Calculus

Problem 25801

Find the exact value of these integrals: (a) $\int_{-2}^{3}|x| dx$ and $\int_{0}^{1}(5x^{3}+2x^{2}-3) dx$ .

See Solution

Problem 25802

Find the derivative $f^{\prime}(x)$ for $f(x)=\ln x^{8}-4 x^{2}$ .

See Solution

Problem 25803

Find the tangent line equation at $x=7$ for $f(x)=6-x^{2}$ . Choose the correct option: A, B, C, or D.

See Solution

Problem 25804

What are the units of the integral $\int_{0}^{20} f(t) dt$ if $y=f(x)$ is in Newtons and $x$ in meters?

See Solution

Problem 25805

Find the min and max of $f(x)=x e^{-2 x}$ for $0 \leq x \leq 2$ . Min at $x=\square$ , Max at $x=$ .

See Solution

Problem 25806

Find the tangent line equation at $x=7$ for $f(x)=6-x^{2}$ in the form $y=m x+b$ . Options: A, B, C, D.

See Solution

Problem 25807

Find $\lim_{x \rightarrow 9000} \frac{C(x)}{x}$ for $C(x)=9000+9x$ . Options: A. 6 B. 10 C. 14 D. Does not exist.

See Solution

Problem 25808

Rewrite the integral using $u$ and $du$ , then evaluate: $\int (u)^{-2} du, \quad u=8x+21$

See Solution

Problem 25809

Find $f(5)$ for the function with $f^{\prime \prime}(x)=5 x+10 \sin (x)$ , given $f(0)=2$ and $f^{\prime}(0)=2$ .

See Solution

Problem 25810

Find the antiderivative $F(t)$ of $f(t)=4 \sec ^{2}(t)-6 t^{2}$ with $F(0)=0$ . Calculate $F(0.9)$ .

See Solution

Problem 25811

C-14 half-life is 5730 years. If a skeleton lost 85% of C-14, find decay rate $k$ and age in years, rounded to whole number.

See Solution

Problem 25812

Differentiate $y=\frac{x^{3}}{x-1}$ and find $\frac{d y}{d x}$ . Choose the correct option from A, B, C, D.

See Solution

Problem 25813

Find the missing expression to make the equation valid: $\frac{d}{d x} \ln \left(x^{7}+5\right)=\frac{1}{x^{7}+5} ?$ The missing expression is $\square$ .

See Solution

Problem 25814

9. Given the death rate model $f(t)$ for SARS in Singapore in 2003, answer the following: (a) What are the units of $\int_{0}^{28} f(t) dt$ ? (b) What does $\int_{0}^{28} f(t) dt$ represent? (c) Estimate $\int_{0}^{28} f(t) dt$ using a Riemann sum.

See Solution

Problem 25815

Find the absolute max and min of $p(x)=x^{2}-2$ on $[-2,3]$ . A. Min is $\square$ at $x=\square$ . B. No min. Max: $\square$ at $x=\square$ .

See Solution

Problem 25816

Given $\int_{2}^{7} f(x) d x=12$ and $\int_{2}^{4}(f(x)+5) d x=8$ , find $\int_{7}^{4} f(x) d x$ .

See Solution

Problem 25817

Rewrite the integral using $u$ and $du$ , then evaluate: $\int x \sqrt{3x+4} \, dx, \quad u=3x+4$

See Solution

Problem 25818

Find the absolute maximum and minimum of $p(x)=x^{2}-2$ on the interval $[-2,3]$ . A. Max is $\square$ at $x=\square$ . B. No min.

See Solution

Problem 25819

Find the limit: $\lim _{x \rightarrow 0} \frac{\int_{x}^{0} \sqrt{t^{2}+4} d t}{x}$ . What is it equal to? Options: $\infty$ , $-2$ , 2, $-\infty$ .

See Solution

Problem 25820

Find the average value of $f(x)=500-60x$ over $[0,2]$ . Graph $f(x)$ in red and its average in blue.

See Solution

Problem 25821

Find the derivative $f^{\prime}(x)$ for $f(x)=6 x^{-2}+8 x^{3}+11 x$ . Choose the correct option.

See Solution

Problem 25822

Find the values of the following integrals given the conditions on $f$ and $g$ :
(a) $\int_{7}^{7} g(x) d x=$ (b) $\int_{1}^{-3} f(x) d x=$ (c) $\int_{1}^{5} g(x) d x=$ (d) $\int_{-3}^{1}(4 f(x)+7 g(x)) d x=$

See Solution

Problem 25823

Find the limit as $x$ approaches -1 for $\frac{6x+5}{5x-6}$ . Choices: A. -11 B. $\frac{1}{11}$ C. 1 D. $-\frac{1}{11}$

See Solution

Problem 25824

Define $\int_{0}^{3} f(x) d x$ as a limit without simplifying or evaluating it.

See Solution

Problem 25825

Find an antiderivative of the function $f(x)=\frac{-15}{\sqrt{1-x^{2}}}$ .

See Solution

Problem 25826

Find $f(\pi / 6)$ given $f^{\prime \prime}(x)=-16 \sin(4x)$ , $f^{\prime}(0)=-2$ , and $f(0)=3$ .

See Solution

Problem 25827

Calculate the integrals: (c) $\int_{1}^{4}\left(1-\frac{3}{x}\right)^{2} dx$ and (d) $\int_{4}^{8} 3 e^{6x-7} dx$ .

See Solution

Problem 25828

Find the average rate of change of the function $g$ from $x = -5$ to $x = -3$ using $g(-5) = -1$ and $g(-3) = 1$ .

See Solution

Problem 25829

Find the absolute max and min of $f(x)=x^{3}-\frac{5}{2} x^{2}-2 x+1$ on the interval $[-1,3]$ .

See Solution

Problem 25830

Find the derivative $\frac{d y}{d x}$ for $y=\frac{1}{3 x^{3}}+\frac{x^{7}}{10}$ . Options: A, B, C, D.

See Solution

Problem 25831

Find the derivative $\frac{d y}{d x}$ for $y=\frac{1}{3 x^{3}}+\frac{x^{7}}{10}$ . Options: A, B, C, D.

See Solution

Problem 25832

Find $S(3)$ and $S^{\prime}(3)$ for $S(t)=\frac{800 t}{t+2}$ . What do the results mean?

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Problem 25833

Find the derivative $f^{\prime}(x)$ for $f(x)=5 e^{x}-4 x^{2}$ . Options: A. $5 e^{x}-8 x$ , B. $5 e^{x}-8 x^{2}$ , C. $5 x e^{x-1}-8 x^{2}$ , D. $5 e^{x}-2 x$ .

See Solution

Problem 25834

Find the derivative $f^{\prime}(x)$ for $f(x)=(4x^{2}+3x)^{2}$ . Choose the correct option.

See Solution

Problem 25835

Evaluate the integral: $\int \frac{9}{x} dx = \square$

See Solution

Problem 25836

Find the derivative of $f(x)=\frac{x^{2}(x-5)^{t}}{(x^{2}+2)^{2}}$ using logarithmic differentiation. Calculate $f'(5)$ .

See Solution

Problem 25837

Find the derivative $\frac{d y}{d x}$ for: 1) $y=\ln\left(\frac{2x-10}{x\sqrt[4]{x^2+1}}\right)$ and 2) $y=x^{\cos(x)}$ .

See Solution

Problem 25838

Find the general antiderivative of $f(x)=e^{x}+3 \sec ^{2}(x)$ for $-\frac{\pi}{2}<x<\frac{\pi}{2}$ . Use $C$ for constants. $F(x)=$

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Problem 25839

Find the expression for $\frac{f(a+h)-f(a)}{h}$ where $f(x)=x^{2}-3x+9$ for $h \neq 0$ . Choices: (A) $h^{2}-3h$ , (B) $h-3$ , (C) $2a+h$ , (D) $2a+h-3$ .

See Solution

Problem 25840

Find the function $f$ where $f^{\prime}(x)=2 \cos (x)+\sec ^{2}(x)$ for $-\pi / 2<x<\pi / 2$ and $f(\pi / 3)=-6$ .

See Solution

Problem 25841

Find the missing expression ? to make the equation valid: $\frac{d}{d x}(7 x+8)^{3}=3(7 x+8)^{2} ?$

See Solution

Problem 25842

Find $f$ given $f'(x) = 2 \cos(x) + \sec^2(x)$ for $-\pi/2 < x < \pi/2$ and $f(\pi/3) = -6$ .

See Solution

Problem 25843

Determine the horizontal asymptotes for the function $f(x)=\frac{4 x^{4}+3 x+5}{5 x^{4}+3 x-2}$ .

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Problem 25844

Find $y^{\prime}$ using implicit differentiation and match it with the correct expression from the options given.

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Problem 25845

Compute $\int_{-1}^{\pi} f(x) dx$ where $f(x) = x^2$ for $x \leq 0$ and $f(x) = -\sin x + C$ for $x > 0$ .

See Solution

Problem 25846

How long for \$8400 to grow to \$14,600 at a 9.4% continuous interest rate? Round to the nearest hundredth.

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Problem 25847

Analyze the function $f(x)=\frac{x^{3}}{x^{2}-4}$ .
A. Identify vertical asymptotes: $x=-2,2$ . B. Find local max at $x_{\max }=\square$ and min at $x_{\min }=\square$ . C. Determine if $f(x)$ is INC or DEC on $(-\infty, \max)$ . D. Find intervals where $f(x)$ is concave up: $\square$ . E. Locate the inflection point at $x=\square$ . F. Sketch and bring the graph to class.

See Solution

Problem 25848

Find the time for \$8400 to grow to \$14,600 at a 9.4% continuous interest rate. Round to the nearest hundredth.

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Problem 25849

Find the rate of change of the base of a triangle given the altitude is 10 cm, area is 84 cm², height increases at 3 cm/min, and area at 5 cm²/min.

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Problem 25850

Evaluate the integral: $\int_{0}^{1} 3 \sqrt[3]{x} \, dx = \square$ (Provide the exact simplified answer.)

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Problem 25851

For the function $f(x)=6 x^{2}-3 x-6$ , find $f(x+h)$ and simplify $\frac{f(x+h)-f(x)}{h}$ .

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Problem 25852

Find the limit: $\lim _{x \rightarrow+\infty}\left(1+\frac{1}{6 x}\right)^{-2 x}=\square$ . Enter "DNE" if it doesn't exist.

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Problem 25853

Find critical numbers $A<B$ for $f(x)=\frac{\ln (x)}{x^{5}}$ and determine if $f^{\prime}(x)$ is + or - in $(A, B)$ and $(B, \infty)$ . Conclude if $f(x)$ has a local MAX or MIN at $B$ .

See Solution

Problem 25854

Find the critical number $A$ for $f(x)=5(x-3)^{\frac{2}{3}}$ and state if it's a local MAX or MIN.

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Problem 25855

Find the time $x$ that maximizes drug concentration given by $K(x)=\frac{3.0 x}{x^{2}+25}$ . Round to nearest tenth.

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Problem 25856

Find the $x$ -coordinate of the absolute minimum for $f(x)=\frac{e^{x}}{x^{4}}$ , where $x>0$ . What is it?

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Problem 25857

Find the $x$ -coordinate of the absolute maximum for $f(x)=\frac{2+6 \ln (x)}{x}$ , where $x>0$ .

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Problem 25858

Find critical values and intervals where $f(x)=x^{3}+3 x^{2}-24 x+6$ is increasing or decreasing.

See Solution

Problem 25859

Find the absolute min and max of $f(x)=x e^{-2 x}$ for $0 \leq x \leq 2$ . What are their values and where do they occur?

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Problem 25860

Find the derivative $f'(x)$ of the function $f(x)=9x^5(x^4-6)$ .

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Problem 25861

Find critical numbers $A$ and $B$ for $f(x)=x^{2} e^{10 x}$ . Determine the sign of $f^{\prime}(x)$ in given intervals and classify extrema.

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Problem 25862

Evaluate the integral from 4 to 4 of $(x^{2}-5x+6)^{10}$ . What is the result?

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Problem 25863

Soit la fonction $f(x)=|x| \arctan (x+1)$ : (a) Écrivez $f(x)$ comme fonction par morceaux. (b) Est-elle continue en $x=0$ ? Justifiez. (c) Est-elle dérivable en $x=0$ ? Justifiez. (d) Que concluez-vous sur le point $(0, f(0))$ ? Justifiez.

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Problem 25864

Check if the Mean Value Theorem applies to $f(x)=3 \sin^{-1} x$ on $[-1,1]$ . If yes, find $c$ values; else, enter DNE.

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Problem 25865

Check if the Mean Value Theorem applies to $f(x)=12 \ln x+7$ on $[1,7]$ . If so, find $c$ values in $[1,7]$ . If not, enter DNE. $c=\square($ Separate multiple answers by commas.)

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Problem 25866

Find the average slope of $f(x)=2 x^{3}-9 x^{2}-60 x+1$ on $[-5,7]$ and list $c$ values where $f'(c)$ equals this slope.

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Problem 25867

Find all values of $\mathrm{c}$ satisfying Rolle's Theorem for $f(x)=5 x \sqrt{x+2}$ on $[-2,0]$ .

See Solution

Problem 25868

Find all values of $\mathrm{c}$ for the Mean Value Theorem for $f(x)=e^{-3x}$ on $[0,3]$ . Use $\mathrm{N}$ for unused blanks.

See Solution

Problem 25869

Which identity about $g(x)=\int f(x) d x$ is TRUE? Options include derivatives involving $f(x)$ and constants.

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Problem 25870

Approximate $\int_{2}^{3} 3 \sin \left(\frac{\pi}{3} x\right) d x$ using midpoint rule with $n=6$ . Round to four decimals.

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Problem 25871

Evaluate the integral $\int_{1}^{\infty} f\left(x^{m}\right) x^{m-1} \, dx$ for $m < 0$ given $F(1)=5$ and $\lim _{x \rightarrow \infty} F(x)=\infty$ .

See Solution

Problem 25872

Which polynomial function $f(x)$ has limits $-\infty$ as $x \to -\infty$ and $x \to \infty$ ? (A) $2 x^{4}+3 x^{3}-2 x-1$ (B) $2 x^{3}-4 x^{2}+3 x+7$ (C) $-2 x^{4}+3 x^{3}-2 x-1$ (D) $-2 x^{3}-4 x^{2}+3 x+7$

See Solution

Problem 25873

Evaluate the integral $\int_{0}^{8} \frac{1}{2\left(x^{1 / 3}\right)} d x$ and determine if it converges or diverges.

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Problem 25874

Given a continuous function $f$ on $(0, \infty)$ with $\lim_{x \to 0^{+}} f(x)=\infty$ and $\lim_{x \to \infty} f(x)=0$ , and $g(x) \geq f(x)$ , which statement is always TRUE?

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Problem 25875

Which statement is TRUE by the Comparison Test regarding integrals? Consider $\int_{e^{10}}^{\infty} \frac{\ln x}{x} dx$ and others.

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Problem 25876

Find the derivative of $G(x)=\int_{x}^{e^{x}}\left(\frac{\cos t}{t+1}+C\right) d t$ .

See Solution

Problem 25877

Solve for $x=f(t)$ in the equation $(t^{2}+4 t) \frac{d x}{d t}=4 x+4$ , given $f(1)=2$ .

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Problem 25878

A guitar string's frequency $f$ changes with a $1.50\%$ increase in tension. Find the new frequency change in percentage.

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Problem 25879

Find critical points of $f(x)=\frac{x^{3}-1}{x}$ and identify local maxima and minima.

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Problem 25880

Find the derivative of $\int_{\sin x}^{\cos x} e^{t} d t$ . What is it?

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Problem 25881

Find $H^{\prime}(x)$ for $H(x)=\int_{1}^{\sin (x)} f(t) dt$ . Options: (a) $-f(\sin (x))$ , (b) $f(x)$ , (c) $f(\sin (x)) \cos (x)$ , (d) $f(\sin (x))-f(1)$ , (e) $f^{\prime}(\sin (x)) \cos ^{2}(x)-f(\sin (x)) \sin (x)$ .

See Solution

Problem 25882

Find the limit: $\lim _{x \rightarrow 0} \frac{x \sin 2 x}{(e^{x}-1)^{2}}$

See Solution

Problem 25883

Calculate the integral $\int_{3}^{5} f^{\prime}(x) \, dx$ using the given values of $f$ and $f^{\prime}$ .

See Solution

Problem 25884

Evaluate the improper integral $\int_{1}^{\infty} \frac{1}{\sqrt{x}} dx$ . What is its value? (a) -1 (b) 0 (c) 1 (d) 2 (e) Diverges.

See Solution

Problem 25885

Evaluate the integral $\int x^{3}(x^{4}-10)^{43} dx$ using the substitution $u=x^{4}-10$ .

See Solution

Problem 25886

Evaluate the integral of $(2x+3)(x^2+3x+4)^4 \, dx$ using the substitution $u=x^2+3x+4$ . $+C$

See Solution

Problem 25887

Approximate the integral $\int_{1}^{3} 3 \sin \left(\frac{\pi}{3} x\right) d x$ using the Trapezoid Rule with $n=6$ . Round to four decimal places.

See Solution

Problem 25888

Find the limit: $\lim _{x \rightarrow 0^{+}} \frac{d}{d x}(\ln (\ln x))$

See Solution

Problem 25889

Find the limit: $\lim _{x \rightarrow 0^{+}} \sin x\left(\frac{d}{d x}(\ln (\ln x))\right)$ .

See Solution

Problem 25890

Find the $x$ -value(s) for local maxima/minima of $f(x)=\frac{\ln (x)}{x}$ . Answer: $x=1$

See Solution

Problem 25891

Find $f(\pi)$ if $f(0)=4$ and $f^{\prime}(x)=\sin (x)+e^{-x}+3 x^{2}$ . Options: $7+e^{\pi}+\pi^{3}$ , $7+e^{-\pi}+\pi^{3}$ , $7-e^{\pi}+\pi^{3}$ , $1+e^{\pi}+\pi^{3}$ , $7-e^{-\pi}+\pi^{3}$ .

See Solution

Problem 25892

Find the $x$ -value(s) where the local maxima/minima of the function $f(x)=-\frac{2}{3} x^{3}+5 x^{2}-8 x-5$ occur. $x=$

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Problem 25893

Find $g'(x)$ for $g(x)=\int_{2}^{x^{3}} \sin (t) e^{t} dt$ . Options include $3 x^{2} \sin (x^{3}) e^{x^{3}}$ and others.

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Problem 25894

Evaluate the integral $\int_{1}^{e^{2}} \frac{5}{x} \mathrm{dx}$ using trapezoid and Simpson's rules with $n=4$ . Find errors.

See Solution

Problem 25895

Approximate $e^{-5}$ to four decimal places.

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Problem 25896

Let $f$ be continuous on $[1, \infty)$ with $F(1)=5$ , $F(0)=-6$ , and $\lim_{x \to \infty} F(x)=\infty$ . If $m<0$ , find the value of $\int_{1}^{\infty} f(x^{m}) x^{m-1} dx$ . Options are: $-\frac{11}{m}$ , $-11$ , diverges, or $0$ .

See Solution

Problem 25897

Evaluate the integral: $\int 3 \sin^{2}(x) \cos(x) \, dx = +C$

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Problem 25898

Evaluate the integral: $\int 3 \sin ^{2}(x) \cos (x) d x=\square+c$

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Problem 25899

Find $y^{\prime \prime}$ given $x^{4}+y^{4}=16$ . First, calculate $y^{\prime}$ .

See Solution

Problem 25900

Approximate the integral $\int_{0}^{\pi} \frac{2 \cos x}{13 / 5-\cos x} d x$ using Simpson's rule. Find the smallest $n$ from $4, 8, 16, 32$ for error < $10^{-3}$ .
$n=\square$

See Solution

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Calculus

Problem 25801

Find the exact value of these integrals: (a) ∫−23∣x∣dx\int_{-2}^{3}|x| dx∫−23​∣x∣dx and ∫01(5x3+2x2−3)dx\int_{0}^{1}(5x^{3}+2x^{2}-3) dx∫01​(5x3+2x2−3)dx.

Problem 25802

Find the derivative f′(x)f^{\prime}(x)f′(x) for f(x)=ln⁡x8−4x2f(x)=\ln x^{8}-4 x^{2}f(x)=lnx8−4x2.

Problem 25803

Find the tangent line equation at x=7x=7x=7 for f(x)=6−x2f(x)=6-x^{2}f(x)=6−x2. Choose the correct option: A, B, C, or D.

Problem 25804

What are the units of the integral ∫020f(t)dt\int_{0}^{20} f(t) dt∫020​f(t)dt if y=f(x)y=f(x)y=f(x) is in Newtons and xxx in meters?

Problem 25805

Find the min and max of f(x)=xe−2xf(x)=x e^{-2 x}f(x)=xe−2x for 0≤x≤20 \leq x \leq 20≤x≤2. Min at x=□x=\squarex=□, Max at x=x=x=.

Problem 25806

Find the tangent line equation at x=7x=7x=7 for f(x)=6−x2f(x)=6-x^{2}f(x)=6−x2 in the form y=mx+by=m x+by=mx+b. Options: A, B, C, D.

Problem 25807

Find lim⁡x→9000C(x)x\lim_{x \rightarrow 9000} \frac{C(x)}{x}limx→9000​xC(x)​ for C(x)=9000+9xC(x)=9000+9xC(x)=9000+9x. Options: A. 6 B. 10 C. 14 D. Does not exist.

Problem 25808

Rewrite the integral using uuu and dududu, then evaluate: ∫(u)−2du,u=8x+21\int (u)^{-2} du, \quad u=8x+21∫(u)−2du,u=8x+21

Problem 25809

Find f(5)f(5)f(5) for the function with f′′(x)=5x+10sin⁡(x)f^{\prime \prime}(x)=5 x+10 \sin (x)f′′(x)=5x+10sin(x), given f(0)=2f(0)=2f(0)=2 and f′(0)=2f^{\prime}(0)=2f′(0)=2.

Problem 25810

Find the antiderivative F(t)F(t)F(t) of f(t)=4sec⁡2(t)−6t2f(t)=4 \sec ^{2}(t)-6 t^{2}f(t)=4sec2(t)−6t2 with F(0)=0F(0)=0F(0)=0. Calculate F(0.9)F(0.9)F(0.9).

Problem 25811

C-14 half-life is 5730 years. If a skeleton lost 85% of C-14, find decay rate kkk and age in years, rounded to whole number.

Problem 25812

Differentiate y=x3x−1y=\frac{x^{3}}{x-1}y=x−1x3​ and find dydx\frac{d y}{d x}dxdy​. Choose the correct option from A, B, C, D.

Problem 25813

Find the missing expression to make the equation valid: ddxln⁡(x7+5)=1x7+5?\frac{d}{d x} \ln \left(x^{7}+5\right)=\frac{1}{x^{7}+5} ?dxd​ln(x7+5)=x7+51​? The missing expression is □\square□.

Problem 25814

Problem 25815

Find the absolute max and min of p(x)=x2−2p(x)=x^{2}-2p(x)=x2−2 on [−2,3][-2,3][−2,3]. A. Min is □\square□ at x=□x=\squarex=□. B. No min. Max: □\square□ at x=□x=\squarex=□.

Problem 25816

Given ∫27f(x)dx=12\int_{2}^{7} f(x) d x=12∫27​f(x)dx=12 and ∫24(f(x)+5)dx=8\int_{2}^{4}(f(x)+5) d x=8∫24​(f(x)+5)dx=8, find ∫74f(x)dx\int_{7}^{4} f(x) d x∫74​f(x)dx.

Problem 25817

Rewrite the integral using uuu and dududu, then evaluate: ∫x3x+4 dx,u=3x+4\int x \sqrt{3x+4} \, dx, \quad u=3x+4∫x3x+4​dx,u=3x+4

Problem 25818

Find the absolute maximum and minimum of p(x)=x2−2p(x)=x^{2}-2p(x)=x2−2 on the interval [−2,3][-2,3][−2,3]. A. Max is □\square□ at x=□x=\squarex=□. B. No min.

Problem 25819

Find the limit: lim⁡x→0∫x0t2+4dtx\lim _{x \rightarrow 0} \frac{\int_{x}^{0} \sqrt{t^{2}+4} d t}{x}x→0lim​x∫x0​t2+4​dt​. What is it equal to? Options: ∞\infty∞, −2-2−2, 2, −∞-\infty−∞.

Problem 25820

Find the average value of f(x)=500−60xf(x)=500-60xf(x)=500−60x over [0,2][0,2][0,2]. Graph f(x)f(x)f(x) in red and its average in blue.

Problem 25821

Find the derivative f′(x)f^{\prime}(x)f′(x) for f(x)=6x−2+8x3+11xf(x)=6 x^{-2}+8 x^{3}+11 xf(x)=6x−2+8x3+11x. Choose the correct option.

Problem 25822

Problem 25823

Find the limit as xxx approaches -1 for 6x+55x−6\frac{6x+5}{5x-6}5x−66x+5​. Choices: A. -11 B. 111\frac{1}{11}111​ C. 1 D. −111-\frac{1}{11}−111​

Problem 25824

Define ∫03f(x)dx\int_{0}^{3} f(x) d x∫03​f(x)dx as a limit without simplifying or evaluating it.

Problem 25825

Find an antiderivative of the function f(x)=−151−x2f(x)=\frac{-15}{\sqrt{1-x^{2}}}f(x)=1−x2​−15​.

Problem 25826

Find f(π/6)f(\pi / 6)f(π/6) given f′′(x)=−16sin⁡(4x)f^{\prime \prime}(x)=-16 \sin(4x)f′′(x)=−16sin(4x), f′(0)=−2f^{\prime}(0)=-2f′(0)=−2, and f(0)=3f(0)=3f(0)=3.

Problem 25827

Calculate the integrals: (c) ∫14(1−3x)2dx\int_{1}^{4}\left(1-\frac{3}{x}\right)^{2} dx∫14​(1−x3​)2dx and (d) ∫483e6x−7dx\int_{4}^{8} 3 e^{6x-7} dx∫48​3e6x−7dx.

Problem 25828

Find the average rate of change of the function ggg from x=−5x = -5x=−5 to x=−3x = -3x=−3 using g(−5)=−1g(-5) = -1g(−5)=−1 and g(−3)=1g(-3) = 1g(−3)=1.

Problem 25829

Find the absolute max and min of f(x)=x3−52x2−2x+1f(x)=x^{3}-\frac{5}{2} x^{2}-2 x+1f(x)=x3−25​x2−2x+1 on the interval [−1,3][-1,3][−1,3].

Problem 25830

Find the derivative dydx\frac{d y}{d x}dxdy​ for y=13x3+x710y=\frac{1}{3 x^{3}}+\frac{x^{7}}{10}y=3x31​+10x7​. Options: A, B, C, D.

Problem 25831

Find the derivative dydx\frac{d y}{d x}dxdy​ for y=13x3+x710y=\frac{1}{3 x^{3}}+\frac{x^{7}}{10}y=3x31​+10x7​. Options: A, B, C, D.

Problem 25832

Find S(3)S(3)S(3) and S′(3)S^{\prime}(3)S′(3) for S(t)=800tt+2S(t)=\frac{800 t}{t+2}S(t)=t+2800t​. What do the results mean?

Problem 25833

Find the derivative f′(x)f^{\prime}(x)f′(x) for f(x)=5ex−4x2f(x)=5 e^{x}-4 x^{2}f(x)=5ex−4x2. Options: A. 5ex−8x5 e^{x}-8 x5ex−8x, B. 5ex−8x25 e^{x}-8 x^{2}5ex−8x2, C. 5xex−1−8x25 x e^{x-1}-8 x^{2}5xex−1−8x2, D. 5ex−2x5 e^{x}-2 x5ex−2x.

Problem 25834

Find the derivative f′(x)f^{\prime}(x)f′(x) for f(x)=(4x2+3x)2f(x)=(4x^{2}+3x)^{2}f(x)=(4x2+3x)2. Choose the correct option.

Problem 25835

Evaluate the integral: ∫9xdx=□\int \frac{9}{x} dx = \square∫x9​dx=□

Problem 25836

Find the derivative of f(x)=x2(x−5)t(x2+2)2f(x)=\frac{x^{2}(x-5)^{t}}{(x^{2}+2)^{2}}f(x)=(x2+2)2x2(x−5)t​ using logarithmic differentiation. Calculate f′(5)f'(5)f′(5).

Problem 25837

Find the derivative dydx\frac{d y}{d x}dxdy​ for: 1) y=ln⁡(2x−10xx2+14)y=\ln\left(\frac{2x-10}{x\sqrt[4]{x^2+1}}\right)y=ln(x4x2+1​2x−10​) and 2) y=xcos⁡(x)y=x^{\cos(x)}y=xcos(x).

Problem 25838

Find the general antiderivative of f(x)=ex+3sec⁡2(x)f(x)=e^{x}+3 \sec ^{2}(x)f(x)=ex+3sec2(x) for −π2<x<π2-\frac{\pi}{2}<x<\frac{\pi}{2}−2π​<x<2π​. Use CCC for constants. F(x)= F(x)= F(x)=

Problem 25839

Find the expression for f(a+h)−f(a)h\frac{f(a+h)-f(a)}{h}hf(a+h)−f(a)​ where f(x)=x2−3x+9f(x)=x^{2}-3x+9f(x)=x2−3x+9 for h≠0h \neq 0h=0. Choices: (A) h2−3hh^{2}-3hh2−3h, (B) h−3h-3h−3, (C) 2a+h2a+h2a+h, (D) 2a+h−32a+h-32a+h−3.

Problem 25840

Find the function fff where f′(x)=2cos⁡(x)+sec⁡2(x)f^{\prime}(x)=2 \cos (x)+\sec ^{2}(x)f′(x)=2cos(x)+sec2(x) for −π/2<x<π/2-\pi / 2<x<\pi / 2−π/2<x<π/2 and f(π/3)=−6f(\pi / 3)=-6f(π/3)=−6.

Problem 25841

Find the exact value of these integrals: (a) $\int_{-2}^{3}|x| dx$ and $\int_{0}^{1}(5x^{3}+2x^{2}-3) dx$ .

Find the derivative $f^{\prime}(x)$ for $f(x)=\ln x^{8}-4 x^{2}$ .

Find the tangent line equation at $x=7$ for $f(x)=6-x^{2}$ . Choose the correct option: A, B, C, or D.

What are the units of the integral $\int_{0}^{20} f(t) dt$ if $y=f(x)$ is in Newtons and $x$ in meters?

Find the min and max of $f(x)=x e^{-2 x}$ for $0 \leq x \leq 2$ . Min at $x=\square$ , Max at $x=$ .

Find the tangent line equation at $x=7$ for $f(x)=6-x^{2}$ in the form $y=m x+b$ . Options: A, B, C, D.

Find $\lim_{x \rightarrow 9000} \frac{C(x)}{x}$ for $C(x)=9000+9x$ . Options: A. 6 B. 10 C. 14 D. Does not exist.

Rewrite the integral using $u$ and $du$ , then evaluate: $\int (u)^{-2} du, \quad u=8x+21$

Find $f(5)$ for the function with $f^{\prime \prime}(x)=5 x+10 \sin (x)$ , given $f(0)=2$ and $f^{\prime}(0)=2$ .

Find the antiderivative $F(t)$ of $f(t)=4 \sec ^{2}(t)-6 t^{2}$ with $F(0)=0$ . Calculate $F(0.9)$ .

C-14 half-life is 5730 years. If a skeleton lost 85% of C-14, find decay rate $k$ and age in years, rounded to whole number.

Differentiate $y=\frac{x^{3}}{x-1}$ and find $\frac{d y}{d x}$ . Choose the correct option from A, B, C, D.

Find the missing expression to make the equation valid: $\frac{d}{d x} \ln \left(x^{7}+5\right)=\frac{1}{x^{7}+5} ?$ The missing expression is $\square$ .

Find the absolute max and min of $p(x)=x^{2}-2$ on $[-2,3]$ . A. Min is $\square$ at $x=\square$ . B. No min. Max: $\square$ at $x=\square$ .

Given $\int_{2}^{7} f(x) d x=12$ and $\int_{2}^{4}(f(x)+5) d x=8$ , find $\int_{7}^{4} f(x) d x$ .

Rewrite the integral using $u$ and $du$ , then evaluate: $\int x \sqrt{3x+4} \, dx, \quad u=3x+4$

Find the absolute maximum and minimum of $p(x)=x^{2}-2$ on the interval $[-2,3]$ . A. Max is $\square$ at $x=\square$ . B. No min.

Find the limit: $\lim _{x \rightarrow 0} \frac{\int_{x}^{0} \sqrt{t^{2}+4} d t}{x}$ . What is it equal to? Options: $\infty$ , $-2$ , 2, $-\infty$ .

Find the average value of $f(x)=500-60x$ over $[0,2]$ . Graph $f(x)$ in red and its average in blue.

Find the derivative $f^{\prime}(x)$ for $f(x)=6 x^{-2}+8 x^{3}+11 x$ . Choose the correct option.

Find the limit as $x$ approaches -1 for $\frac{6x+5}{5x-6}$ . Choices: A. -11 B. $\frac{1}{11}$ C. 1 D. $-\frac{1}{11}$

Define $\int_{0}^{3} f(x) d x$ as a limit without simplifying or evaluating it.

Find an antiderivative of the function $f(x)=\frac{-15}{\sqrt{1-x^{2}}}$ .

Find $f(\pi / 6)$ given $f^{\prime \prime}(x)=-16 \sin(4x)$ , $f^{\prime}(0)=-2$ , and $f(0)=3$ .

Calculate the integrals: (c) $\int_{1}^{4}\left(1-\frac{3}{x}\right)^{2} dx$ and (d) $\int_{4}^{8} 3 e^{6x-7} dx$ .

Find the average rate of change of the function $g$ from $x = -5$ to $x = -3$ using $g(-5) = -1$ and $g(-3) = 1$ .

Find the absolute max and min of $f(x)=x^{3}-\frac{5}{2} x^{2}-2 x+1$ on the interval $[-1,3]$ .

Find the derivative $\frac{d y}{d x}$ for $y=\frac{1}{3 x^{3}}+\frac{x^{7}}{10}$ . Options: A, B, C, D.

Find the derivative $\frac{d y}{d x}$ for $y=\frac{1}{3 x^{3}}+\frac{x^{7}}{10}$ . Options: A, B, C, D.

Find $S(3)$ and $S^{\prime}(3)$ for $S(t)=\frac{800 t}{t+2}$ . What do the results mean?

Find the derivative $f^{\prime}(x)$ for $f(x)=5 e^{x}-4 x^{2}$ . Options: A. $5 e^{x}-8 x$ , B. $5 e^{x}-8 x^{2}$ , C. $5 x e^{x-1}-8 x^{2}$ , D. $5 e^{x}-2 x$ .

Find the derivative $f^{\prime}(x)$ for $f(x)=(4x^{2}+3x)^{2}$ . Choose the correct option.

Evaluate the integral: $\int \frac{9}{x} dx = \square$

Find the derivative of $f(x)=\frac{x^{2}(x-5)^{t}}{(x^{2}+2)^{2}}$ using logarithmic differentiation. Calculate $f'(5)$ .

Find the derivative $\frac{d y}{d x}$ for: 1) $y=\ln\left(\frac{2x-10}{x\sqrt[4]{x^2+1}}\right)$ and 2) $y=x^{\cos(x)}$ .

Find the general antiderivative of $f(x)=e^{x}+3 \sec ^{2}(x)$ for $-\frac{\pi}{2}<x<\frac{\pi}{2}$ . Use $C$ for constants. $F(x)=$

Find the expression for $\frac{f(a+h)-f(a)}{h}$ where $f(x)=x^{2}-3x+9$ for $h \neq 0$ . Choices: (A) $h^{2}-3h$ , (B) $h-3$ , (C) $2a+h$ , (D) $2a+h-3$ .

Find the function $f$ where $f^{\prime}(x)=2 \cos (x)+\sec ^{2}(x)$ for $-\pi / 2<x<\pi / 2$ and $f(\pi / 3)=-6$ .

Find the missing expression ? to make the equation valid: $\frac{d}{d x}(7 x+8)^{3}=3(7 x+8)^{2} ?$

Find $f$ given $f'(x) = 2 \cos(x) + \sec^2(x)$ for $-\pi/2 < x < \pi/2$ and $f(\pi/3) = -6$ .

Determine the horizontal asymptotes for the function $f(x)=\frac{4 x^{4}+3 x+5}{5 x^{4}+3 x-2}$ .

Find $y^{\prime}$ using implicit differentiation and match it with the correct expression from the options given.

Compute $\int_{-1}^{\pi} f(x) dx$ where $f(x) = x^2$ for $x \leq 0$ and $f(x) = -\sin x + C$ for $x > 0$ .

Evaluate the integral: $\int_{0}^{1} 3 \sqrt[3]{x} \, dx = \square$ (Provide the exact simplified answer.)

For the function $f(x)=6 x^{2}-3 x-6$ , find $f(x+h)$ and simplify $\frac{f(x+h)-f(x)}{h}$ .

Find the limit: $\lim _{x \rightarrow+\infty}\left(1+\frac{1}{6 x}\right)^{-2 x}=\square$ . Enter "DNE" if it doesn't exist.

Find critical numbers $A<B$ for $f(x)=\frac{\ln (x)}{x^{5}}$ and determine if $f^{\prime}(x)$ is + or - in $(A, B)$ and $(B, \infty)$ . Conclude if $f(x)$ has a local MAX or MIN at $B$ .

Find the critical number $A$ for $f(x)=5(x-3)^{\frac{2}{3}}$ and state if it's a local MAX or MIN.

Find the time $x$ that maximizes drug concentration given by $K(x)=\frac{3.0 x}{x^{2}+25}$ . Round to nearest tenth.

Find the $x$ -coordinate of the absolute minimum for $f(x)=\frac{e^{x}}{x^{4}}$ , where $x>0$ . What is it?

Find the $x$ -coordinate of the absolute maximum for $f(x)=\frac{2+6 \ln (x)}{x}$ , where $x>0$ .

Find critical values and intervals where $f(x)=x^{3}+3 x^{2}-24 x+6$ is increasing or decreasing.

Find the absolute min and max of $f(x)=x e^{-2 x}$ for $0 \leq x \leq 2$ . What are their values and where do they occur?

Find the derivative $f'(x)$ of the function $f(x)=9x^5(x^4-6)$ .

Find critical numbers $A$ and $B$ for $f(x)=x^{2} e^{10 x}$ . Determine the sign of $f^{\prime}(x)$ in given intervals and classify extrema.

Evaluate the integral from 4 to 4 of $(x^{2}-5x+6)^{10}$ . What is the result?

Check if the Mean Value Theorem applies to $f(x)=3 \sin^{-1} x$ on $[-1,1]$ . If yes, find $c$ values; else, enter DNE.

Check if the Mean Value Theorem applies to $f(x)=12 \ln x+7$ on $[1,7]$ . If so, find $c$ values in $[1,7]$ . If not, enter DNE. $c=\square($ Separate multiple answers by commas.)

Find the average slope of $f(x)=2 x^{3}-9 x^{2}-60 x+1$ on $[-5,7]$ and list $c$ values where $f'(c)$ equals this slope.

Find all values of $\mathrm{c}$ satisfying Rolle's Theorem for $f(x)=5 x \sqrt{x+2}$ on $[-2,0]$ .

Find all values of $\mathrm{c}$ for the Mean Value Theorem for $f(x)=e^{-3x}$ on $[0,3]$ . Use $\mathrm{N}$ for unused blanks.

Which identity about $g(x)=\int f(x) d x$ is TRUE? Options include derivatives involving $f(x)$ and constants.

Approximate $\int_{2}^{3} 3 \sin \left(\frac{\pi}{3} x\right) d x$ using midpoint rule with $n=6$ . Round to four decimals.

Evaluate the integral $\int_{1}^{\infty} f\left(x^{m}\right) x^{m-1} \, dx$ for $m < 0$ given $F(1)=5$ and $\lim _{x \rightarrow \infty} F(x)=\infty$ .

Evaluate the integral $\int_{0}^{8} \frac{1}{2\left(x^{1 / 3}\right)} d x$ and determine if it converges or diverges.

Given a continuous function $f$ on $(0, \infty)$ with $\lim_{x \to 0^{+}} f(x)=\infty$ and $\lim_{x \to \infty} f(x)=0$ , and $g(x) \geq f(x)$ , which statement is always TRUE?