Calculus

Problem 2001

Find the limit: $\lim _{x \rightarrow-\infty} \frac{x+3}{x^{2}+16}$ .

See Solution

Problem 2002

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

See Solution

Problem 2003

Find the limit: $\lim _{x \rightarrow \infty} e^{x}=\square$ . Provide the exact answer.

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Problem 2004

Find the limit as $x$ approaches infinity for $e^{-x}$ .

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Problem 2005

Find the limit: $\lim _{x \rightarrow \infty} \frac{4 e^{-x}}{5+4 e^{-x}}$ . What is the value?

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Problem 2006

Determine the truth of the following statements about the function $f(x)$ defined as:
$f(x)=\left\{\begin{array}{ll} \frac{x^{2}-5 x+6}{x-2} & \text { if } x \neq 2 \\ -1 & \text { if } x=2 \end{array}\right.$
1. $f(x)$ has a discontinuity at $x=2$ because $\lim _{x \rightarrow 2^{+}} f^{\prime}(x) \neq \lim _{x \rightarrow 2^{-}} f(x)$
2. $f(x)$ is continuous at $x=2$ since $\lim _{x \rightarrow 2} f(x)=f(2)$
3. $f(x)$ has a discontinuity at $x=2$ since $\lim _{x \rightarrow 2} f(x) \neq f(2)$
4. $f(x)$ is continuous at $x=2$ since $\lim _{x \rightarrow 2^{+}} f(x)=\lim _{x \rightarrow 2^{-}} f(x)$

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Problem 2007

Find $\lim _{x \rightarrow 0} g(x)$ given $3-2 x^{2} \leq g(x) \leq 3 \cos x$ .

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Problem 2008

Find the limit using the squeeze theorem: $\lim _{x \rightarrow 0} x^{2} \sin \left(\frac{4}{x}\right) =$

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Problem 2009

Show that $\lim _{x \rightarrow 0} x^{4} \cos \frac{1}{x}=0$ using the sandwich theorem by bounding $\cos \frac{1}{x}$ .

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Problem 2010

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin (3 x)}{18 x}=$ .

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Problem 2011

Find the limit: $\lim _{x \rightarrow 0} \frac{1-\cos ^{2}(3 x)}{9 x^{2}}$ .

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Problem 2012

Determine if $f(x)=2x+4\cos(x)$ is continuous or discontinuous. Options: continuous or discontinuous.

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Problem 2013

Find where the function $f(x)=\frac{1}{x-5}$ is discontinuous. Identify the type and continuity.

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Problem 2014

Find the derivative of $f(x) = -x + \frac{3}{x} + 1$ .

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Problem 2015

Find the derivative of the function: $f(x)=9 x^{0.5}+4 x^{-0.5}$ . What is $f^{\prime}(x)$ ?

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Problem 2016

Find where the function $f(x)=\cos\left(\frac{1}{x-5}\right)$ for $x \neq 5$ and $0$ for $x=5$ is discontinuous. What type?

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Problem 2017

Analyze the continuity of the function $g(x)$ at $x=-1$ . Is it continuous from both sides, one side, or neither?

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Problem 2018

Maximize the area of a rectangle with fencing costs of \$8/ft (east/west) and \$4/ft (north/south) under a \$96 budget. Use Lagrange multipliers.

See Solution

Problem 2019

Find the limit as $x$ approaches -1 for the expression $8x^3 - 2$ . What is the result?

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Problem 2020

Evaluate the limit: $\lim _{x \rightarrow \pi} \cos \left(\frac{x}{3}-\frac{\pi}{2}\right)$ .

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Problem 2021

Find critical points of $f(x, y)=x^{2}+2xy+2y^{2}-6x+8y$ and classify as max, min, or saddle. Enter DNE if none.

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Problem 2022

Find critical points of $g(x, y)=1-2x^{2}-x-4y^{2}+y$ and classify them as max, min, or saddle.

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Problem 2023

Find the derivative of $g(x)=\frac{1}{x^{7}}+\frac{1}{x^{8}}$ . What is $g^{\prime}(x)$ ?

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Problem 2024

Find the derivative of $h(x)=\frac{4}{x^{2}}+\frac{8}{x^{3}}$ .

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Problem 2025

Determine if the function $f(x)=5 x+3 \cos (x)$ is continuous using the Laws of Continuity.

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Problem 2026

Find the derivative of $r(x)=\frac{2}{3 x^{2}}+\frac{1}{x^{3.9}}$ .

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Problem 2027

Find the derivative of $s(x)=\sqrt[3]{x}+\frac{11}{\sqrt[3]{x}}$ .

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Problem 2028

Evaluate the limit: $\lim _{x \rightarrow-1}\left(1-14 x^{3}\right)^{3 / 2}$ using substitution.

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Problem 2029

Minimize the cost function $C(x, y)=3,000+400 x^{2}+500 y^{2}$ for $x$ (sulfur lbs/day) and $y$ (lead lbs/day).

See Solution

Problem 2030

Find the tangent line equation for $f(x)=5x^2$ at $x=0$ and sketch the curve with the tangent line.

See Solution

Problem 2031

Find the derivative of $f(x)=3.57 x^{2}-30.5 x+81$ for $4.5 \leq x \leq 8.5$ .

See Solution

Problem 2032

Find the discontinuities of the function $f(x)=\begin{cases}\cos \left(\frac{1}{x-5}\right) & x \neq 5 \\ 0 & x=5\end{cases}$ and classify them.

See Solution

Problem 2033

Find the derivative of $r(x)=\frac{2}{3 x^{2}}+\frac{1}{x^{3.9}}$ . What is $r^{\prime}(x)$ ?

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Problem 2034

Find the derivative of the expression $a t^{2}+b t+c$ with respect to $t$ , where $a, b, c$ are constants.

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Problem 2035

Find the slope of the tangent to $g(x)=2 x^{4}$ at the point (-2,32).

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Problem 2036

Find the slope of the tangent for $f(x)=\frac{x}{8}-1$ at the point $(-16,-3)$ . Calculate $f'(-16)$ .

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Problem 2037

Find the derivative $P^{\prime}(t)$ of the oil price function $P(t)=0.45 t^{2}-12 t+105$ and compute $P^{\prime}(23)$ . What does this indicate about oil prices?

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Problem 2038

Find the price $p$ that maximizes profit, given $q = 476000 - 17000p$ and cost per copy is \$4. Use Lagrange multipliers.

See Solution

Problem 2039

Hitung $\int(f(x)-g(x)) d x$ untuk $f(x)=-5+(x-3)^{2}$ dan $g(x)=8-9 x$ . Pilih jawaban yang tepat.

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Problem 2040

Find the limit as $x$ approaches -1 for the expression $\frac{2 x^{2}+3 x+1}{x^{2}+2 x-3}$ .

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Problem 2041

Find the tangent line equation for $f(x)=\frac{3}{x^{2}}$ at $x=1$ . $y=$

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Problem 2042

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

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Problem 2043

Evaluate the limit: $\lim _{x \rightarrow \pi} \cos \left(\frac{x}{2}-\frac{2 \pi}{3}\right)$ .

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Problem 2044

Maximize $f(x, y)=x y$ with constraint $y=3-x^{2}$ . Find $f_{\text {max }}$ and point $(x, y)$ .

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Problem 2045

Minimize $f(x, y)=x^{2}+y^{2}$ with constraint $x+2y=15$ . Find $f_{\min }$ and the point $(x, y)$ .

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Problem 2046

Find the derivative $f^{\prime}(x)$ of the function $f(x)=3.57 x^{2}-30.5 x+81$ for $4.5 \leq x \leq 8.5$ .

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Problem 2047

Find the integral of $\sin x \cos^{6} x \, dx$ . Choose the correct answer from the options provided.

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Problem 2048

Find the integral $\int u \sqrt{2u-9} \, du$ and choose the correct answer from the options given.

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Problem 2049

Find the integral: $\int \frac{9 x^{2}}{e^{x}} d x$ . Choose the correct answer from the options provided.

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Problem 2050

Find the integral: $\int u \sqrt{2 u-9} du$ . Choose the correct answer from the options provided.

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Problem 2051

Evaluate the integral: $\int u \sqrt{2 u-9} d u$ and choose the correct answer from the options given.

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Problem 2052

Find the integral of $\frac{13 q}{q-12}$ with respect to $q$ . Choose the correct answer from the options.

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Problem 2053

Evaluate the integral: $\int \tan ^{3} \frac{\pi x}{7} \sec ^{2} \frac{\pi x}{7} d x$ . Find the correct answer from the options.

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Problem 2054

Evaluate the integral $\int e^{9 x} \sqrt{1-e^{18 x}} d x$ and identify the correct expression from the options.

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Problem 2055

Calculate the integral: $\int \frac{x^{2}}{x-7} d x$ and choose the correct answer from the options.

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Problem 2056

Find the integral: $\int \sin ^{3} \frac{x}{7} d x$ . Choose the correct answer from the options provided.

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Problem 2057

Evaluate the integral: $\int \frac{\cos ^{3} 11 \theta}{\sqrt{\sin 11 \theta}} d \theta$ . Choose the correct answer from options (A) to (E).

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Problem 2058

Evaluate the integral: $\int \frac{\cos^{3} 11 \theta}{\sqrt{\sin 11 \theta}} d \theta$ . Choose the correct answer from options (A)-(E).

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Problem 2059

Find the integral $\int \sec ^{6} 8 x \, dx$ and choose the correct answer from the options provided.

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Problem 2060

Find the integral $\int 4 \sec ^{2} x e^{\tan x} \mathrm{dx}$ . Choose the correct answer: (A) $4 \tan x+c$ , (B) $4 e^{\tan x}+C$ , (C) $4 \tan x \sec x+C$ , (D) $4 \sec x+C$ , (E) $e^{5 \tan x}+C$ .

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Problem 2061

Find the integral of $\sin^{2}(7x) \, dx$ . Which is the correct answer? (A) $\frac{7x - \sin(7x) \cos(7x)}{7} + C$ (B) $\frac{7x - \sin(7x) \cos(7x)}{14} + C$ (C) $\frac{7x + \sin(7x) \cos(7x)}{14} + C$ (D) $\frac{7x + \sin(7x) \cos(7x)}{7} + C$ (E) $\frac{7x - \sin^{2}(7x) \cos(7x)}{14} + C$

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Problem 2062

Find the integral $\int 4 \sec ^{2} x e^{\tan x} \mathrm{~d} x$ . Choose the correct answer from the options.

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Problem 2063

Evaluate the integral $\int \cos ^{3} 5 x \sin ^{2} 5 x \, dx$ and choose the correct answer from the options.

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Problem 2064

Calculate the integral $\int 4 \sec ^{2} x e^{\tan x} \mathrm{~d} x$ . Choose the correct answer from the options.

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Problem 2065

Find the integral: $\int \tan^{3} 5x \, dx$ . Choose the correct answer from the options provided.

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Problem 2066

Calculate the integral: $\int \frac{4 x+3}{e^{7 x}} d x$ . Choose the correct result from the options provided.

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Problem 2067

Find the integral of $x^{7}$ : $\int x^{7} d x$ . Choose the correct answer: (A) (B) (C) (D) (E)

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Problem 2068

Evaluate the integral: $\int x(5 x^{2}+2)^{3} dx$ . Choose the correct answer from the options provided.

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Problem 2069

Find $f(x)$ with $f(0)=0$ and slope of tangent line as $x\left(x^{2}+2\right)^{3}$ . Choose from options (A)-(E).

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Problem 2070

Find the integral: $\int(6 x^{23}-x^{-6}+2) dx$ . Which option is correct?

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Problem 2071

Evaluate the integral: $\int 2 x \sqrt{4 x^{2}-4 d x}$ . Choose the correct answer from the options provided.

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Problem 2072

Evaluate the integral: $\int 2 x \sqrt{4 x^{2}-4} d x$ . Choose the correct answer from the options.

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Problem 2073

Find the integral of $(3x+5)^{5}$ with respect to $x$ . Choose the correct answer from the options.

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Problem 2074

Find the integral of $\frac{x^{2.6}}{5}-3.4$ with respect to $x$ .

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Problem 2075

Find the integral: $\int\left((6 x-1) e^{6 x^{2}-2 x}+5 x e^{x^{2}}\right) d x$ . Which option is correct?

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Problem 2076

Evaluate the integral: $\int \frac{-2 x-7}{\left(x^{2}+7 x+7\right)^{3}} d x$ . Choose the correct option.

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Problem 2077

Find the integral of $e^{-4x} \, dx$ using $u = -4x$ . What is the result?

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Problem 2078

Evaluate the integral: $\int x(x+9)^{6} d x$ . Choose the correct answer from the options provided.

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Problem 2079

Find the expression for distance $s$ in terms of time $t$ given $v=t(t^{2}+4)^{4}+5t$ and $s=0$ at $t=0$ .

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Problem 2080

Calculate the integral: $\int(1+24.3 e^{2.7 x-4}) dx$ . Choose the correct answer from the options.

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Problem 2081

Find $f(x)$ given $f(4)=-8$ and slope of tangent line is $7 e^{x}+4$ . Choose the correct function from options (A) to (E).

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Problem 2082

Find the average rate of change of $g(x)=-3 x^{2}+3$ between $x=4$ and $x=9$ . Simplify your answer.

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Problem 2083

Jeremy's parachute jump shows velocity from 0 to 50 m/s in 14 seconds, then constant at 50 m/s.
a) Find acceleration at $t=10 \mathrm{~s}$ (1 decimal place).
b) Calculate average speed for this segment (2 significant figures).

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Problem 2084

How long to double an investment at $16\%$ compounded continuously? Round to two decimal places.

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Problem 2085

Show that $f(x)=x^{3}+x-3$ has a solution in $(1,2)$ using the Intermediate Value Theorem. Is $f(x)$ continuous? A, B, C, or D?

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Problem 2086

Use the Intermediate Value Theorem to find a solution for $-x^{3}+5 x^{2}-3 x=2$ in the interval $(-1,5)$ . Calculate $f(5)$ .

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Problem 2087

Find the limit: $\lim _{h \rightarrow 0} \frac{\frac{4}{5+h}-\frac{4}{5}}{h}$ or state if it doesn't exist.

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Problem 2088

Find the limit: $\lim _{x \rightarrow 2} \frac{x-2}{\sqrt{7 x+2}-4}$ or state it does not exist. Options: A. Answer = $\square$ B. Limit does not exist.

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Problem 2089

Calcula la integral definida de $\sin(x)$ desde $0$ hasta $\pi$ : $\int_0^{\pi} \sin(x) \, dx$ .

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Problem 2090

Find the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{289+h}-17}{h}$ . Choose A for the limit value or B if it doesn't exist.

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Problem 2091

Find the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{289+h}-17}{h}$ . Choose A for the limit value or B if it doesn't exist.

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Problem 2092

Find the derivative of $f(x)=-3x+1$ at $x=4$ . What is $f^{\prime}(4)=\square$ ?

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Problem 2093

Find the tangent line equation for $f(x)=x^{2}+7$ at the point $(-5,32)$ . $y=$

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Problem 2094

Find the tangent line equation to $y=9 \sqrt{x}$ at $(1,9)$ . The equation is $y=$

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Problem 2095

Find the derivative $f'$ of $f(x)=2x^2+3x-2$ and the tangent line at $(0, f(0))$ .

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Problem 2096

Find $f^{\prime}(x)$ and the tangent line at $x=1$ for $f(x)=-3 x^{2}-5 x+1$ .

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Problem 2097

Find the derivative $f^{\prime}$ of $f(x)=\sqrt{3x+1}$ and the tangent line at $(5, f(5))$ .

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Problem 2098

Find the derivative $f^{\prime}$ of $f(x)=\frac{2}{3x+5}$ and the tangent line at $a=-1$ .

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Problem 2099

Find the slope of the tangent line to $f(x)=x^{4}+3 x^{3}-x^{2}+7$ at $x=-1$ .

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Problem 2100

Find where the tangent line of $f(x)=x^{4}+x^{2}+3$ is horizontal.

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Calculus

Problem 2001

Find the limit: lim⁡x→−∞x+3x2+16\lim _{x \rightarrow-\infty} \frac{x+3}{x^{2}+16}limx→−∞​x2+16x+3​.

Problem 2002

Find the limit: lim⁡x→−∞ex\lim _{x \rightarrow-\infty} e^{x}limx→−∞​ex.

Problem 2003

Find the limit: lim⁡x→∞ex=□\lim _{x \rightarrow \infty} e^{x}=\squarelimx→∞​ex=□. Provide the exact answer.

Problem 2004

Find the limit as xxx approaches infinity for e−xe^{-x}e−x.

Problem 2005

Find the limit: lim⁡x→∞4e−x5+4e−x\lim _{x \rightarrow \infty} \frac{4 e^{-x}}{5+4 e^{-x}}x→∞lim​5+4e−x4e−x​. What is the value?

Problem 2006

Problem 2007

Find lim⁡x→0g(x)\lim _{x \rightarrow 0} g(x)limx→0​g(x) given 3−2x2≤g(x)≤3cos⁡x3-2 x^{2} \leq g(x) \leq 3 \cos x3−2x2≤g(x)≤3cosx.

Problem 2008

Find the limit using the squeeze theorem: lim⁡x→0x2sin⁡(4x)=\lim _{x \rightarrow 0} x^{2} \sin \left(\frac{4}{x}\right) = limx→0​x2sin(x4​)=

Problem 2009

Show that lim⁡x→0x4cos⁡1x=0\lim _{x \rightarrow 0} x^{4} \cos \frac{1}{x}=0limx→0​x4cosx1​=0 using the sandwich theorem by bounding cos⁡1x\cos \frac{1}{x}cosx1​.

Problem 2010

Find the limit: lim⁡x→0sin⁡(3x)18x=\lim _{x \rightarrow 0} \frac{\sin (3 x)}{18 x}=x→0lim​18xsin(3x)​=.

Problem 2011

Find the limit: lim⁡x→01−cos⁡2(3x)9x2\lim _{x \rightarrow 0} \frac{1-\cos ^{2}(3 x)}{9 x^{2}}limx→0​9x21−cos2(3x)​.

Problem 2012

Determine if f(x)=2x+4cos⁡(x)f(x)=2x+4\cos(x)f(x)=2x+4cos(x) is continuous or discontinuous. Options: continuous or discontinuous.

Problem 2013

Find where the function f(x)=1x−5f(x)=\frac{1}{x-5}f(x)=x−51​ is discontinuous. Identify the type and continuity.

Problem 2014

Find the derivative of f(x)=−x+3x+1f(x) = -x + \frac{3}{x} + 1f(x)=−x+x3​+1.

Problem 2015

Find the derivative of the function: f(x)=9x0.5+4x−0.5f(x)=9 x^{0.5}+4 x^{-0.5}f(x)=9x0.5+4x−0.5. What is f′(x)f^{\prime}(x)f′(x)?

Problem 2016

Find where the function f(x)=cos⁡(1x−5)f(x)=\cos\left(\frac{1}{x-5}\right)f(x)=cos(x−51​) for x≠5x \neq 5x=5 and 000 for x=5x=5x=5 is discontinuous. What type?

Problem 2017

Analyze the continuity of the function g(x)g(x)g(x) at x=−1x=-1x=−1. Is it continuous from both sides, one side, or neither?

Problem 2018

Maximize the area of a rectangle with fencing costs of \$8/ft (east/west) and \$4/ft (north/south) under a \$96 budget. Use Lagrange multipliers.

Problem 2019

Find the limit as xxx approaches -1 for the expression 8x3−28x^3 - 28x3−2. What is the result?

Problem 2020

Evaluate the limit: lim⁡x→πcos⁡(x3−π2)\lim _{x \rightarrow \pi} \cos \left(\frac{x}{3}-\frac{\pi}{2}\right)x→πlim​cos(3x​−2π​).

Problem 2021

Find critical points of f(x,y)=x2+2xy+2y2−6x+8yf(x, y)=x^{2}+2xy+2y^{2}-6x+8yf(x,y)=x2+2xy+2y2−6x+8y and classify as max, min, or saddle. Enter DNE if none.

Problem 2022

Find critical points of g(x,y)=1−2x2−x−4y2+yg(x, y)=1-2x^{2}-x-4y^{2}+yg(x,y)=1−2x2−x−4y2+y and classify them as max, min, or saddle.

Problem 2023

Find the derivative of g(x)=1x7+1x8g(x)=\frac{1}{x^{7}}+\frac{1}{x^{8}}g(x)=x71​+x81​. What is g′(x)g^{\prime}(x)g′(x)?

Problem 2024

Find the derivative of h(x)=4x2+8x3h(x)=\frac{4}{x^{2}}+\frac{8}{x^{3}}h(x)=x24​+x38​.

Problem 2025

Determine if the function f(x)=5x+3cos⁡(x)f(x)=5 x+3 \cos (x)f(x)=5x+3cos(x) is continuous using the Laws of Continuity.

Problem 2026

Find the derivative of r(x)=23x2+1x3.9r(x)=\frac{2}{3 x^{2}}+\frac{1}{x^{3.9}}r(x)=3x22​+x3.91​.

Problem 2027

Find the derivative of s(x)=x3+11x3s(x)=\sqrt[3]{x}+\frac{11}{\sqrt[3]{x}}s(x)=3x​+3x​11​.

Problem 2028

Evaluate the limit: lim⁡x→−1(1−14x3)3/2\lim _{x \rightarrow-1}\left(1-14 x^{3}\right)^{3 / 2}x→−1lim​(1−14x3)3/2 using substitution.

Problem 2029

Minimize the cost function C(x,y)=3,000+400x2+500y2C(x, y)=3,000+400 x^{2}+500 y^{2}C(x,y)=3,000+400x2+500y2 for xxx (sulfur lbs/day) and yyy (lead lbs/day).

Problem 2030

Find the tangent line equation for f(x)=5x2f(x)=5x^2f(x)=5x2 at x=0x=0x=0 and sketch the curve with the tangent line.

Problem 2031

Find the derivative of f(x)=3.57x2−30.5x+81f(x)=3.57 x^{2}-30.5 x+81f(x)=3.57x2−30.5x+81 for 4.5≤x≤8.54.5 \leq x \leq 8.54.5≤x≤8.5.

Problem 2032

Find the discontinuities of the function f(x)={cos⁡(1x−5)x≠50x=5f(x)=\begin{cases}\cos \left(\frac{1}{x-5}\right) & x \neq 5 \\ 0 & x=5\end{cases}f(x)={cos(x−51​)0​x=5x=5​ and classify them.

Problem 2033

Find the derivative of r(x)=23x2+1x3.9r(x)=\frac{2}{3 x^{2}}+\frac{1}{x^{3.9}}r(x)=3x22​+x3.91​. What is r′(x)r^{\prime}(x)r′(x)?

Problem 2034

Find the derivative of the expression at2+bt+ca t^{2}+b t+cat2+bt+c with respect to ttt, where a,b,ca, b, ca,b,c are constants.

Problem 2035

Find the slope of the tangent to g(x)=2x4g(x)=2 x^{4}g(x)=2x4 at the point (-2,32).

Problem 2036

Find the slope of the tangent for f(x)=x8−1f(x)=\frac{x}{8}-1f(x)=8x​−1 at the point (−16,−3)(-16,-3)(−16,−3). Calculate f′(−16)f'(-16)f′(−16).

Problem 2037

Find the derivative P′(t)P^{\prime}(t)P′(t) of the oil price function P(t)=0.45t2−12t+105P(t)=0.45 t^{2}-12 t+105P(t)=0.45t2−12t+105 and compute P′(23)P^{\prime}(23)P′(23). What does this indicate about oil prices?

Problem 2038

Find the price ppp that maximizes profit, given q=476000−17000pq = 476000 - 17000pq=476000−17000p and cost per copy is \$4. Use Lagrange multipliers.

Problem 2039

Hitung ∫(f(x)−g(x))dx\int(f(x)-g(x)) d x∫(f(x)−g(x))dx untuk f(x)=−5+(x−3)2f(x)=-5+(x-3)^{2}f(x)=−5+(x−3)2 dan g(x)=8−9xg(x)=8-9 xg(x)=8−9x. Pilih jawaban yang tepat.

Problem 2040

Find the limit as xxx approaches -1 for the expression 2x2+3x+1x2+2x−3\frac{2 x^{2}+3 x+1}{x^{2}+2 x-3}x2+2x−32x2+3x+1​.

Find the limit: $\lim _{x \rightarrow-\infty} \frac{x+3}{x^{2}+16}$ .

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

Find the limit: $\lim _{x \rightarrow \infty} e^{x}=\square$ . Provide the exact answer.

Find the limit as $x$ approaches infinity for $e^{-x}$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{4 e^{-x}}{5+4 e^{-x}}$ . What is the value?

Find $\lim _{x \rightarrow 0} g(x)$ given $3-2 x^{2} \leq g(x) \leq 3 \cos x$ .

Find the limit using the squeeze theorem: $\lim _{x \rightarrow 0} x^{2} \sin \left(\frac{4}{x}\right) =$

Show that $\lim _{x \rightarrow 0} x^{4} \cos \frac{1}{x}=0$ using the sandwich theorem by bounding $\cos \frac{1}{x}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin (3 x)}{18 x}=$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{1-\cos ^{2}(3 x)}{9 x^{2}}$ .

Determine if $f(x)=2x+4\cos(x)$ is continuous or discontinuous. Options: continuous or discontinuous.

Find where the function $f(x)=\frac{1}{x-5}$ is discontinuous. Identify the type and continuity.

Find the derivative of $f(x) = -x + \frac{3}{x} + 1$ .

Find the derivative of the function: $f(x)=9 x^{0.5}+4 x^{-0.5}$ . What is $f^{\prime}(x)$ ?

Find where the function $f(x)=\cos\left(\frac{1}{x-5}\right)$ for $x \neq 5$ and $0$ for $x=5$ is discontinuous. What type?

Analyze the continuity of the function $g(x)$ at $x=-1$ . Is it continuous from both sides, one side, or neither?

Find the limit as $x$ approaches -1 for the expression $8x^3 - 2$ . What is the result?

Evaluate the limit: $\lim _{x \rightarrow \pi} \cos \left(\frac{x}{3}-\frac{\pi}{2}\right)$ .

Find critical points of $f(x, y)=x^{2}+2xy+2y^{2}-6x+8y$ and classify as max, min, or saddle. Enter DNE if none.

Find critical points of $g(x, y)=1-2x^{2}-x-4y^{2}+y$ and classify them as max, min, or saddle.

Find the derivative of $g(x)=\frac{1}{x^{7}}+\frac{1}{x^{8}}$ . What is $g^{\prime}(x)$ ?

Find the derivative of $h(x)=\frac{4}{x^{2}}+\frac{8}{x^{3}}$ .

Determine if the function $f(x)=5 x+3 \cos (x)$ is continuous using the Laws of Continuity.

Find the derivative of $r(x)=\frac{2}{3 x^{2}}+\frac{1}{x^{3.9}}$ .

Find the derivative of $s(x)=\sqrt[3]{x}+\frac{11}{\sqrt[3]{x}}$ .

Evaluate the limit: $\lim _{x \rightarrow-1}\left(1-14 x^{3}\right)^{3 / 2}$ using substitution.

Minimize the cost function $C(x, y)=3,000+400 x^{2}+500 y^{2}$ for $x$ (sulfur lbs/day) and $y$ (lead lbs/day).

Find the tangent line equation for $f(x)=5x^2$ at $x=0$ and sketch the curve with the tangent line.

Find the derivative of $f(x)=3.57 x^{2}-30.5 x+81$ for $4.5 \leq x \leq 8.5$ .

Find the discontinuities of the function $f(x)=\begin{cases}\cos \left(\frac{1}{x-5}\right) & x \neq 5 \\ 0 & x=5\end{cases}$ and classify them.

Find the derivative of $r(x)=\frac{2}{3 x^{2}}+\frac{1}{x^{3.9}}$ . What is $r^{\prime}(x)$ ?

Find the derivative of the expression $a t^{2}+b t+c$ with respect to $t$ , where $a, b, c$ are constants.

Find the slope of the tangent to $g(x)=2 x^{4}$ at the point (-2,32).

Find the slope of the tangent for $f(x)=\frac{x}{8}-1$ at the point $(-16,-3)$ . Calculate $f'(-16)$ .

Find the derivative $P^{\prime}(t)$ of the oil price function $P(t)=0.45 t^{2}-12 t+105$ and compute $P^{\prime}(23)$ . What does this indicate about oil prices?

Find the price $p$ that maximizes profit, given $q = 476000 - 17000p$ and cost per copy is \$4. Use Lagrange multipliers.

Hitung $\int(f(x)-g(x)) d x$ untuk $f(x)=-5+(x-3)^{2}$ dan $g(x)=8-9 x$ . Pilih jawaban yang tepat.

Find the limit as $x$ approaches -1 for the expression $\frac{2 x^{2}+3 x+1}{x^{2}+2 x-3}$ .

Find the tangent line equation for $f(x)=\frac{3}{x^{2}}$ at $x=1$ . $y=$

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

Evaluate the limit: $\lim _{x \rightarrow \pi} \cos \left(\frac{x}{2}-\frac{2 \pi}{3}\right)$ .

Maximize $f(x, y)=x y$ with constraint $y=3-x^{2}$ . Find $f_{\text {max }}$ and point $(x, y)$ .

Minimize $f(x, y)=x^{2}+y^{2}$ with constraint $x+2y=15$ . Find $f_{\min }$ and the point $(x, y)$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=3.57 x^{2}-30.5 x+81$ for $4.5 \leq x \leq 8.5$ .

Find the integral of $\sin x \cos^{6} x \, dx$ . Choose the correct answer from the options provided.

Find the integral $\int u \sqrt{2u-9} \, du$ and choose the correct answer from the options given.

Find the integral: $\int \frac{9 x^{2}}{e^{x}} d x$ . Choose the correct answer from the options provided.

Find the integral: $\int u \sqrt{2 u-9} du$ . Choose the correct answer from the options provided.

Evaluate the integral: $\int u \sqrt{2 u-9} d u$ and choose the correct answer from the options given.

Find the integral of $\frac{13 q}{q-12}$ with respect to $q$ . Choose the correct answer from the options.

Evaluate the integral: $\int \tan ^{3} \frac{\pi x}{7} \sec ^{2} \frac{\pi x}{7} d x$ . Find the correct answer from the options.

Evaluate the integral $\int e^{9 x} \sqrt{1-e^{18 x}} d x$ and identify the correct expression from the options.

Calculate the integral: $\int \frac{x^{2}}{x-7} d x$ and choose the correct answer from the options.

Find the integral: $\int \sin ^{3} \frac{x}{7} d x$ . Choose the correct answer from the options provided.

Evaluate the integral: $\int \frac{\cos ^{3} 11 \theta}{\sqrt{\sin 11 \theta}} d \theta$ . Choose the correct answer from options (A) to (E).

Evaluate the integral: $\int \frac{\cos^{3} 11 \theta}{\sqrt{\sin 11 \theta}} d \theta$ . Choose the correct answer from options (A)-(E).

Find the integral $\int \sec ^{6} 8 x \, dx$ and choose the correct answer from the options provided.

Find the integral $\int 4 \sec ^{2} x e^{\tan x} \mathrm{~d} x$ . Choose the correct answer from the options.

Evaluate the integral $\int \cos ^{3} 5 x \sin ^{2} 5 x \, dx$ and choose the correct answer from the options.

Calculate the integral $\int 4 \sec ^{2} x e^{\tan x} \mathrm{~d} x$ . Choose the correct answer from the options.

Find the integral: $\int \tan^{3} 5x \, dx$ . Choose the correct answer from the options provided.

Calculate the integral: $\int \frac{4 x+3}{e^{7 x}} d x$ . Choose the correct result from the options provided.

Find the integral of $x^{7}$ : $\int x^{7} d x$ . Choose the correct answer: (A) (B) (C) (D) (E)

Evaluate the integral: $\int x(5 x^{2}+2)^{3} dx$ . Choose the correct answer from the options provided.

Find $f(x)$ with $f(0)=0$ and slope of tangent line as $x\left(x^{2}+2\right)^{3}$ . Choose from options (A)-(E).

Find the integral: $\int(6 x^{23}-x^{-6}+2) dx$ . Which option is correct?

Evaluate the integral: $\int 2 x \sqrt{4 x^{2}-4 d x}$ . Choose the correct answer from the options provided.

Evaluate the integral: $\int 2 x \sqrt{4 x^{2}-4} d x$ . Choose the correct answer from the options.

Find the integral of $(3x+5)^{5}$ with respect to $x$ . Choose the correct answer from the options.

Find the integral of $\frac{x^{2.6}}{5}-3.4$ with respect to $x$ .