Problem 24017

Graph f(x)=x3−13x2+44x−27f(x)=x^{3}-13 x^{2}+44 x-27f(x)=x3−13x2+44x−27 and find ccc for the Mean Value Theorem on [0,5][0, 5][0,5].

Problem 24018

Problem 24019

Find the minimum value of f(x)=x4+1xf(x)=\frac{x}{4}+\frac{1}{x}f(x)=4x​+x1​ on the interval [1,5][1,5][1,5]. What is xxx?

Problem 24020

Find a function fff and a number aaa such that: lim⁡h→0(1+h)7−1h\lim _{h \rightarrow 0} \frac{(1+h)^{7}-1}{h}h→0lim​h(1+h)7−1​.

Problem 24021

Calculate the average value of f(x)=x2−11f(x)=x^{2}-11f(x)=x2−11 over the interval from x=0x=0x=0 to x=3x=3x=3.

Problem 24022

Problem 24023

Find the derivative of x2+ef(x)+ln⁡(f(x))x^{2}+e^{f(x)}+\ln(f(x))x2+ef(x)+ln(f(x)) at x=2x=2x=2 given f(2)=4f(2)=4f(2)=4 and f′(2)=3f^{\prime}(2)=3f′(2)=3. Round to three decimal places.

Problem 24024

Find the average value of f(x)=1xf(x)=\frac{1}{x}f(x)=x1​ from x=111x=\frac{1}{11}x=111​ to x=11x=11x=11. The average value is □\square□.

Problem 24025

Problem 24026

Plot the line segment for f(x)=x3+10x2+17x−18f(x)=x^{3}+10 x^{2}+17 x-18f(x)=x3+10x2+17x−18 between x=−4x=-4x=−4 and x=1x=1x=1. Find all ccc for the Mean Value Theorem on [−4,1][-4,1][−4,1].

Problem 24027

Given the function h(x)=x2+2x−3x+2h(x) = \frac{x^{2} + 2x - 3}{x + 2}h(x)=x+2x2+2x−3​, find its intercepts, asymptotes, and use limits for vertical asymptote behavior.

Problem 24028

Evaluate ∫344f(x)dx\int_{3}^{4} 4 f(x) d x∫34​4f(x)dx given ∫37f(x)dx=16\int_{3}^{7} f(x) dx=16∫37​f(x)dx=16 and ∫47f(x)dx=8\int_{4}^{7} f(x) dx=8∫47​f(x)dx=8.

Problem 24029

Plot the line segment on f(x)=x3+10x2+17x−18f(x)=x^{3}+10x^{2}+17x-18f(x)=x3+10x2+17x−18 between x=−4x=-4x=−4 and x=1x=1x=1. Find values of ccc for the Mean Value Theorem on [−4,1][-4, 1][−4,1].

Problem 24030

Compute the integral ∫−16f(x) dx\int_{-1}^{6} f(x) \, \mathrm{d}x∫−16​f(x)dx using the graph of fff described by the user.

Problem 24031

Plot a line segment on f(x)=x3−13x2+47x−25f(x)=x^{3}-13x^{2}+47x-25f(x)=x3−13x2+47x−25 between x=0x=0x=0 and x=5x=5x=5. Find all ccc for the Mean Value Theorem in [0,5][0, 5][0,5].

Problem 24032

Calculate the consumer's surplus for the demand curve p=5−x15p=5-\frac{x}{15}p=5−15x​ at x=30x=30x=30. The result is $□\$ \square$□.

Problem 24033

Find the average temperature over 15 hours given T(t)=43+6t−13t2T(t)=43+6t-\frac{1}{3}t^{2}T(t)=43+6t−31​t2. Average temp: □∘\square^{\circ}□∘.

Problem 24034

Find the value of ∫472g(x)dx\int_{4}^{7} 2 g(x) d x∫47​2g(x)dx given the other integrals.

Problem 24035

Problem 24036

Evaluate the integrals based on given functions fff and ggg on [3,7][3,7][3,7] with specified values.

Problem 24037

Find a function fff and a number aaa such that lim⁡h→0(1+h)7−1h \lim _{h \rightarrow 0} \frac{(1+h)^{7}-1}{h} h→0lim​h(1+h)7−1​ represents the derivative of fff at aaa.

Problem 24038

Evaluate the integrals based on given functions fff and ggg over [3,7][3,7][3,7] with specified integral values.

Problem 24039

Calculate the volume of the solid formed by revolving y=25−x2y=\sqrt{25-x^{2}}y=25−x2​ from x=−5x=-5x=−5 to x=5x=5x=5 around the xxx-axis. The volume is □\square□ cubic units.

Problem 24040

Find a function fff and a number aaa such that the limit equals the derivative at that point: lim⁡x→π/4tan⁡x−1x−π/4\lim _{x \rightarrow \pi / 4} \frac{\tan x-1}{x-\pi / 4}x→π/4lim​x−π/4tanx−1​

Problem 24041

Find the volume of the solid formed by revolving the area between y=2x2y=2 x^{2}y=2x2 and the xxx-axis from x=0x=0x=0 to x=3x=3x=3 around the xxx-axis. The volume is □\square□ cubic units.

Problem 24042

Find the marginal revenue and profit functions for UB Spectrum, where revenue per copy is \ $0.70 and cost is$ C(x)=90+0.5x+0.001x^{2}$.

Evaluate the derivative of $(f(x))^{2}$ at $x=5$ , given $f(5)=4$ and $f'(5)=3$ . What is the value?

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{5 x^{2}-x+2}{4 x^{2}+5 x-1}$

Evaluate the limits of $f(x)=\frac{8}{x^{3}-1}$ as $x$ approaches 1 from the left and right: $\lim _{x \rightarrow 1^{-}} f(x)$ and $\lim _{x \rightarrow 1^{+}} f(x)$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{9 x^{6}-x}}{x^{3}+6}$ . If none exists, enter DNE.

Find the area of the shaded region between the curves $x=24y^2-8y^3$ and $x=6y^2-18y$ . Set up the integral(s).

Find the limit: $\lim _{x \rightarrow \infty} \sqrt{x^{2}+7}$ . If none, write DNE.

Cost of airing $x$ Super Bowl commercials is $C(x)=80+3000x+0.05x^{2}$ . Find $\bar{C}(4)$ and interpret the average cost.

Calculate the area between the curves $y=x(2-x)$ and $y=4$ from $x=0$ to $x=2$ . Area = $\square$ square units.

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{2-e^{x}}{2+9 e^{x}}$

Calculate the area between the curves $y=x^{2}$ and $y=3x$ at their intersection points.

Find the limit: $\lim _{t \rightarrow \infty} \frac{\sqrt{t}+t^{2}}{4 t-t^{2}}$ . If none, enter DNE.

Determine the horizontal and vertical asymptotes for the curve $y=\frac{2+x^{4}}{x^{2}-x^{4}}$ .

Find intervals where the function $f$ is increasing given that $f^{\prime}(x)=-x^{4}-12 x^{3}+28 x^{2}$ .

Find the area between $\mathrm{MP}_{1}(x)=-x^{2}+20 x-84$ and $\mathrm{MP}_{2}(x)=-x^{2}+18 x-72$ from $x=6$ to $x=8$ .

Find $G'(a)$ for $G(x)=x^{2}-5x+5$ and use it to get tangent line equations at $(0,5)$ and $(4,1)$ .

Graph $f(x)=x^{3}-13 x^{2}+44 x-27$ and find $c$ for the Mean Value Theorem on $[0, 5]$ .

Find the minimum value of $f(x)=\frac{x}{4}+\frac{1}{x}$ on the interval $[1,5]$ . What is $x$ ?

Find a function $f$ and a number $a$ such that: $\lim _{h \rightarrow 0} \frac{(1+h)^{7}-1}{h}$ .

Calculate the average value of $f(x)=x^{2}-11$ over the interval from $x=0$ to $x=3$ .

Find the derivative of $x^{2}+e^{f(x)}+\ln(f(x))$ at $x=2$ given $f(2)=4$ and $f^{\prime}(2)=3$ . Round to three decimal places.

Find the average value of $f(x)=\frac{1}{x}$ from $x=\frac{1}{11}$ to $x=11$ . The average value is $\square$ .

Plot the line segment for $f(x)=x^{3}+10 x^{2}+17 x-18$ between $x=-4$ and $x=1$ . Find all $c$ for the Mean Value Theorem on $[-4,1]$ .

Given the function $h(x) = \frac{x^{2} + 2x - 3}{x + 2}$ , find its intercepts, asymptotes, and use limits for vertical asymptote behavior.

Evaluate $\int_{3}^{4} 4 f(x) d x$ given $\int_{3}^{7} f(x) dx=16$ and $\int_{4}^{7} f(x) dx=8$ .

Plot the line segment on $f(x)=x^{3}+10x^{2}+17x-18$ between $x=-4$ and $x=1$ . Find values of $c$ for the Mean Value Theorem on $[-4, 1]$ .

Compute the integral $\int_{-1}^{6} f(x) \, \mathrm{d}x$ using the graph of $f$ described by the user.

Plot a line segment on $f(x)=x^{3}-13x^{2}+47x-25$ between $x=0$ and $x=5$ . Find all $c$ for the Mean Value Theorem in $[0, 5]$ .

Calculate the consumer's surplus for the demand curve $p=5-\frac{x}{15}$ at $x=30$ . The result is $\$ \square$ .

Find the average temperature over 15 hours given $T(t)=43+6t-\frac{1}{3}t^{2}$ . Average temp: $\square^{\circ}$ .

Find the value of $\int_{4}^{7} 2 g(x) d x$ given the other integrals.

Evaluate the integrals based on given functions $f$ and $g$ on $[3,7]$ with specified values.

Find a function $f$ and a number $a$ such that
$\lim _{h \rightarrow 0} \frac{(1+h)^{7}-1}{h}$
represents the derivative of $f$ at $a$ .

Evaluate the integrals based on given functions $f$ and $g$ over $[3,7]$ with specified integral values.

Calculate the volume of the solid formed by revolving $y=\sqrt{25-x^{2}}$ from $x=-5$ to $x=5$ around the $x$ -axis. The volume is $\square$ cubic units.

Find a function $f$ and a number $a$ such that the limit equals the derivative at that point: $\lim _{x \rightarrow \pi / 4} \frac{\tan x-1}{x-\pi / 4}$

Find the volume of the solid formed by revolving the area between $y=2 x^{2}$ and the $x$ -axis from $x=0$ to $x=3$ around the $x$ -axis. The volume is $\square$ cubic units.

Find a function $f$ and a number $a$ such that the limit equals the derivative: $\lim _{h \rightarrow 0} \frac{\cos (\pi+h)+1}{h}$

Given $g^{\prime}(x)=-\frac{x(x-3)}{(x-5)e^{2x}}$ , find if $x=3$ and $x=0$ are local max or min.

Find the derivative $f^{\prime}(a)$ for the function $f(t)=t^{4}-6t$ .

Calculate the volume of the solid formed by revolving $y=\sqrt{x}$ from $x=0$ to $x=4$ around the $x$ -axis. Volume = $\square$ .

Find $f^{\prime}(a)$ for $f(x)=\frac{1}{\sqrt{x+9}}$ when $f^{\prime}(a)=1$ .

Calculate the integral $\int_{0}^{3}\left(4 x-2 x^{3}\right) d x$ given that $I=\frac{45}{4}$ .

Find the derivative of $g(x)=\sqrt{1-x}$ using the definition of derivative: $g^{\prime}(x)=\square$ .

Evaluate the integrals:
a. $\int_{0}^{3}(4x-2x^{3})dx$
b. $\int_{3}^{0}(2x-x^{3})dx$

Evaluate the integral $\int_{0}^{3}(5 x+1) d x$ using right Riemann sums. Simplify your answer.

Find the derivative of $g(x)=x^{2/3}$ at $a \neq 0$ : $g^{\prime}(a)=\ ?$

Calculate the area between the curve $f(x)=e^{2 x}-12$ and the $x$ -axis from $0$ to $\frac{\ln 12}{2}$ .

Differentiate the function $f(t)=4-\frac{2}{3} t$ and find $f^{\prime}(t)=\square$ .

Calculate the area between the curve $f(x)=x(x^{2}-25)$ and the $x$ -axis from $x=-5$ to $x=5$ . Area = $\square$ square units.

Differentiate the function $f(x)=x^{5}-2x+7$ and find $f^{\prime}(x)=\square$ .

Calculate the area between $y=x^{2}-2x$ and the $x$ -axis for these intervals: (a) 0 to 2, (b) 0 to 3, (c) -2 to 2.

Compute $\frac{\partial f}{\partial s}$ for $f(x, y)=x^{4} y^{2}$ where $x=s t, y=s^{2} t$ . Options: A, B, C, D, E.

Calculate the area between $y=x^{2}-2x$ and the $x$ -axis for these intervals: (a) 0 to 2, (b) 0 to 3, (c) -2 to 2.

Find the rate of change of daily oil revenue at the start of 2010, given $q(t)=0.015 t^{2}-0.1 t+5.11$ and oil price decreasing from \$85 at \$19/year.

A company makes alternators with laborers ( $x$ ) and robots ( $y$ ) satisfying $xy=58500$ . With 195 robots increasing by 13/month, how fast are laborers laid off?

Differentiate the function $g(x)=x^{2}(1-2 x)$ .

Differentiate the function $A(s)=-\frac{12}{s^{5}}$ .

Find the intervals where the function is increasing, decreasing, and constant. Given: increasing on $(-4,2)$ . Decreasing: $\square$ .

Find the slope and equation of the tangent line to $16 x^{2}+8 y^{2}=24$ at point $(1,-1)$ .

Evaluate the integral: $\int\left(\sec ^{2} x+7\right) d x$

Find the garden's width that minimizes the area of a $150 \mathrm{~m}^{2}$ garden with a $1 \mathrm{~m}$ and $2 \mathrm{~m}$ border. The height is $5 \sqrt{3} \mathrm{~m}$ ; find the width.

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

Differentiate the function $y=\sqrt{x}(x-10)$ and find $y^{\prime}=\square$ .

Differentiate the function: $y=6 e^{x}+\frac{2}{\sqrt[3]{x}}$ .

Calculate the area between the curve $f(x)=e^{3x}-5$ and the $x$ -axis from $0$ to $\frac{\ln 5}{3}$ . Area is $\square$ square unit(s).

Find the integral of the function: $\int\left(\sec ^{2} x+3\right) d x$ .

Calculate the integral $\int(6 x^{2}-5 x+1)(2 x^{3}-\frac{5}{2} x^{2}+x-2305)^{11} dx$ .