Calculus

Problem 1001

Find the interval of convergence for the series $\sum_{n=1}^{\infty} \frac{(x-1)^{n}}{3^{n}}$ .

See Solution

Problem 1002

Identify the true statement about the function $f(x)=(x-3)(x-7)^{2}$ from the options given.

See Solution

Problem 1003

Sketch the graph of the piecewise function and find the limits:
1. For $f(x) = \begin{cases} \sin x & x < 0 \\ x^2 & 0 \leq x < 2 \\ x & x \geq 2 \end{cases}$ , find: i. $\lim_{x \to 0} f(x)$ ii. $\lim_{x \to 2} f(x)$
2. For $f(x) = \begin{cases} e^x & x \leq 0 \\ |x| + 1 & 0 < x < 1 \\ \ln x & x \geq 1 \end{cases}$ , find: i. $\lim_{x \to 0} f(x)$ ii. $\lim_{x \to 1} f(x)$

See Solution

Problem 1004

Find the slope of the tangent line to $f^{-1}$ at the point $(1,-1)$ for $f(x)=(x+2)^{2}$ where $x \geq-2$ . The slope is $\square$ .

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Problem 1005

Find the limit: $\lim _{x \rightarrow-4} \frac{x^{2}-2 x-24}{6 x^{2}+23 x-4}$ .

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Problem 1006

Find the derivative $f'(90^\circ)$ for the function $f(x) = \sin x - 3 \cos x$ .

See Solution

Problem 1007

Find $f'(90^{\circ})$ for $f(x) = \sin x - 3\cos x$ and evaluate $3 \lim_{x \rightarrow 0} \frac{\cos 2x - 11}{\cos x - 1}$ .

See Solution

Problem 1008

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

See Solution

Problem 1009

Find the limit as $x$ approaches infinity of $\left(\frac{x+4}{2-3 x}\right)^{3}$ .

See Solution

Problem 1010

Find the limit: $\lim _{a \rightarrow 1} \frac{a^{3}-a}{a^{2}-1}$ .

See Solution

Problem 1011

Find the limit as $x$ approaches -3 for the expression $\frac{x^{2}+9x+18}{x+3}$ .

See Solution

Problem 1012

Find the limit: $\lim _{x \rightarrow 0} \frac{-3 x}{\sqrt{3 x+64}-8}$ .

See Solution

Problem 1013

Find the limit: $\lim _{x \rightarrow-3} \frac{x^{2}-9}{4 x+12}$ .

See Solution

Problem 1014

수렴하는 수열을 고르세요: ᄀ. $\left\{\frac{1}{2 n+1}\right\}$ , ㄴ. $\left\{\frac{(-1)^{n}}{n}\right\}$ , ᄃ. $\left\{(-1)^{n}+2\right\}$ .

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Problem 1015

A conical water tank has a semi-vertical angle of $\tan^{-1} \frac{3}{4}$ . Water is added at 6 m³/hour. Find the rate of increase of the wet curved surface area (m²/hour) when the water depth is 4 m.

See Solution

Problem 1016

Find $f^{\prime}(-12)$ if the tangent line at $(-12,-9)$ passes through $(-1,46)$ . $f^{\prime}(-12)=$

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Problem 1017

Find the average rate of change of $f(x)=\frac{1}{x+2}$ on $[8,8+h]$ . Express your answer in terms of $h$ .

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Problem 1018

Find the derivative $f^{\prime}(0)$ for the function $f(x)=6+2x-3x^{2}$ using the definition of the derivative.

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Problem 1019

Find the slope of the tangent line to $f(x)=3x^{2}$ at $x=3$ using the limit definition of the derivative. Evaluate:
1. $f(3+h)=$
2. $f(3+h)-f(3)=$
3. $\frac{f(3+h)-f(3)}{h}=$
4. $\lim_{h \rightarrow 0} \frac{f(3+h)-f(3)}{h}=$
Then, find $f^{\prime}(3)=$

See Solution

Problem 1020

Find the average rate of change of $f(x)=x^{3}-6 x+4$ for the intervals: (a) -7 to -4, (b) -2 to 2, (c) 2 to 7.

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Problem 1021

Given $F(x)=-x^{4}+2 x^{2}+224$ , find if $F$ is even/odd, a second local max, and the area from $x=-4$ to $x=0$ .

See Solution

Problem 1022

Find the instantaneous velocity of the object at $t=2$ for the location function $L(t)=-3 t^{2}-5 t-3$ . Compute $L(2+h)$ , average velocity, and then the limit as $h \to 0$ .

See Solution

Problem 1023

Find the instantaneous velocity of the object at $t=7$ for $s(t)=\frac{1}{2t-5}$ using limits.

See Solution

Problem 1024

Find the instantaneous velocity of the object at $t=5$ for $s(t)=\sqrt{4t-3}$ using limits and exact values.

See Solution

Problem 1025

Find the instantaneous velocity of the object at $t=4$ for $s(t)=-2t^{2}-5t-2$ . Compute $s(4+h)$ and average velocity.

See Solution

Problem 1026

Find the slope of the tangent line to $f(x)=3 x^{2}$ at $x=4$ using the limit definition of the derivative.

See Solution

Problem 1027

For the function $F(x)=-x^{4}+4 x^{2}+192$ , find if $F$ is even/odd, a second local max, and area from $x=-4$ to $x=0$ .

See Solution

Problem 1028

A ball is dropped from a 94 ft building. Find the time to fall half the distance and to ground level: (a) $t=$ sec, (b) $t=$ sec.

See Solution

Problem 1029

A ball drops from a 94 ft building. (Round to 3 decimal places.) (a) Time to fall half distance? $t=1.8$ (b) Time to reach ground? $t=2.4$

See Solution

Problem 1030

Find all numbers $x$ in $(-1,2)$ where $f'(x)=3x^2+4x-1$ equals the average rate of change of $f(x)=x^3+2x^2-x+6$ .

See Solution

Problem 1031

Show that if $y=\frac{5+\sqrt{x}}{\sqrt{x}}$ , then $6 \frac{d y}{d x}+4 x \frac{d^{2} y}{d x^{2}}=0$ .

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Problem 1032

Differentiate the following functions:
1. Constants and powers: (a) $\frac{d}{d x}(7)$ (b) $\frac{d}{d x}(\ln 3)$ (c) $\frac{d}{d x}(x^{3})$ (d) $\frac{d}{d x}(\frac{1}{x^{8}})$ (e) $\frac{d}{d x}(4x^{5})$
2. Find $\frac{d y}{d x}$ for: (a) $y=(x+3)(2x^{2}+4)$ (b) $y=\frac{3x^{2}-7x-4}{x^{3}}$ (c) $y=(4x-1)^{2}(x^{2}-3)$ (product rule) (d) $y=\frac{x-2}{x+4}$ (quotient rule) (e) $y=\frac{e^{3x}}{x}$ (f) $y=\log_{2} x$ (g) $y=\ln(4+9x^{2})$
3. For $y=\sqrt{3x-7}$ , find: (a) $\frac{d y}{d x}$ (b) $\frac{d^{2} y}{d x^{2}}$

See Solution

Problem 1033

Find the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for these functions: (a) $f(x, y)=(x^{2}-1)(y+2)$ (b) $f(x, y)=e^{x+y+1}$ (c) $f(x, y)=\ln (x+y)$

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Problem 1034

Find the limit as $x$ approaches infinity for $\frac{x^{2}}{2^{x}}$ . Evaluate $f(x)=\frac{x^{2}}{2^{x}}$ for $x=0,1,\ldots,100$ .

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Problem 1035

Find the limit as $x$ approaches infinity for $\sqrt{x^{2}+6}$ .

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Problem 1036

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: $v=\frac{0.17[\mathrm{~S}]}{0.021+[\mathrm{S}]}$ .

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Problem 1037

Find the limit: $\lim _{x \rightarrow \infty}\left(\sqrt{25 x^{2}+x}-5 x\right)$ . If it doesn't exist, enter DNE.

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Problem 1038

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: $v=\frac{0.17[S]}{0.021+[S]}$ . What does it indicate?

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Problem 1039

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: $v=\frac{0.17[\mathrm{~S}]}{0.021+[\mathrm{S}]}$ . What does it signify?

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Problem 1040

Bertalanffy growth function:
1. Find $\lim_{t \rightarrow \infty} L(t)$ and interpret.
2. With $L_{0}=1 \mathrm{~cm}$ , $L_{T}=38 \mathrm{~cm}$ , graph $L(t)$ for various $k$ .

See Solution

Problem 1041

Find $\lim_{t \rightarrow \infty} B(t)$ for the biomass model $B(t)=\frac{9 \times 10^{7}}{1+3 e^{-0.72 t}}$ . What does this limit mean?

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Problem 1042

Evaluate $f(x)=\frac{x^{2}-3 x}{x^{2}-x-6}$ at $x=3.5,3.1,...,2.999$ and find $\lim_{x \to 3} f(x) \approx 0.600000$ .

See Solution

Problem 1043

Evaluate $f(x)=\frac{x^{2}-8 x}{x^{2}-7 x-8}$ at $x=0,-0.5,-0.9,-0.95,-0.99,-0.999,-2,-1.5,-1.1,-1.01,-1.001$ . Find $\lim_{x \to -1} f(x)$ to six decimal places.

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Problem 1044

Find the tangent line equation for the polar curve $r=3 \cos \theta$ at $\theta=\frac{\pi}{3}$ .

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Problem 1045

Find the slope of the tangent line for $r=3 \cos \theta$ at $\theta=\frac{\pi}{3}$ .

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Problem 1046

Find the average rate of change of $f(x)=-2 x^{3}+4 x+7$ on $[-1,3]$ . Is it even, odd, or neither?

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Problem 1047

Find $a$ so that $\lim_{{x \rightarrow \infty}} \frac{ax^{3}+4x^{2}-x-1}{3x^{3}+x^{2}-9} = -5$ .

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Problem 1048

An automobile accelerated from rest to $345.6 \mathrm{~m/s}$ in $3.07 \mathrm{~s}$ . What distance did it travel?

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Problem 1049

How long does it take for a football to fall $1719 \mathrm{~m}$ on Mars? $(\mathrm{g}=3.72 \mathrm{~m}/\mathrm{s}^2)$ ?

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Problem 1050

Find the integral of $\frac{\cos (t)}{\sin ^{2}(t)}$ with respect to $t$ .

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Problem 1051

Find the integral of $\frac{\sin (t)}{\cos ^{2}(t)}$ with respect to $t$ .

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Problem 1052

Evaluate the integral $\int t^{5} \ln (t) \, dt$ .

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Problem 1053

Find the limit: $\lim _{x \rightarrow 2}[f(x)+5 g(x)]$ given $\lim _{x \rightarrow 2} f(x)=9$ and $\lim _{x \rightarrow 2} g(x)=-5$ .

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Problem 1054

Evaluate the limit: $\lim _{x \rightarrow-3} \frac{x^{2}+3 x}{x^{2}-5 x-24}$ (Enter DNE if it doesn't exist.)

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Problem 1055

Evaluate the limit: $\lim _{x \rightarrow 2} \frac{x^{2}+6 x+4}{x-2}$ . If it doesn't exist, enter DNE.

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Problem 1056

Find the limit: $\lim _{u \rightarrow-5} \frac{u+5}{u^{3}+125}$ . Simplify and evaluate the limit, or state DNE.

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Problem 1057

Find the limit: $\lim _{u \rightarrow-5} \frac{u+5}{u^{3}+125}$ and simplify the expression.

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Problem 1058

Find the limit: $\lim _{x \rightarrow 2} \frac{2-x}{\sqrt{x+2}-2}$ . Simplify and evaluate if possible.

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Problem 1059

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{(h-4)^{2}-16}{h}$ . If it doesn't exist, write DNE.

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Problem 1060

Find the limit: $\lim _{x \rightarrow 2} \frac{2-x}{\sqrt{x+2}-2}$ and simplify the expression.

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Problem 1061

Find the limit: $\lim _{x \rightarrow 2} \frac{2-x}{\sqrt{x+2}-2}$ and simplify to $\lim _{x \rightarrow 2}(\sqrt{x+2}+2 x)$ .

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Problem 1062

Evaluate the limits and value for the piecewise function $g(x)$ at $x=1$ and $x=2$ .

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Problem 1063

Find the limit as $t$ approaches 0 from the right of $\log(t)$ .

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Problem 1064

Find $\delta$ such that if $|x-4|<\delta$ , then $|\sqrt{x}-2|<0.4$ for the function $f(x)=\sqrt{x}$ .

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Problem 1065

Find the limit: $\lim _{t \rightarrow 0^{-}} \ln (-t)$

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Problem 1066

Calculate the limit: $\lim _{t \rightarrow 5^{+}} \ln (t-5)$ .

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Problem 1067

Find the largest $\delta$ for $\varepsilon=0.2$ and $\varepsilon=0.1$ in the limit $\lim _{x \rightarrow 2}(x^{3}-5 x+5)=3$ . Round to four decimal places.

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Problem 1068

Find $\delta$ for $\varepsilon=0.1$ in the limit: $\lim _{x \rightarrow 6}-3 x-4=-22$ .

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Problem 1069

Prove that $\lim _{x \rightarrow 8}\left(\frac{1}{7} x+6\right)=\frac{50}{7}$ by finding $\delta$ in terms of $\varepsilon$ .

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Problem 1070

Find $\delta$ for $f(x)=x^{\frac{2}{3}}$ as $x \to 1$ with $\varepsilon=0.001$ such that $|f(x)-L|<\varepsilon$ .

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Problem 1071

Prove that $\lim _{x \rightarrow 6}\left(\frac{1}{7} x-9\right)=-\frac{57}{7}$ by finding $\delta$ for any $\varepsilon>0$ .

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Problem 1072

Find the largest $\delta$ for $\lim _{x \rightarrow 2}\left(x^{3}-5 x+5\right)=3$ with $\varepsilon=0.2$ and $\varepsilon=0.1$ . Round to four decimal places.

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Problem 1073

Find the limit: $\lim _{x \rightarrow 2}\left(\frac{9 x-6}{3 x^{2}-10 x+8}-\frac{9 x+24}{4 x^{2}-9 x+2}\right)$ .

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Problem 1074

Find the values of $x$ where $f(x)=8x+5$ has a local maximum. Options: A. $x=$ (integers), B. No local maximum.

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Problem 1075

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=-6x+5$ , with $h \neq 0$ . Simplify your answer.

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Problem 1076

Find the difference quotient of $f(x)=\frac{1}{x+2}$ and simplify.

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Problem 1077

Determine the marginal cost from the cost function $C(x, y) = 240,000 + 6,000x + 4,000y$ .

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Problem 1078

Given the function $f(x)=\left\{\begin{array}{lll}2 x+18 & \text { if } & x<-6 \\ \sqrt{x+42} & \text { if } & x>-6 \\ 2 & \text { if } & x=-6\end{array}\right.$ , determine the truth of the following statements about $f(-6)$ and its limits.

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Problem 1079

Find $h^{\prime}(-5)$ if $h(x)=f(x)-g(x)$ , given $f(-5)=8, f'(-5)=-2, g(-5)=3, g'(-5)=5$ .

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Problem 1080

Find the integral of $\frac{1}{x^{2}-x+1}$ with respect to $x$ .

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Problem 1081

Find the limit of the fish length model $L(t)=L_{T}-\left(L_{T}-L_{0}\right) e^{-k t}$ as $t \to \infty$ .

See Solution

Problem 1082

Un cohete "Pioneer" alcanzó $125.000 \mathrm{~km}$ . ¿Cuál es su velocidad al regresar a la atmósfera a $130 \mathrm{~km}$ ?

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Problem 1083

Demostrar que la serie $\sum_{n=1}^{\infty}(\operatorname{sen} n x) / n^{2}$ converge para todo $x$ , y que $f(x)$ es continua en $[0, \pi]$ . Luego, probar que $\int_{0}^{\pi} f(x) d x=2 \sum_{n=1}^{\infty} \frac{1}{(2 n-1)^{3}}$ .

See Solution

Problem 1084

Find the limit as $x$ approaches 1 for the expression $2 - 3x$ .

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Problem 1085

Evaluate the limit: $107 \lim _{x \rightarrow 1}(2-3 x)=-1$

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Problem 1086

Peter has 2000 yards of fencing. Find the rectangle dimensions that maximize the area and state the maximum area.

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Problem 1087

Calculate the horizontal distance a ball travels from a height of 7 feet before hitting the ground, rounding to the nearest tenth.

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Problem 1088

Find the value of $k$ for the function $f(x)=\left\{\begin{array}{ll}\frac{\sqrt{x+3}-\sqrt{3 x-1}}{x-2} & x \neq 2 \\ k & x=2\end{array}\right.$ to be continuous.

See Solution

Problem 1089

Minimize marginal cost for $C(x) = x^{2} - 140x + 7700$ . Find optimal $x$ and minimum cost.

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Problem 1090

Find $\lim _{x \rightarrow 0} \frac{\cos 2 x}{3}$ . Choose from: A) 0 B) $\frac{1}{3}$ C) $\frac{2}{3}$ D) does not exist.

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Problem 1091

Find the limit: $271 \lim _{x \rightarrow+\infty} \frac{4 x-1}{2 x+1}$ . What is the result?

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Problem 1092

Find the limit as $x$ approaches infinity for $\frac{4x - 1}{2x + 1}$ .

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Problem 1093

Find the slope of the tangent line to $y=x^{3}$ at the point (1,1).

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Problem 1094

Estimate the slope of the tangent line to $y=x^{1/2}$ at the point $(1,1)$ .

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Problem 1095

Find $\frac{d y}{d x}$ using implicit differentiation for $xy=35$ at the point $(-5,-7)$ . What is the slope?

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Problem 1096

Calculate the integral $\int_{1}^{5}\left(x+\frac{2}{x^{2}}\right) d x$ using the trapezium rule at $x=1,2,3,4,5$ , rounded to two decimal places.

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Problem 1097

Find the limit as $x$ approaches -2 for $\frac{x^{3}+8}{x+2}$ .

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Problem 1098

Does Lagrange's mean value theorem apply to $f(x)=x^{1/3}$ on $[-1, 1]$ ? What can we conclude?

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Problem 1099

Calculate the limit: $\lim _{x \rightarrow+\infty} \frac{2-x}{3}(x^{3}-1)$ .

See Solution

Problem 1100

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=x^{2}-7x+2$ , where $h \neq 0$ .

See Solution

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Calculus

Problem 1001

Find the interval of convergence for the series ∑n=1∞(x−1)n3n\sum_{n=1}^{\infty} \frac{(x-1)^{n}}{3^{n}}∑n=1∞​3n(x−1)n​.

Problem 1002

Identify the true statement about the function f(x)=(x−3)(x−7)2f(x)=(x-3)(x-7)^{2}f(x)=(x−3)(x−7)2 from the options given.

Problem 1003

Problem 1004

Find the slope of the tangent line to f−1f^{-1}f−1 at the point (1,−1)(1,-1)(1,−1) for f(x)=(x+2)2f(x)=(x+2)^{2}f(x)=(x+2)2 where x≥−2x \geq-2x≥−2. The slope is □\square□.

Problem 1005

Find the limit: lim⁡x→−4x2−2x−246x2+23x−4\lim _{x \rightarrow-4} \frac{x^{2}-2 x-24}{6 x^{2}+23 x-4}limx→−4​6x2+23x−4x2−2x−24​.

Problem 1006

Find the derivative f′(90∘)f'(90^\circ)f′(90∘) for the function f(x)=sin⁡x−3cos⁡xf(x) = \sin x - 3 \cos xf(x)=sinx−3cosx.

Problem 1007

Find f′(90∘)f'(90^{\circ})f′(90∘) for f(x)=sin⁡x−3cos⁡xf(x) = \sin x - 3\cos xf(x)=sinx−3cosx and evaluate 3lim⁡x→0cos⁡2x−11cos⁡x−13 \lim_{x \rightarrow 0} \frac{\cos 2x - 11}{\cos x - 1}3limx→0​cosx−1cos2x−11​.

Problem 1008

Find the limit: lim⁡x→∞(x+42−3x)3\lim _{x \rightarrow \infty}\left(\frac{x+4}{2-3 x}\right)^{3}limx→∞​(2−3xx+4​)3.

Problem 1009

Find the limit as xxx approaches infinity of (x+42−3x)3\left(\frac{x+4}{2-3 x}\right)^{3}(2−3xx+4​)3.

Problem 1010

Find the limit: lim⁡a→1a3−aa2−1\lim _{a \rightarrow 1} \frac{a^{3}-a}{a^{2}-1}lima→1​a2−1a3−a​.

Problem 1011

Find the limit as xxx approaches -3 for the expression x2+9x+18x+3\frac{x^{2}+9x+18}{x+3}x+3x2+9x+18​.

Problem 1012

Find the limit: lim⁡x→0−3x3x+64−8\lim _{x \rightarrow 0} \frac{-3 x}{\sqrt{3 x+64}-8}limx→0​3x+64​−8−3x​.

Problem 1013

Find the limit: lim⁡x→−3x2−94x+12\lim _{x \rightarrow-3} \frac{x^{2}-9}{4 x+12}limx→−3​4x+12x2−9​.

Problem 1014

수렴하는 수열을 고르세요: ᄀ. {12n+1}\left\{\frac{1}{2 n+1}\right\}{2n+11​}, ㄴ. {(−1)nn}\left\{\frac{(-1)^{n}}{n}\right\}{n(−1)n​}, ᄃ. {(−1)n+2}\left\{(-1)^{n}+2\right\}{(−1)n+2}.

Problem 1015

A conical water tank has a semi-vertical angle of tan⁡−134\tan^{-1} \frac{3}{4}tan−143​. Water is added at 6 m³/hour. Find the rate of increase of the wet curved surface area (m²/hour) when the water depth is 4 m.

Problem 1016

Find f′(−12)f^{\prime}(-12)f′(−12) if the tangent line at (−12,−9)(-12,-9)(−12,−9) passes through (−1,46)(-1,46)(−1,46). f′(−12)=f^{\prime}(-12)=f′(−12)=

Problem 1017

Find the average rate of change of f(x)=1x+2f(x)=\frac{1}{x+2}f(x)=x+21​ on [8,8+h][8,8+h][8,8+h]. Express your answer in terms of hhh.

Problem 1018

Find the derivative f′(0)f^{\prime}(0)f′(0) for the function f(x)=6+2x−3x2f(x)=6+2x-3x^{2}f(x)=6+2x−3x2 using the definition of the derivative.

Problem 1019

Problem 1020

Find the average rate of change of f(x)=x3−6x+4f(x)=x^{3}-6 x+4f(x)=x3−6x+4 for the intervals: (a) -7 to -4, (b) -2 to 2, (c) 2 to 7.

Problem 1021

Given F(x)=−x4+2x2+224F(x)=-x^{4}+2 x^{2}+224F(x)=−x4+2x2+224, find if FFF is even/odd, a second local max, and the area from x=−4x=-4x=−4 to x=0x=0x=0.

Problem 1022

Find the instantaneous velocity of the object at t=2t=2t=2 for the location function L(t)=−3t2−5t−3L(t)=-3 t^{2}-5 t-3L(t)=−3t2−5t−3. Compute L(2+h)L(2+h)L(2+h), average velocity, and then the limit as h→0h \to 0h→0.

Problem 1023

Find the instantaneous velocity of the object at t=7t=7t=7 for s(t)=12t−5s(t)=\frac{1}{2t-5}s(t)=2t−51​ using limits.

Problem 1024

Find the instantaneous velocity of the object at t=5t=5t=5 for s(t)=4t−3s(t)=\sqrt{4t-3}s(t)=4t−3​ using limits and exact values.

Problem 1025

Find the instantaneous velocity of the object at t=4t=4t=4 for s(t)=−2t2−5t−2s(t)=-2t^{2}-5t-2s(t)=−2t2−5t−2. Compute s(4+h)s(4+h)s(4+h) and average velocity.

Problem 1026

Find the slope of the tangent line to f(x)=3x2f(x)=3 x^{2}f(x)=3x2 at x=4x=4x=4 using the limit definition of the derivative.

Problem 1027

For the function F(x)=−x4+4x2+192F(x)=-x^{4}+4 x^{2}+192F(x)=−x4+4x2+192, find if FFF is even/odd, a second local max, and area from x=−4x=-4x=−4 to x=0x=0x=0.

Problem 1028

A ball is dropped from a 94 ft building. Find the time to fall half the distance and to ground level: (a) t=t=t= sec, (b) t=t=t= sec.

Problem 1029

A ball drops from a 94 ft building. (Round to 3 decimal places.) (a) Time to fall half distance? t=1.8 t=1.8 t=1.8 (b) Time to reach ground? t=2.4 t=2.4 t=2.4

Problem 1030

Find all numbers xxx in (−1,2)(-1,2)(−1,2) where f′(x)=3x2+4x−1f'(x)=3x^2+4x-1f′(x)=3x2+4x−1 equals the average rate of change of f(x)=x3+2x2−x+6f(x)=x^3+2x^2-x+6f(x)=x3+2x2−x+6.

Problem 1031

Show that if y=5+xxy=\frac{5+\sqrt{x}}{\sqrt{x}}y=x​5+x​​, then 6dydx+4xd2ydx2=06 \frac{d y}{d x}+4 x \frac{d^{2} y}{d x^{2}}=06dxdy​+4xdx2d2y​=0.

Problem 1032

Problem 1033

Problem 1034

Find the limit as xxx approaches infinity for x22x\frac{x^{2}}{2^{x}}2xx2​. Evaluate f(x)=x22xf(x)=\frac{x^{2}}{2^{x}}f(x)=2xx2​ for x=0,1,…,100x=0,1,\ldots,100x=0,1,…,100.

Problem 1035

Find the limit as xxx approaches infinity for x2+6\sqrt{x^{2}+6}x2+6​.

Problem 1036

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: v=0.17[ S]0.021+[S]v=\frac{0.17[\mathrm{~S}]}{0.021+[\mathrm{S}]}v=0.021+[S]0.17[ S]​.

Problem 1037

Find the limit: lim⁡x→∞(25x2+x−5x)\lim _{x \rightarrow \infty}\left(\sqrt{25 x^{2}+x}-5 x\right)x→∞lim​(25x2+x​−5x). If it doesn't exist, enter DNE.

Problem 1038

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: v=0.17[S]0.021+[S]v=\frac{0.17[S]}{0.021+[S]}v=0.021+[S]0.17[S]​. What does it indicate?

Problem 1039

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: v=0.17[ S]0.021+[S]v=\frac{0.17[\mathrm{~S}]}{0.021+[\mathrm{S}]}v=0.021+[S]0.17[ S]​. What does it signify?

Problem 1040

Bertalanffy growth function: 1. Find lim⁡t→∞L(t)\lim_{t \rightarrow \infty} L(t)limt→∞​L(t) and interpret.2. With L0=1 cmL_{0}=1 \mathrm{~cm}L0​=1 cm, LT=38 cmL_{T}=38 \mathrm{~cm}LT​=38 cm, graph L(t)L(t)L(t) for various kkk.

Problem 1041

Find lim⁡t→∞B(t)\lim_{t \rightarrow \infty} B(t)limt→∞​B(t) for the biomass model B(t)=9×1071+3e−0.72tB(t)=\frac{9 \times 10^{7}}{1+3 e^{-0.72 t}}B(t)=1+3e−0.72t9×107​. What does this limit mean?

Problem 1042

Find the interval of convergence for the series $\sum_{n=1}^{\infty} \frac{(x-1)^{n}}{3^{n}}$ .

Identify the true statement about the function $f(x)=(x-3)(x-7)^{2}$ from the options given.

Find the slope of the tangent line to $f^{-1}$ at the point $(1,-1)$ for $f(x)=(x+2)^{2}$ where $x \geq-2$ . The slope is $\square$ .

Find the limit: $\lim _{x \rightarrow-4} \frac{x^{2}-2 x-24}{6 x^{2}+23 x-4}$ .

Find the derivative $f'(90^\circ)$ for the function $f(x) = \sin x - 3 \cos x$ .

Find $f'(90^{\circ})$ for $f(x) = \sin x - 3\cos x$ and evaluate $3 \lim_{x \rightarrow 0} \frac{\cos 2x - 11}{\cos x - 1}$ .

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

Find the limit as $x$ approaches infinity of $\left(\frac{x+4}{2-3 x}\right)^{3}$ .

Find the limit: $\lim _{a \rightarrow 1} \frac{a^{3}-a}{a^{2}-1}$ .

Find the limit as $x$ approaches -3 for the expression $\frac{x^{2}+9x+18}{x+3}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{-3 x}{\sqrt{3 x+64}-8}$ .

Find the limit: $\lim _{x \rightarrow-3} \frac{x^{2}-9}{4 x+12}$ .

수렴하는 수열을 고르세요: ᄀ. $\left\{\frac{1}{2 n+1}\right\}$ , ㄴ. $\left\{\frac{(-1)^{n}}{n}\right\}$ , ᄃ. $\left\{(-1)^{n}+2\right\}$ .

A conical water tank has a semi-vertical angle of $\tan^{-1} \frac{3}{4}$ . Water is added at 6 m³/hour. Find the rate of increase of the wet curved surface area (m²/hour) when the water depth is 4 m.

Find $f^{\prime}(-12)$ if the tangent line at $(-12,-9)$ passes through $(-1,46)$ . $f^{\prime}(-12)=$

Find the average rate of change of $f(x)=\frac{1}{x+2}$ on $[8,8+h]$ . Express your answer in terms of $h$ .

Find the derivative $f^{\prime}(0)$ for the function $f(x)=6+2x-3x^{2}$ using the definition of the derivative.

Find the average rate of change of $f(x)=x^{3}-6 x+4$ for the intervals: (a) -7 to -4, (b) -2 to 2, (c) 2 to 7.

Given $F(x)=-x^{4}+2 x^{2}+224$ , find if $F$ is even/odd, a second local max, and the area from $x=-4$ to $x=0$ .

Find the instantaneous velocity of the object at $t=2$ for the location function $L(t)=-3 t^{2}-5 t-3$ . Compute $L(2+h)$ , average velocity, and then the limit as $h \to 0$ .

Find the instantaneous velocity of the object at $t=7$ for $s(t)=\frac{1}{2t-5}$ using limits.

Find the instantaneous velocity of the object at $t=5$ for $s(t)=\sqrt{4t-3}$ using limits and exact values.

Find the instantaneous velocity of the object at $t=4$ for $s(t)=-2t^{2}-5t-2$ . Compute $s(4+h)$ and average velocity.

Find the slope of the tangent line to $f(x)=3 x^{2}$ at $x=4$ using the limit definition of the derivative.

For the function $F(x)=-x^{4}+4 x^{2}+192$ , find if $F$ is even/odd, a second local max, and area from $x=-4$ to $x=0$ .

A ball is dropped from a 94 ft building. Find the time to fall half the distance and to ground level: (a) $t=$ sec, (b) $t=$ sec.

A ball drops from a 94 ft building. (Round to 3 decimal places.) (a) Time to fall half distance? $t=1.8$ (b) Time to reach ground? $t=2.4$

Find all numbers $x$ in $(-1,2)$ where $f'(x)=3x^2+4x-1$ equals the average rate of change of $f(x)=x^3+2x^2-x+6$ .

Show that if $y=\frac{5+\sqrt{x}}{\sqrt{x}}$ , then $6 \frac{d y}{d x}+4 x \frac{d^{2} y}{d x^{2}}=0$ .

Find the limit as $x$ approaches infinity for $\frac{x^{2}}{2^{x}}$ . Evaluate $f(x)=\frac{x^{2}}{2^{x}}$ for $x=0,1,\ldots,100$ .

Find the limit as $x$ approaches infinity for $\sqrt{x^{2}+6}$ .

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: $v=\frac{0.17[\mathrm{~S}]}{0.021+[\mathrm{S}]}$ .

Find the limit: $\lim _{x \rightarrow \infty}\left(\sqrt{25 x^{2}+x}-5 x\right)$ . If it doesn't exist, enter DNE.

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: $v=\frac{0.17[S]}{0.021+[S]}$ . What does it indicate?

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: $v=\frac{0.17[\mathrm{~S}]}{0.021+[\mathrm{S}]}$ . What does it signify?

Bertalanffy growth function:
1. Find $\lim_{t \rightarrow \infty} L(t)$ and interpret.
2. With $L_{0}=1 \mathrm{~cm}$ , $L_{T}=38 \mathrm{~cm}$ , graph $L(t)$ for various $k$ .

Find $\lim_{t \rightarrow \infty} B(t)$ for the biomass model $B(t)=\frac{9 \times 10^{7}}{1+3 e^{-0.72 t}}$ . What does this limit mean?

Evaluate $f(x)=\frac{x^{2}-3 x}{x^{2}-x-6}$ at $x=3.5,3.1,...,2.999$ and find $\lim_{x \to 3} f(x) \approx 0.600000$ .

Find the tangent line equation for the polar curve $r=3 \cos \theta$ at $\theta=\frac{\pi}{3}$ .

Find the slope of the tangent line for $r=3 \cos \theta$ at $\theta=\frac{\pi}{3}$ .

Find the average rate of change of $f(x)=-2 x^{3}+4 x+7$ on $[-1,3]$ . Is it even, odd, or neither?

Find $a$ so that $\lim_{{x \rightarrow \infty}} \frac{ax^{3}+4x^{2}-x-1}{3x^{3}+x^{2}-9} = -5$ .

An automobile accelerated from rest to $345.6 \mathrm{~m/s}$ in $3.07 \mathrm{~s}$ . What distance did it travel?

How long does it take for a football to fall $1719 \mathrm{~m}$ on Mars? $(\mathrm{g}=3.72 \mathrm{~m}/\mathrm{s}^2)$ ?

Find the integral of $\frac{\cos (t)}{\sin ^{2}(t)}$ with respect to $t$ .

Find the integral of $\frac{\sin (t)}{\cos ^{2}(t)}$ with respect to $t$ .

Evaluate the integral $\int t^{5} \ln (t) \, dt$ .

Find the limit: $\lim _{x \rightarrow 2}[f(x)+5 g(x)]$ given $\lim _{x \rightarrow 2} f(x)=9$ and $\lim _{x \rightarrow 2} g(x)=-5$ .

Evaluate the limit: $\lim _{x \rightarrow-3} \frac{x^{2}+3 x}{x^{2}-5 x-24}$ (Enter DNE if it doesn't exist.)

Evaluate the limit: $\lim _{x \rightarrow 2} \frac{x^{2}+6 x+4}{x-2}$ . If it doesn't exist, enter DNE.

Find the limit: $\lim _{u \rightarrow-5} \frac{u+5}{u^{3}+125}$ . Simplify and evaluate the limit, or state DNE.

Find the limit: $\lim _{u \rightarrow-5} \frac{u+5}{u^{3}+125}$ and simplify the expression.

Find the limit: $\lim _{x \rightarrow 2} \frac{2-x}{\sqrt{x+2}-2}$ . Simplify and evaluate if possible.

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{(h-4)^{2}-16}{h}$ . If it doesn't exist, write DNE.

Find the limit: $\lim _{x \rightarrow 2} \frac{2-x}{\sqrt{x+2}-2}$ and simplify the expression.

Find the limit: $\lim _{x \rightarrow 2} \frac{2-x}{\sqrt{x+2}-2}$ and simplify to $\lim _{x \rightarrow 2}(\sqrt{x+2}+2 x)$ .

Evaluate the limits and value for the piecewise function $g(x)$ at $x=1$ and $x=2$ .

Find the limit as $t$ approaches 0 from the right of $\log(t)$ .

Find $\delta$ such that if $|x-4|<\delta$ , then $|\sqrt{x}-2|<0.4$ for the function $f(x)=\sqrt{x}$ .

Find the limit: $\lim _{t \rightarrow 0^{-}} \ln (-t)$

Calculate the limit: $\lim _{t \rightarrow 5^{+}} \ln (t-5)$ .

Find the largest $\delta$ for $\varepsilon=0.2$ and $\varepsilon=0.1$ in the limit $\lim _{x \rightarrow 2}(x^{3}-5 x+5)=3$ . Round to four decimal places.

Find $\delta$ for $\varepsilon=0.1$ in the limit: $\lim _{x \rightarrow 6}-3 x-4=-22$ .

Prove that $\lim _{x \rightarrow 8}\left(\frac{1}{7} x+6\right)=\frac{50}{7}$ by finding $\delta$ in terms of $\varepsilon$ .

Find $\delta$ for $f(x)=x^{\frac{2}{3}}$ as $x \to 1$ with $\varepsilon=0.001$ such that $|f(x)-L|<\varepsilon$ .

Prove that $\lim _{x \rightarrow 6}\left(\frac{1}{7} x-9\right)=-\frac{57}{7}$ by finding $\delta$ for any $\varepsilon>0$ .

Find the largest $\delta$ for $\lim _{x \rightarrow 2}\left(x^{3}-5 x+5\right)=3$ with $\varepsilon=0.2$ and $\varepsilon=0.1$ . Round to four decimal places.

Find the limit: $\lim _{x \rightarrow 2}\left(\frac{9 x-6}{3 x^{2}-10 x+8}-\frac{9 x+24}{4 x^{2}-9 x+2}\right)$ .

Find the values of $x$ where $f(x)=8x+5$ has a local maximum. Options: A. $x=$ (integers), B. No local maximum.