Calculus

Problem 20501

Calculate the integral: $=\left(\frac{1}{6}\right) \int_{0}^{6}\left(8 e^{t}\right) d t$

See Solution

Problem 20502

The volume of a cube is increasing at $1200 \mathrm{~cm}^{3} / \mathrm{min}$ when edges are $20 \mathrm{~cm}$ . Find the edge rate change.

See Solution

Problem 20503

Find the hourly growth rate of a bacteria population that grows from 2200 to 2346 in 2.5 hours. Express as a percentage.

See Solution

Problem 20504

Calculate $\left(\frac{1}{3}\right) \int_{0}^{3} 8 e^{t} \, dt$ .

See Solution

Problem 20505

A radioactive substance starts at $864 \mathrm{~kg}$ and decays at $3\%$ per day. Find its mass after 2 days.

See Solution

Problem 20506

Sketch the area between $y=e^{2x}$ , $y=e^{6x}$ , and $x=1$ . Calculate the enclosed area.

See Solution

Problem 20507

Calculate the integral $\int_{0}^{7}(9 e^{t}) dt$ and multiply by $\frac{1}{7}$ .

See Solution

Problem 20508

A conical tank is 24 ft high and 10 ft radius. Water flows in at 20 cu ft/min. Find the depth increase rate when water is 16 ft deep.

See Solution

Problem 20509

Calculate the area under the curve of $f$ from $x=3$ to $x=5$ , where $f(x)=x+9$ for $x \leq 4$ and $f(x)=17-\frac{1}{2} x$ for $x>4$ .

See Solution

Problem 20510

Calculate the area between the function $f(x)$ and the $x$ -axis on the interval $[-6,4]$ where: $f(x)=\begin{cases} x^{2}+2x+2 & x<2 \\ 3x+4 & x \geq 2 \end{cases}$

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Problem 20511

Find the limit: $\lim _{x \rightarrow-\infty} \sqrt{4 x^{2}-x^{2}+1}+2 x$ .

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Problem 20512

Find the derivative of $f(x)=\frac{3x-2}{2x+3}$ . Is $f'(x) = -\frac{13}{(2x+3)^{2}}$ true?

See Solution

Problem 20513

If $\int_{1}^{5} f(x) d x=2$ , find $\int_{-4}^{0} 3 f(x+5)$ .

See Solution

Problem 20514

A cone has a radius of 5 in and height of 7 in. When water is 3 in deep and falling at $-7 \mathrm{in/sec}$ , find the drain rate.

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Problem 20515

Given $f$ and its derivative in the table, find $g^{\prime}(3)$ for the inverse function $g$ . Options: (A) $\frac{1}{13}$ (B) $\frac{1}{4}$ (C) 1 (D) 4 (E) 13.

See Solution

Problem 20516

Find the limit: $\lim _{x \rightarrow-\infty} \frac{\sqrt{4 x^{2}-x+1}+2 x}{1}$ .

See Solution

Problem 20517

Find total profit $P(10)$ over the first 10 days given $R'(t)=140 e^{t}$ and $C'(t)=140-0.2t$ . Round to the nearest cent.

See Solution

Problem 20518

Find $f^{\prime}(2)$ if $f(x)=e^{2 x}(x^{3}+1)$ . Options: (A) $6 e^{4}$ , (B) $21 e^{4}$ , (C) $24 e^{4}$ , (D) $30 e^{4}$ .

See Solution

Problem 20519

Given the function $h(x)=\frac{x^{3}+\sin x}{x}$ , show the steps for the quotient rule to find $h^{\prime}(x)$ and verify it's $h^{\prime}(x)=\frac{2 x^{3}+x \cdot \cos x-\sin x}{x^{2}}$ . Then, argue algebraically why the slope approaches 0 as $x$ approaches 0 without using graphs.

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Problem 20520

Given $h(x)=\frac{x^{3}+\sin x}{x}$ , show $h^{\prime}(x)=\frac{2 x^{3}+x \cdot \cos x-\sin x}{x^{2}}$ using the quotient rule. Then, argue why the slope approaches 0 as $x \to 0$ without using graphs or L'Hopital's rule.

See Solution

Problem 20521

Find the partial derivatives $f_{x}(x, y)$ , $f_{y}(x, y)$ , $f_{xx}(x, y)$ , and $f_{xy}(x, y)$ for $f(x, y)=-6 x^{6}-3 x^{2} y^{2}+4 y^{5}$ .

See Solution

Problem 20522

For small $h$ , which function approximates $f(x)=\frac{\tan (x+h)-\tan x}{h}$ best? (A) $\sin x$ (B) $\frac{\sin x}{x}$ (C) $\frac{\tan x}{\sim}$ (D) $\sec x$

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Problem 20523

Identify which series are suitable for the integral test: a) $\sum_{n=1}^{\infty} \frac{n}{e^n}$ b) $\sum_{n=1}^{\infty} \frac{1}{n \ln n}$ c) $\sum_{n=1}^{\infty} \frac{n^2}{n^2 + 1}$ d) $\sum_{n=1}^{\infty} \frac{\sin n}{n^2}$ e) $\sum_{n=1}^{\infty} \tan\left(\frac{1}{n}\right)$

See Solution

Problem 20524

Find $\frac{d y}{d x}$ in terms of $y$ for the equation $3 x - \tan y = 4$ .

See Solution

Problem 20525

Evaluate the integral $\int \sqrt{x} \ln \left(x^{4}\right) d x$ using integration by parts. Choose $u$ and $d v$ .

See Solution

Problem 20526

Given continuous functions $f(x)$ and $g(x)$ with $f(x) \geq g(x) \geq 0$ , determine which statements about their integrals are true.

See Solution

Problem 20527

Calculate the future value of a 14-year continuous income of \$160,000 at a continuous compounding rate of 4.9\%.

See Solution

Problem 20528

Find $\lim _{x \rightarrow 0} \frac{4 x^{2}}{e^{4 x}-4 x-1}$ . Choose from (A) 0, (B) $1 / 2$ , (C) 8, (D) non existent.

See Solution

Problem 20529

Identify which series are suitable for the integral test: a) $\sum_{n=1}^{\infty} \frac{n}{e^n}$ b) $\sum_{n=1}^{\infty} \frac{1}{n \ln(n)}$ c) $\sum_{n=1}^{\infty} \frac{n^3}{n^2 + 1}$ d) $\sum_{n=1}^{\infty} \frac{\sin(n)}{n^2}$ e) $\sum_{n=1}^{\infty} \tan\left(\frac{1}{n}\right)$

See Solution

Problem 20530

For the function $f(x)=\frac{x^{3}}{x^{2}-36}$ on $[-18,16]$ , find vertical asymptotes and inflection points.

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Problem 20531

Find the equilibrium quantity where $d(x)=980-0.3 x^{2}$ and $s(x)=0.5 x^{2}$ . Then, calculate consumer surplus.

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Problem 20532

Find the period of a pendulum of length 0.1016 m under gravity $9.81 \, \text{m/s}^2$ . Use the formula $T = 2\pi \sqrt{\frac{L}{g}}$ .

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Problem 20533

Find the area represented by the integral $\int_{-4}^{4} \sqrt{16-x^{2}} d x$ using a sketch of the region.

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Problem 20534

Find the area represented by the integral $\int_{-2}^{2} \sqrt{4-x^{2}} d x$ by sketching the region.

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Problem 20535

Find the area represented by the integral $\int_{-1}^{7}(7-2 x) d x$ .

See Solution

Problem 20536

Find the object's position at $t=190$ using the area under $v(t)$ , rounded to four decimal places.

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Problem 20537

Find the object's position at $t=220$ using the area under the velocity function. Round to four decimal places.

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Problem 20538

Find the area under the graph of $f(x)$ from $x = -5$ to $x = 7$ using the points (-5, 3), (-3, 3), (4, 8), (7, 8).

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Problem 20539

Find the equation of the line perpendicular to the tangent of $f(x)=\frac{1}{2} e^{x}$ at $x=\ln 3$ .

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Problem 20540

Evaluate the integral: $\int \sin^{2}\left(\theta-\frac{\pi}{23}\right) d\theta = \square$ (Type exact answer.)

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Problem 20541

Find the limit: $\lim _{x \rightarrow 0^{+}} 8 \sin (x) \ln (x)$ . Use I'Hôpital's rule if needed.

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Problem 20542

For the sine function $f$ through (0, 0), (2, 4.4), (5, 0), (6.4, -1.8), (8, 0), order A, B, C, D by value.

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Problem 20543

Shade the area under $f(x)=-\frac{1}{2} x-1$ for $\int_{-5}^{1} f(x) \, dx$ and evaluate it using the area of two triangles.

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Problem 20544

Evaluate the following integrals as areas under the graph of function $f(x)$ :
a) $\int_{0}^{2} f(x) d x$
b) $\int_{0}^{5} f(x) d x$
c) $\int_{5}^{7} f(x) d x$
d) $\int_{0}^{9} f(x) d x$

See Solution

Problem 20545

Evaluate the integral $\int_{-12}^{10} f(x) d x$ for the points $(-12,-4)$ , $(-8,0)$ , $(-3,5)$ , $(5,5)$ , $(10,-20)$ .

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Problem 20546

Evaluate the integral $\int_{0}^{3 / 2}\left(\frac{9}{2} x^{2}-2 x^{2} y^{2}-\frac{x^{4}}{4}\right) \bigg|_{0}^{\sqrt{-9+4 y^{2}}} dy$ .

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Problem 20547

Evaluate $\int_{-12}^{10} f(x) dx$ as the area under the curve, counting below the x-axis as negative.

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Problem 20548

Evaluate the limit: $\lim _{x \rightarrow 0^{+}} x^{6 \sin (x)}$ using L'Hospital's rule if needed.

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Problem 20549

Explain why the function $y=e^{x}+e^{2 x}$ has no local extrema using algebra. Then graph it to confirm your findings.

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Problem 20550

Find the function $f(x)$ given that $f^{\prime \prime}(x)=8x+9\sin(x)$ , with $f(0)=3$ and $f^{\prime}(0)=2$ .

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Problem 20551

Approximate $\int_{1}^{3} \frac{1}{x} dx$ using the trapezoidal method with $\Delta x=0.5$ . Why is the result larger than $\ln 3$ ?
How long for liquid depth $h$ to rise from 0.1 ft to 0.5 ft in a tank with base area 4 ft², inflow 0.08 ft³/sec, outflow 0.12 ft³/sec?

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Problem 20552

Evaluate the integral $\int \frac{48 x^{2}}{(x-18)(x+6)^{2}} d x$ and find its partial fraction decomposition.

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Problem 20553

Find the general antiderivative of $f(x)=-8 x-3(1-x^{2})^{-1/2}$ for $-1<x<1$ . What is $F(x)=?$

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Problem 20554

Evaluate the integral $\int_{0}^{\pi} 2 x \cos x \, dx$ using integration by parts. Choose the simpler integral option.

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Problem 20555

Find the general antiderivative of $f(x)=4 e^{x}+8 \sec ^{2}(x)$ for $-\frac{\pi}{2}<x<\frac{\pi}{2}$ . $F(x) =$

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Problem 20556

Evaluate the integral $\int x^{2} e^{6 x} \mathrm{dx}$ using integration by parts. Choose the correct answer.

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Problem 20557

Evaluate the integral $\int t^{2} e^{-8 t} dt$ using integration by parts. Choose the correct simplification: A, B, C, or D.

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Problem 20558

Show that $\int_{1}^{2} \frac{1}{x} dx = \int_{-5}^{-10} \frac{1}{x} dx$ without decimals. Find $a$ and $c$ for $\frac{x+2}{x^{2}-3x} = \frac{a}{x}+\frac{c}{x-3}$ to evaluate $\int_{4}^{7} \frac{x+2}{x^{2}-3x} dx$ .

See Solution

Problem 20559

Evaluate $\mathrm{I}$ with the substitution $u=x-7$ : $\int \frac{x^{2}}{x-7} d x=\square$

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Problem 20560

Show that $f(x)=x^{6}+3x+1$ has exactly one root in $[-4,-1]$ using the intermediate value and Rolle's theorems.

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Problem 20561

Evaluate the integral $I=\int \frac{x^{2}}{x-7} d x$ using the substitution $u=x-7$ . What is $I$ ?

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Problem 20562

Evaluate the integral $I=\int \frac{x^{2}}{x-7} dx$ using substitution $u=x-7$ and long division. Reconcile results.

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Problem 20563

Untersuchen Sie das Verhalten der Funktionen an den Definitionslücken für $x_{0}=3$ und $x_{0}=0$ in den folgenden Fällen: a) $f(x)=\frac{x^{2}-9}{2 x-6}$ b) $f(x)=\frac{x+1}{x}$ c) $f(x)=\frac{x+1}{x^{2}}$

See Solution

Problem 20564

Untersuchen Sie die Funktion $f(x)=-0,125 x^{3}+0,75 x^{2}+1,5 x$ : Verhalten für $|x|$ , Symmetrie, Achsenschnittpunkte, Extrempunkte, Wendepunkte, Graph skizzieren, Wendetangente berechnen.

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Problem 20565

Find the general antiderivative of $f(x)=-8x-3(1-x^{2})^{-1/2}$ for $-1<x<1$ .

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Problem 20566

Calculate the integral: $\int (3 x^{2}+2 x^{3}) \, dx$ .

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Problem 20567

Calculate the integral: $\int (4 x^{3}-5 x^{2}+\frac{7}{2} x) \, dx$

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Problem 20568

Evaluate the integral from -7 to -2: $\int_{-7}^{-2} (6u^{2} - 6u + 8) \, du =$ (round to three decimal places).

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Problem 20569

Evaluate the integral $\int t^{2} e^{-8 t} dt$ using integration by parts. Choose the correct answer for the new integral.

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Problem 20570

Evaluate the integral from -3 to -2: $\int_{-3}^{-2} \frac{y^{5}-6 y^{2}}{y^{4}} d y$

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Problem 20571

Find the total words memorized in 13 minutes using $M^{\prime}(t)=-0.001 t^{2}+0.3 t$ . Round to the nearest whole word.

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Problem 20572

Evaluate the integral $I=\int \frac{x^{2}}{x-7} dx$ . Use substitution $u=x-7$ and perform long division on the integrand.

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Problem 20573

Evaluate the integral: $\int \frac{x^{2}-7}{x^{3}-2 x^{2}+x} dx$ and find its partial fraction decomposition.

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Problem 20574

Is $m(8)=12$ guaranteed if $m(3)=10$ , $\int_{3}^{8} m^{\prime}(x) dx=2$ , and $\int_{3}^{8} m(x) dx=2$ ?

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Problem 20575

Evaluate the integral $I=\int \frac{x^{2}}{x-7} dx$ using substitution $u=x-7$ and long division.

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Problem 20576

a. Show that $\int f^{-1}(x) dx=\int y f'(y) dy$ . b. Prove $\int f^{-1}(x) dx=y f(y)-\int f(y) dy$ if $f'$ is continuous. c. Find $\int \ln x dx$ in terms of $x$ . d. Find $\int \sin^{-1} x dx$ in terms of $x$ . e. Find $\int \tan^{-1} x dx$ in terms of $x$ .

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Problem 20577

Calculate the area under the curve $y=5 x^{3}$ from $x=0$ to $x=4$ in fractional form.

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Problem 20578

Find the area under $f(x)=2x+6$ from $x=17$ to $x=22$ using the Fundamental Theorem of Calculus. Round to the nearest tenth.

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Problem 20579

Find the area under $f(x)=-x^{2}+9x$ from $x=3$ to $x=7$ using the Fundamental Theorem of Calculus. Enter as a fraction.

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Problem 20580

If $\int_{n}^{v} w(x) d x=h(v)-h(n)$ , what can we conclude about $h$ and $w$ ? Choose true statements:
1. $h$ is an antiderivative of $w$
2. $w$ is an antiderivative of $h$
3. $w$ is the derivative of $h$
4. $h$ is the derivative of $w$
5. none of these

See Solution

Problem 20581

Calculate the area under the curve $y=\frac{8}{x}$ from $x=1$ to $x=4$ .

See Solution

Problem 20582

Evaluate the integral: $\int_{-1}^{-3} \frac{y^{5}-6 y^{2}}{y^{5}} d y$

See Solution

Problem 20583

Evaluate the integral: $\int_{-2}^{-3} \frac{y^{6}-6 y}{y^{4}} d y$ .

See Solution

Problem 20584

Calculate the integral from -2 to -1: $\int_{-2}^{-1} \frac{y^{5}-7 y}{y^{4}} d y$ .

See Solution

Problem 20585

Find $f'(x)$ for $f(x)=\int_{4}^{x} t^{4} dt$ and calculate $f'(-2)$ .

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Problem 20586

Berechne $\int \ln x \cdot x \, dx$ mit einfacher partieller Integration.

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Problem 20587

Find $f'(x)$ and $f''(x)$ if $f(x)=\int_{0}^{x} (t^{3}+6 t^{2}+1) dt$ .

See Solution

Problem 20588

Find $h^{\prime}(x)$ for $h(x)=\int_{0}^{x^{2}} \sqrt{9+t^{3}} d t$ .

See Solution

Problem 20589

Find the derivatives using the Fundamental Theorem of Calculus:
1. $f(x)=\int_{-2}^{x} \sqrt{t^{3}+8} dt$ , $f^{\prime}(x)=$
2. $g(x)=\int_{5}^{x} \frac{1}{1+t^{4}} dt$ , $g^{\prime}(x)=$

See Solution

Problem 20590

Find the area under $f(x)=-x^{2}+7x$ from $x=2$ to $x=5$ using the Fundamental Theorem of Calculus. Answer as a fraction.

See Solution

Problem 20591

A cone has radius 4.5 cm and height 9 cm. A cylinder fits inside.
(a) Express height $h$ in terms of radius $r$ and show $V=9 \pi r^{2}-2 \pi r^{3}$ . (b) Find the stationary value of $V$ as $r$ varies.

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Problem 20592

Find the definite integral for the area under $y=-x+12$ , above the $x$ -axis, from $x=-3$ to $x=2$ . $\int_{-3}^{2} (-x+12) \, dx$

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Problem 20593

Berechnen Sie die Integrale mit partieller Integration: a) $\int x \cdot \cos x \, dx$ b) $\int \ln x \cdot x \, dx$

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Problem 20594

Find the bounds $a$ and $b$ for the integral where $\int_{a}^{b} f(x) dx=\int_{-2}^{8} f(x) dx+\int_{8}^{11} f(u) du-\int_{-2}^{2} f(t) dt.$

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Problem 20595

Find the derivative of $f(x)=\int_{x}^{18} t^{7} dt$ and $g(x)=\int_{x}^{3} \sin(t^{2}) dt$ .

See Solution

Problem 20596

Evaluate the following integrals using geometry:
1. $\int_{-3}^{6} 5 \, dx = \square$
2. $\int_{2}^{7} (5x - 20) \, dx = \square$

See Solution

Problem 20597

Bestimme den Wert der Reihe $\sum_{k=0}^{\infty} \frac{1}{\left((-1)^{k}+4\right)^{k}}$ .

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Problem 20598

Berechne $\int x^{3} \cdot e^{x} d x$ mit dreifacher partieller Integration.

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Problem 20599

Calculate $\int_{-51}^{-21}[4 f(x)+5 g(x)-h(x)] d x$ given $\int_{-51}^{-21} f(x) d x=8$ , $\int_{-51}^{-21} g(x) d x=11$ , $\int_{-51}^{-21} h(x) d x=25$ .

See Solution

Problem 20600

Find the value of $\int_{4}^{1} f(s) d s$ given that $\int_{1}^{4} f(x) d x=\frac{16}{7}$ .

See Solution

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Calculus

Problem 20501

Calculate the integral: =(16)∫06(8et)dt=\left(\frac{1}{6}\right) \int_{0}^{6}\left(8 e^{t}\right) d t=(61​)∫06​(8et)dt

Problem 20502

The volume of a cube is increasing at 1200 cm3/min1200 \mathrm{~cm}^{3} / \mathrm{min}1200 cm3/min when edges are 20 cm20 \mathrm{~cm}20 cm. Find the edge rate change.

Problem 20503

Find the hourly growth rate of a bacteria population that grows from 2200 to 2346 in 2.5 hours. Express as a percentage.

Problem 20504

Calculate (13)∫038et dt\left(\frac{1}{3}\right) \int_{0}^{3} 8 e^{t} \, dt(31​)∫03​8etdt.

Problem 20505

A radioactive substance starts at 864 kg864 \mathrm{~kg}864 kg and decays at 3%3\%3% per day. Find its mass after 2 days.

Problem 20506

Sketch the area between y=e2xy=e^{2x}y=e2x, y=e6xy=e^{6x}y=e6x, and x=1x=1x=1. Calculate the enclosed area.

Problem 20507

Calculate the integral ∫07(9et)dt\int_{0}^{7}(9 e^{t}) dt∫07​(9et)dt and multiply by 17\frac{1}{7}71​.

Problem 20508

A conical tank is 24 ft high and 10 ft radius. Water flows in at 20 cu ft/min. Find the depth increase rate when water is 16 ft deep.

Problem 20509

Calculate the area under the curve of fff from x=3x=3x=3 to x=5x=5x=5, where f(x)=x+9f(x)=x+9f(x)=x+9 for x≤4x \leq 4x≤4 and f(x)=17−12xf(x)=17-\frac{1}{2} xf(x)=17−21​x for x>4x>4x>4.

Problem 20510

Calculate the area between the function f(x)f(x)f(x) and the xxx-axis on the interval [−6,4][-6,4][−6,4] where: f(x)={x2+2x+2x<23x+4x≥2 f(x)=\begin{cases} x^{2}+2x+2 & x<2 \\ 3x+4 & x \geq 2 \end{cases} f(x)={x2+2x+23x+4​x<2x≥2​

Problem 20511

Find the limit: lim⁡x→−∞4x2−x2+1+2x\lim _{x \rightarrow-\infty} \sqrt{4 x^{2}-x^{2}+1}+2 xlimx→−∞​4x2−x2+1​+2x.

Problem 20512

Find the derivative of f(x)=3x−22x+3f(x)=\frac{3x-2}{2x+3}f(x)=2x+33x−2​. Is f′(x)=−13(2x+3)2f'(x) = -\frac{13}{(2x+3)^{2}}f′(x)=−(2x+3)213​ true?

Problem 20513

If ∫15f(x)dx=2\int_{1}^{5} f(x) d x=2∫15​f(x)dx=2, find ∫−403f(x+5)\int_{-4}^{0} 3 f(x+5)∫−40​3f(x+5).

Problem 20514

A cone has a radius of 5 in and height of 7 in. When water is 3 in deep and falling at −7in/sec-7 \mathrm{in/sec}−7in/sec, find the drain rate.

Problem 20515

Given fff and its derivative in the table, find g′(3)g^{\prime}(3)g′(3) for the inverse function ggg. Options: (A) 113\frac{1}{13}131​ (B) 14\frac{1}{4}41​ (C) 1 (D) 4 (E) 13.

Problem 20516

Find the limit: lim⁡x→−∞4x2−x+1+2x1\lim _{x \rightarrow-\infty} \frac{\sqrt{4 x^{2}-x+1}+2 x}{1}limx→−∞​14x2−x+1​+2x​.

Problem 20517

Find total profit P(10)P(10)P(10) over the first 10 days given R′(t)=140etR'(t)=140 e^{t}R′(t)=140et and C′(t)=140−0.2tC'(t)=140-0.2tC′(t)=140−0.2t. Round to the nearest cent.

Problem 20518

Find f′(2)f^{\prime}(2)f′(2) if f(x)=e2x(x3+1)f(x)=e^{2 x}(x^{3}+1)f(x)=e2x(x3+1). Options: (A) 6e46 e^{4}6e4, (B) 21e421 e^{4}21e4, (C) 24e424 e^{4}24e4, (D) 30e430 e^{4}30e4.

Problem 20519

Problem 20520

Problem 20521

Find the partial derivatives fx(x,y)f_{x}(x, y)fx​(x,y), fy(x,y)f_{y}(x, y)fy​(x,y), fxx(x,y)f_{xx}(x, y)fxx​(x,y), and fxy(x,y)f_{xy}(x, y)fxy​(x,y) for f(x,y)=−6x6−3x2y2+4y5f(x, y)=-6 x^{6}-3 x^{2} y^{2}+4 y^{5}f(x,y)=−6x6−3x2y2+4y5.

Problem 20522

For small hhh, which function approximates f(x)=tan⁡(x+h)−tan⁡xhf(x)=\frac{\tan (x+h)-\tan x}{h}f(x)=htan(x+h)−tanx​ best? (A) sin⁡x\sin xsinx (B) sin⁡xx\frac{\sin x}{x}xsinx​ (C) tan⁡x∼\frac{\tan x}{\sim}∼tanx​ (D) sec⁡x\sec xsecx

Problem 20523

Problem 20524

Find dydx\frac{d y}{d x}dxdy​ in terms of yyy for the equation 3x−tan⁡y=43 x - \tan y = 43x−tany=4.

Problem 20525

Evaluate the integral ∫xln⁡(x4)dx\int \sqrt{x} \ln \left(x^{4}\right) d x∫x​ln(x4)dx using integration by parts. Choose uuu and dvd vdv.

Problem 20526

Given continuous functions f(x)f(x)f(x) and g(x)g(x)g(x) with f(x)≥g(x)≥0f(x) \geq g(x) \geq 0f(x)≥g(x)≥0, determine which statements about their integrals are true.

Problem 20527

Calculate the future value of a 14-year continuous income of \$160,000 at a continuous compounding rate of 4.9\%.

Problem 20528

Find lim⁡x→04x2e4x−4x−1\lim _{x \rightarrow 0} \frac{4 x^{2}}{e^{4 x}-4 x-1}limx→0​e4x−4x−14x2​. Choose from (A) 0, (B) 1/21 / 21/2, (C) 8, (D) non existent.

Problem 20529

Problem 20530

For the function f(x)=x3x2−36f(x)=\frac{x^{3}}{x^{2}-36}f(x)=x2−36x3​ on [−18,16][-18,16][−18,16], find vertical asymptotes and inflection points.

Problem 20531

Find the equilibrium quantity where d(x)=980−0.3x2d(x)=980-0.3 x^{2}d(x)=980−0.3x2 and s(x)=0.5x2s(x)=0.5 x^{2}s(x)=0.5x2. Then, calculate consumer surplus.

Problem 20532

Find the period of a pendulum of length 0.1016 m under gravity 9.81 m/s29.81 \, \text{m/s}^29.81m/s2. Use the formula T=2πLgT = 2\pi \sqrt{\frac{L}{g}}T=2πgL​​.

Problem 20533

Find the area represented by the integral ∫−4416−x2dx\int_{-4}^{4} \sqrt{16-x^{2}} d x∫−44​16−x2​dx using a sketch of the region.

Problem 20534

Find the area represented by the integral ∫−224−x2dx\int_{-2}^{2} \sqrt{4-x^{2}} d x∫−22​4−x2​dx by sketching the region.

Problem 20535

Find the area represented by the integral ∫−17(7−2x)dx\int_{-1}^{7}(7-2 x) d x∫−17​(7−2x)dx.

Problem 20536

Find the object's position at t=190t=190t=190 using the area under v(t)v(t)v(t), rounded to four decimal places.

Problem 20537

Find the object's position at t=220t=220t=220 using the area under the velocity function. Round to four decimal places.

Problem 20538

Find the area under the graph of f(x)f(x)f(x) from x=−5x = -5x=−5 to x=7x = 7x=7 using the points (-5, 3), (-3, 3), (4, 8), (7, 8).

Problem 20539

Find the equation of the line perpendicular to the tangent of f(x)=12exf(x)=\frac{1}{2} e^{x}f(x)=21​ex at x=ln⁡3x=\ln 3x=ln3.

Problem 20540

Evaluate the integral: ∫sin⁡2(θ−π23)dθ=□\int \sin^{2}\left(\theta-\frac{\pi}{23}\right) d\theta = \square∫sin2(θ−23π​)dθ=□ (Type exact answer.)

Problem 20541

Find the limit: lim⁡x→0+8sin⁡(x)ln⁡(x)\lim _{x \rightarrow 0^{+}} 8 \sin (x) \ln (x)limx→0+​8sin(x)ln(x). Use I'Hôpital's rule if needed.

Problem 20542

Calculate the integral: $=\left(\frac{1}{6}\right) \int_{0}^{6}\left(8 e^{t}\right) d t$

The volume of a cube is increasing at $1200 \mathrm{~cm}^{3} / \mathrm{min}$ when edges are $20 \mathrm{~cm}$ . Find the edge rate change.

Calculate $\left(\frac{1}{3}\right) \int_{0}^{3} 8 e^{t} \, dt$ .

A radioactive substance starts at $864 \mathrm{~kg}$ and decays at $3\%$ per day. Find its mass after 2 days.

Sketch the area between $y=e^{2x}$ , $y=e^{6x}$ , and $x=1$ . Calculate the enclosed area.

Calculate the integral $\int_{0}^{7}(9 e^{t}) dt$ and multiply by $\frac{1}{7}$ .

Calculate the area under the curve of $f$ from $x=3$ to $x=5$ , where $f(x)=x+9$ for $x \leq 4$ and $f(x)=17-\frac{1}{2} x$ for $x>4$ .

Calculate the area between the function $f(x)$ and the $x$ -axis on the interval $[-6,4]$ where: $f(x)=\begin{cases} x^{2}+2x+2 & x<2 \\ 3x+4 & x \geq 2 \end{cases}$

Find the limit: $\lim _{x \rightarrow-\infty} \sqrt{4 x^{2}-x^{2}+1}+2 x$ .

Find the derivative of $f(x)=\frac{3x-2}{2x+3}$ . Is $f'(x) = -\frac{13}{(2x+3)^{2}}$ true?

If $\int_{1}^{5} f(x) d x=2$ , find $\int_{-4}^{0} 3 f(x+5)$ .

A cone has a radius of 5 in and height of 7 in. When water is 3 in deep and falling at $-7 \mathrm{in/sec}$ , find the drain rate.

Given $f$ and its derivative in the table, find $g^{\prime}(3)$ for the inverse function $g$ . Options: (A) $\frac{1}{13}$ (B) $\frac{1}{4}$ (C) 1 (D) 4 (E) 13.

Find the limit: $\lim _{x \rightarrow-\infty} \frac{\sqrt{4 x^{2}-x+1}+2 x}{1}$ .

Find total profit $P(10)$ over the first 10 days given $R'(t)=140 e^{t}$ and $C'(t)=140-0.2t$ . Round to the nearest cent.

Find $f^{\prime}(2)$ if $f(x)=e^{2 x}(x^{3}+1)$ . Options: (A) $6 e^{4}$ , (B) $21 e^{4}$ , (C) $24 e^{4}$ , (D) $30 e^{4}$ .

Find the partial derivatives $f_{x}(x, y)$ , $f_{y}(x, y)$ , $f_{xx}(x, y)$ , and $f_{xy}(x, y)$ for $f(x, y)=-6 x^{6}-3 x^{2} y^{2}+4 y^{5}$ .

For small $h$ , which function approximates $f(x)=\frac{\tan (x+h)-\tan x}{h}$ best? (A) $\sin x$ (B) $\frac{\sin x}{x}$ (C) $\frac{\tan x}{\sim}$ (D) $\sec x$

Find $\frac{d y}{d x}$ in terms of $y$ for the equation $3 x - \tan y = 4$ .

Evaluate the integral $\int \sqrt{x} \ln \left(x^{4}\right) d x$ using integration by parts. Choose $u$ and $d v$ .

Given continuous functions $f(x)$ and $g(x)$ with $f(x) \geq g(x) \geq 0$ , determine which statements about their integrals are true.

Find $\lim _{x \rightarrow 0} \frac{4 x^{2}}{e^{4 x}-4 x-1}$ . Choose from (A) 0, (B) $1 / 2$ , (C) 8, (D) non existent.

For the function $f(x)=\frac{x^{3}}{x^{2}-36}$ on $[-18,16]$ , find vertical asymptotes and inflection points.

Find the equilibrium quantity where $d(x)=980-0.3 x^{2}$ and $s(x)=0.5 x^{2}$ . Then, calculate consumer surplus.

Find the period of a pendulum of length 0.1016 m under gravity $9.81 \, \text{m/s}^2$ . Use the formula $T = 2\pi \sqrt{\frac{L}{g}}$ .

Find the area represented by the integral $\int_{-4}^{4} \sqrt{16-x^{2}} d x$ using a sketch of the region.

Find the area represented by the integral $\int_{-2}^{2} \sqrt{4-x^{2}} d x$ by sketching the region.

Find the area represented by the integral $\int_{-1}^{7}(7-2 x) d x$ .

Find the object's position at $t=190$ using the area under $v(t)$ , rounded to four decimal places.

Find the object's position at $t=220$ using the area under the velocity function. Round to four decimal places.

Find the area under the graph of $f(x)$ from $x = -5$ to $x = 7$ using the points (-5, 3), (-3, 3), (4, 8), (7, 8).

Find the equation of the line perpendicular to the tangent of $f(x)=\frac{1}{2} e^{x}$ at $x=\ln 3$ .

Evaluate the integral: $\int \sin^{2}\left(\theta-\frac{\pi}{23}\right) d\theta = \square$ (Type exact answer.)

Find the limit: $\lim _{x \rightarrow 0^{+}} 8 \sin (x) \ln (x)$ . Use I'Hôpital's rule if needed.

For the sine function $f$ through (0, 0), (2, 4.4), (5, 0), (6.4, -1.8), (8, 0), order A, B, C, D by value.

Shade the area under $f(x)=-\frac{1}{2} x-1$ for $\int_{-5}^{1} f(x) \, dx$ and evaluate it using the area of two triangles.

Evaluate the following integrals as areas under the graph of function $f(x)$ :
a) $\int_{0}^{2} f(x) d x$
b) $\int_{0}^{5} f(x) d x$
c) $\int_{5}^{7} f(x) d x$
d) $\int_{0}^{9} f(x) d x$

Evaluate the integral $\int_{-12}^{10} f(x) d x$ for the points $(-12,-4)$ , $(-8,0)$ , $(-3,5)$ , $(5,5)$ , $(10,-20)$ .

Evaluate the integral $\int_{0}^{3 / 2}\left(\frac{9}{2} x^{2}-2 x^{2} y^{2}-\frac{x^{4}}{4}\right) \bigg|_{0}^{\sqrt{-9+4 y^{2}}} dy$ .

Evaluate $\int_{-12}^{10} f(x) dx$ as the area under the curve, counting below the x-axis as negative.

Evaluate the limit: $\lim _{x \rightarrow 0^{+}} x^{6 \sin (x)}$ using L'Hospital's rule if needed.

Explain why the function $y=e^{x}+e^{2 x}$ has no local extrema using algebra. Then graph it to confirm your findings.

Find the function $f(x)$ given that $f^{\prime \prime}(x)=8x+9\sin(x)$ , with $f(0)=3$ and $f^{\prime}(0)=2$ .

Evaluate the integral $\int \frac{48 x^{2}}{(x-18)(x+6)^{2}} d x$ and find its partial fraction decomposition.

Find the general antiderivative of $f(x)=-8 x-3(1-x^{2})^{-1/2}$ for $-1<x<1$ . What is $F(x)=?$

Evaluate the integral $\int_{0}^{\pi} 2 x \cos x \, dx$ using integration by parts. Choose the simpler integral option.

Find the general antiderivative of $f(x)=4 e^{x}+8 \sec ^{2}(x)$ for $-\frac{\pi}{2}<x<\frac{\pi}{2}$ . $F(x) =$

Evaluate the integral $\int x^{2} e^{6 x} \mathrm{dx}$ using integration by parts. Choose the correct answer.

Evaluate the integral $\int t^{2} e^{-8 t} dt$ using integration by parts. Choose the correct simplification: A, B, C, or D.

Evaluate $\mathrm{I}$ with the substitution $u=x-7$ : $\int \frac{x^{2}}{x-7} d x=\square$

Show that $f(x)=x^{6}+3x+1$ has exactly one root in $[-4,-1]$ using the intermediate value and Rolle's theorems.

Evaluate the integral $I=\int \frac{x^{2}}{x-7} d x$ using the substitution $u=x-7$ . What is $I$ ?

Evaluate the integral $I=\int \frac{x^{2}}{x-7} dx$ using substitution $u=x-7$ and long division. Reconcile results.

Untersuchen Sie die Funktion $f(x)=-0,125 x^{3}+0,75 x^{2}+1,5 x$ : Verhalten für $|x|$ , Symmetrie, Achsenschnittpunkte, Extrempunkte, Wendepunkte, Graph skizzieren, Wendetangente berechnen.

Find the general antiderivative of $f(x)=-8x-3(1-x^{2})^{-1/2}$ for $-1<x<1$ .

Calculate the integral: $\int (3 x^{2}+2 x^{3}) \, dx$ .

Calculate the integral: $\int (4 x^{3}-5 x^{2}+\frac{7}{2} x) \, dx$

Evaluate the integral from -7 to -2: $\int_{-7}^{-2} (6u^{2} - 6u + 8) \, du =$ (round to three decimal places).

Evaluate the integral $\int t^{2} e^{-8 t} dt$ using integration by parts. Choose the correct answer for the new integral.

Evaluate the integral from -3 to -2: $\int_{-3}^{-2} \frac{y^{5}-6 y^{2}}{y^{4}} d y$

Find the total words memorized in 13 minutes using $M^{\prime}(t)=-0.001 t^{2}+0.3 t$ . Round to the nearest whole word.

Evaluate the integral $I=\int \frac{x^{2}}{x-7} dx$ . Use substitution $u=x-7$ and perform long division on the integrand.

Evaluate the integral: $\int \frac{x^{2}-7}{x^{3}-2 x^{2}+x} dx$ and find its partial fraction decomposition.

Is $m(8)=12$ guaranteed if $m(3)=10$ , $\int_{3}^{8} m^{\prime}(x) dx=2$ , and $\int_{3}^{8} m(x) dx=2$ ?