Calculus

Problem 16501

Find $F(x)=\int_{2}^{x}(t^{3}+4t-4) dt$ and evaluate $F(2)$ , $F(5)$ , and $F(8)$ .

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Problem 16502

Find $c$ in $\left[-\frac{\pi}{3}, \frac{\pi}{3}\right]$ for $f(x)=\cos x$ satisfying $f'(c) = \frac{f(b) - f(a)}{b - a}$ .

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Problem 16503

An object at $190^{\circ} \mathrm{F}$ is in water at $40^{\circ} \mathrm{F}$ . Find $F(t)$ with cooling constant $k=-0.6$ . $F(t)=$

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Problem 16504

Find the derivative $F^{\prime}(x)$ for $F(x)=\int_{x}^{x+2}(6 t+3) d t$ .

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Problem 16505

Calculate the present value of a continuous income stream $F(t)=30+7t$ over 20 years with a $1.9\%$ continuous interest rate.

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Problem 16506

Find the derivative of the function $y=\int_{7}^{\tan (x)} \sqrt{3 t+\sqrt{t}} \, dt$ . What is $y'$ ?

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Problem 16507

Find the derivative $F^{\prime}(x)$ where $F(x)=\int_{0}^{\sin (x)} 3 \sqrt{t} dt$ .

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Problem 16508

Q1. Simplify the rate formula $R$ and find $R$ for $t=0$ and $t=4$ .
Q2. Calculate $E$ in $\mathrm{J}$ if $\log E=-18.49$ .
Q3. Given $\mathrm{c}=\mathrm{a}+\mathrm{bT}$ , find constants $a$ and $b$ using given heat capacities.
Q4. A baseball is ejected at $42.5 \mathrm{~m/s}$ . Find its height and velocity at $t=0.43 \mathrm{~s}$ .

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Problem 16509

Find the derivative $F^{\prime}(x)$ if $F(x)=\int_{1}^{x^{2}} \frac{1}{t^{3}} dt$ .

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Problem 16510

Resuelve las siguientes integrales usando integración por partes o tabulación:
1. $\int x^{3} \ln x \, dx$
2. $\int \cos^{-1} x \, dx$
3. $\int (t^{4}+2t^{3}+t^{2}-t+5) e^{3t} \, dt$
4. $\int \sin^{6} \theta \cos^{3} \theta \, d\theta$
5. $\int \sin^{3} \theta \cos^{2} (3\theta) \, d\theta$
6. $\int \sec^{4} (3x) \tan^{3} (3x) \, dx$
7. $\int \frac{\sec^{2} t}{\tan^{2} t} \, dt$

See Solution

Problem 16511

Evaluate the integral $\int_{0}^{2}\left(\frac{3}{3 x+2}+\frac{2}{x^{2}+4}\right) d x$ .

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Problem 16512

Calculez les dérivées des fonctions suivantes : a) $f(x)=\log_{\pi}(3x^{2}+1)+\pi^{(3x^{2}+1)}$ b) $f(x)=e^{2x+1}\ln(2x+1)$ c) $f(x)=(\sqrt{x+1})\tan(2+\sqrt{x+1})$ d) $f(x)=\frac{\arcsin(x^{2}+1)}{x}$ e) $f(x)=\arctan(x^{2}+x+e)$ f) $f(x)=e^{\ln(x+1)}-\ln(e^{(x+1)})$

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Problem 16513

Evaluate the integral: $\int \sin (48 t) \sec ^{2}(\cos (48 t)) dt$ (use C for the constant).

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Problem 16514

Evaluate the integral: $\int\left(t^{4}+2 t^{3}+t^{2}-t+5\right) e^{3 t} d t$

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Problem 16515

Evaluate the integral: $\int \frac{\cos (x)}{\sin ^{6}(x)} dx$ and include the constant $C$ .

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Problem 16516

A radioactive sample starts with $618.8 \mathrm{mg}$ and has a half-life of 14 days.
(a) Write the formula $y = \square e^{\mathbb{D}}$ for amount $y$ after $t$ days.
(b) Find the amount after 16 days, rounded to the nearest tenth in $\mathrm{mg}$ .

See Solution

Problem 16517

Calculez les dérivées des fonctions suivantes : 1. $f(x)=\mathrm{e}^{-\sqrt{x}}+\ln(x^{3}+1)$ 2. $f(x)=\log_{2}(1-x^{2})-\log_{2}(1+x)$ 3. $f(x)=\tan\left(\frac{1}{x+1}\right)$ 4. $f(x)=\arccos x+\arcsin(2x)$ 5. $f(x)=\pi^{x^{3}+2x-7}$

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Problem 16518

Evaluate the integral using the substitution $u=1/x^{9}$ . Find $\int \frac{\sec^{2}(u)}{x^{10}} dx + C$ .

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Problem 16519

Evaluate the derivative $y'$ at the point $P=(0, \frac{\pi}{2})$ from the equation $y' \cos(y+x+x^2) - y \sin(y+x+x^2)(y'+1+2x) = 3x^2$ .

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Problem 16520

Find the area between the curves $y=x^{2}+3$ and $y=6x-6$ from $x=-1$ to $x=2$ . Round to three decimal places.

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Problem 16521

Find the area between the curves $y=e^{x}$ and $y=-\frac{1}{x}$ from $x=1$ to $x=2$ . Area = $\square$ square units.

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Problem 16522

Find the area between $y=e^{0.75 x}$ and $y=-\frac{1}{x}$ from $x=1$ to $x=2$ . The area is $\square$ square units.

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Problem 16523

Find the instantaneous velocity using $s=\frac{s}{t^{2}}$ at $t_{1}=-2$ .

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Problem 16524

Evaluate the limit: $\lim _{n \rightarrow+\infty} \sum_{i=1}^{n} \frac{\sin (i \pi / n)}{n}$ using a definite integral on $[0,1]$ .

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Problem 16525

Find the tangent line equation to $\cos(x) + 10y^2 = xy^9 + 31$ at $(0, \sqrt{3})$ .

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Problem 16526

Evaluate the following integrals using $\int_{0}^{4} x^{2} d x=\frac{64}{3}$ : (a) $\int_{-4}^{0} x^{2} d x$ (b) $\int_{-4}^{4} x^{2} d x$ (c) $\int_{0}^{4}-x^{2} d x$ (d) $\int_{-4}^{0} 3 x^{2} d x$

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Problem 16527

Find the remaining amount $Q$ of a \ $6 million grant after 70 days, given$ \frac{dQ}{dt} \propto (100-t)^2$.

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Problem 16528

Calculate the integral $\int \sin^{6} \theta \cos^{3} \theta \, d\theta$ .

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Problem 16529

Check if the series $\sum_{n=2}^{\infty} \frac{1}{n \sqrt{\ln n}}$ converges or diverges.

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Problem 16530

Evaluate the integral: $\int_{1}^{27} \frac{3}{x \cdot \sqrt[3]{x}} dx$

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Problem 16531

Kenny wants to withdraw \$33,000 annually for 25 years at 8.7% interest. How much must he have at age 65?

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Problem 16532

Determine where $f(x)$ is increasing, decreasing, and find local extrema for $f(x)=-2 x^{2}-20 x-21$ .

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Problem 16533

Determine where the function $f(x) = -2x^2 - 20x - 20$ is increasing, decreasing, and find local extrema.

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Problem 16534

Given the sequence defined by $a_{1}=2$ and $a_{n+1}=\frac{1}{2}(a_{n}+6)$ , prove it's nondecreasing & bounded by 6, then find $\lim_{n \to \infty} a_{n}$ .

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Problem 16535

Evaluate the integral: $\int_{0}^{2 \pi} \frac{\sin ^{2}(\theta)}{5+4 \cos (\theta)} d \theta$

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Problem 16536

Find the absolute maximum and minimum of $f(x)=2 x^{3}-6 x$ on the interval $[0,3]$ .

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Problem 16537

Find the tangent line equation for $y=f(x)$ at $x=e$ , where $f(x)=x-8 \ln x$ .

See Solution

Problem 16538

Find the tangent line equation to $y=f(x)$ at $x=e$ , where $f(x)=x-8 \ln x$ .

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Problem 16539

Evaluate the integral $\int_{-1}^{2}\left(\frac{-1}{x-3}+\frac{2}{x+2}\right) d x$ .

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Problem 16540

Find where the function $f(x) = -3x^2 - 12x - 20$ is increasing, decreasing, and its local extrema.

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Problem 16541

Find the derivatives: a. $\frac{d z}{d w}$ for $z=w^{e}+e^{\pi}+\pi^{w}$ , b. $f^{\prime}(x)$ for $f(x)=e^{6}+\ln (9)$ , c. $f^{\prime}(x)$ for $f(x)=\frac{\log _{8}(x)}{4}$ , d. $g^{\prime}(s)$ for $g(s)=\frac{6^{s}}{\ln s}$ .

See Solution

Problem 16542

Determine where the function $f(x)$ is increasing, decreasing, and find the local extrema for $f(x)=-3 x^{2}-18 x-27$ .

See Solution

Problem 16543

Find where $f(x)$ is increasing, decreasing, and its local extrema for $f(x)=-3 x^{2}-18 x-20$ .

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Problem 16544

Evaluate the limit: $\lim _{x \rightarrow-1^{+}} \frac{5 \cos ^{-1}(x)-5 \pi}{14 \sin ^{-1}(x)+7 \pi}$ .

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Problem 16545

Find the dimensions of a Norman window with a perimeter of 90 feet that maximize its area. $(r, y)=(\square) \mathrm{ft}$

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Problem 16546

Find the critical points of the function $f(x)=x^{4} \ln x$ by determining where $f^{\prime}(x)=0$ .

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Problem 16547

Find the difference quotient for the function $f(x)=5 x^{2}+2 x-2$ : $\frac{f(x+h)-f(x)}{h}$ .

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Problem 16548

Maximize the area of a window with a semicircle and rectangle. Total height is 90 ft; width is $2r$ . Find $r$ and $y$ .

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Problem 16549

Find the maximum volume of a cylinder formed by rotating a rectangle around the $x$ -axis, bounded by $f(x)=\frac{x}{x^{2}+1}$ .

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Problem 16550

Rewrite $f(x)=e^{8 x+5}$ as $f(x)=C e^{k x}$ and $f(x)=C \cdot b^{x}$ . Find $f'(x)$ . Do the same for $g(x)=7^{-3 x}$ . Find $g'(x)$ .

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Problem 16551

Calculate the series sum: $\sum_{n=1}^{\infty}\left(\frac{5}{n(n+1)}-\frac{1}{2^{n}}\right)$ .

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Problem 16552

Find the area under the standard normal curve from $z=0$ to $z=0.89$ . Round to four decimal places.

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Problem 16553

Evaluate the integral and verify by differentiating: $\int \frac{9}{8+9 x} d x, x \neq -\frac{8}{9}$

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Problem 16554

Find the difference quotient, $\frac{f(x+h)-f(x)}{h}$ , for the function $f(x)=-3x-7$ .

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Problem 16555

Prove that if the series $\sum_{n \in \mathbb{N}} a_{n}$ and $\sum_{n \in \mathbb{N}} b_{n}$ converge, then $\sum_{n \in \mathbb{N}} a_{n} b_{n}$ converges.

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Problem 16556

Determine the local min and max of the polynomial $P(x)=\frac{4}{9} x^{3}-2 x^{2}$ .

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Problem 16557

Prove that if $\sum_{n \in \mathbb{N}} a_{n}$ and $\sum_{n \in \mathbb{N}} b_{n}$ converge, then $\sum_{n \in \mathbb{N}} a_{n} b_{n}$ converges. Also, show $\sum_{n \in \mathbb{N}} a_{n}^{2}$ converges if $\sum_{n \in \mathbb{N}} a_{n}$ converges.

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Problem 16558

Invest \$16,037 at 5.5% interest, compounded continuously. Find the function for amount over time, balances after 1, 2, 5, 10 years, and doubling time.

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Problem 16559

In a town of 4000, the infected after $t$ days is $N(t)=\frac{4000}{1+15.1 e^{-0.6 t}}$ . Find $N(0)$ , $N(2)$ , $N(5)$ , $N(8)$ , $N(12)$ , $N(16)$ , and discuss if all 4000 will be infected.

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Problem 16560

Find the initial temperature and the temperature after 20 minutes of a soda cooling in a cooler, given $T(x)=-4+24 e^{-0.034 x}$ .

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Problem 16561

Find the population size $P(t)=\frac{800}{1+4 e^{-0.4 t}}$ after 6 and 8 years. Round to the nearest whole number.

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Problem 16562

Austin invested \ $92,000 at$ 5 \frac{7}{8} \% $daily, and Josiah at$ 5 \frac{3}{4} \%$ continuously. After 17 years, find the difference.

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Problem 16563

Invest \ $10,000 at$ 9\% $daily or$ 8.92\%$ continuously. Which gives more in 3 years?

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Problem 16564

Find the difference quotient for the function $f(x)=-3 x^{2}+x-8$ : $\frac{f(x+h)-f(x)}{h}$ .

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Problem 16565

Insect population $P(t)=500 e^{0.02 t}$ : (a) Find $P(0)$ . (b) Determine growth rate. (c) Find $P(10)$ . (d) When is $P=650$ ? (e) When does $P$ double?

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Problem 16566

Find the value of $\int_{3}^{3} f(x) d x$ given $\int_{0}^{1} f(x) d x=1$ , $\int_{0}^{2} f(x) d x=3$ , and $\int_{2}^{5} f(x) d x=2$ .

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Problem 16567

Find the mosquito population after 4 days if it grows from 1000 to 1900 in 1 day. Approximately $\square$ mosquitoes.

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Problem 16568

Caroline invested \ $900 at an interest rate of$ 8 \frac{7}{8} \% $compounded continuously. Jaxon invested \$900 at$ 8 \frac{5}{8} \%$ compounded daily. After 8 years, find the difference in their account balances.

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Problem 16569

Find $g^{\prime}\left(\frac{5}{2}\right)$ given $g(x)-9 x \sin (g(x))=6 x^{2}-x-35$ and $g\left(\frac{5}{2}\right)=0$ .

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Problem 16570

Evaluate the integral: $\int_{-\infty}^{\infty} \frac{3 x^{2}}{(x^{2}+2 x+2)(x^{2}+1)^{2}} dx$

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Problem 16571

Temukan titik ekstrem dari $y=\frac{1}{3} x^{3}+8 x^{2}+60 x$ .

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Problem 16572

How long will it take for \$2,400 to grow to \$2,930 at a continuous interest rate of 3.4%? Round to the nearest tenth of a year.

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Problem 16573

Elijah invested \$280 at a 2.1% continuous interest rate. How long until it grows to \$360? Round to the nearest tenth of a year.

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Problem 16574

Find the derivatives of these functions:
1) $f(x)=\frac{1}{8}(x-1)(x^{2}-12 x+16)$
2) $K(x)=a x^{3}+b x^{2}+c x+d$
3) $f(x)=-7 e^{x}+3 e$
4) $f(x)=-\frac{3}{4} e^{x}-e \cdot x$
5) $f(x)=\frac{1}{4} e^{x}-5 x^{3}+6 e^{3}$
6) $f(t)=\frac{1}{2} t^{4}-t+5 \cos (t)$
7) $A(u)=-\frac{1}{2} \sin (u)+\cos \left(\frac{\pi}{3}\right)$
8) $f(x)=t^{2} x(x^{2}-8 x)$
9) $f(x)=\sin \left(\frac{\pi}{6}\right) \cdot e^{x}+e^{3} x^{2}$
10) $f(x)=x^{3}-2 x^{2}+4 x+\frac{3}{x}-5 \sqrt{x}$

See Solution

Problem 16575

Calculate $\lim _{x \rightarrow 3^{-}} \frac{-3|x-3|}{x-3}$ . Options: A. +3 B. -3 C. $-\infty$ D. None

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Problem 16576

Untersuchen Sie das Verhalten der Funktion $f(x)=\frac{1}{2} x^{3}-2 x^{2}$ für $x \rightarrow \pm \infty$ und ändern Sie die Koeffizienten für gewünschte Limiten.

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Problem 16577

Find the limit of $a_{n}=n\left(\sqrt[3]{8 n^{3}+2 n-1}-2 n-1\right)$ .

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Problem 16578

Find the hourly growth rate of a bacteria population that grows from 2200 to 2371 in 2 hours. Express as a percentage.

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Problem 16579

Find the half-life of a radioactive substance with a decay rate of $6.7\%$ per day using the continuous decay model.

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Problem 16580

A radioactive substance with a half-life of 4 min starts with $882.9 \mathrm{~g}$ .
(a) Write the formula $y = \square e^{\text{(II)} t}$ .
(b) Find the amount after 23 min: $\square \mathrm{g}$ .

See Solution

Problem 16581

Oblicz granice ciągów:
1. $\lim_{n \to \infty} \sqrt[n]{3^{n}+4^{n}+n^{3}+n^{4}}$
2. $\lim_{n \to \infty} n^{2}\left(\frac{\sqrt{n^{2}+1}-\sqrt{n^{2}-1}}{n+1}\right)$
3. $\lim_{n \to \infty} (e-\sqrt[n]{n})^{n}$
4. $\lim_{n \to \infty} \left(\frac{n^{2}+4 n+1}{n^{2}+4 n+4}\right)^{n(n+1)}$

See Solution

Problem 16582

c) $\lim _{x \rightarrow 1} \frac{x^{3}-x}{x-1}$ ; d) $\lim _{x \rightarrow-2} \frac{x^{4}-16}{x+2}$ ; Untersuchen Sie $f(x)$ um $x_{0}$ : a) $f(x)=\frac{x^{2}-9}{2 x-6}, x_{0}=3$ ; b) $f(x)=\frac{x+1}{x}, x_{0}=0$ ; c) $f(x)=\frac{x+1}{x^{2}}, x_{0}=0$ ; Grenzwertberechnung: a) $\lim _{x \rightarrow 4} \frac{x^{2}-16}{x-4}$ ; b) $\lim _{x \rightarrow-1} \frac{x^{3}-x}{x+1}$ ; c) $\lim _{x \rightarrow 3} \frac{3-x}{2 x^{2}-6 x}$ ; d) $\lim _{x \rightarrow 2} \frac{x^{4}-16}{x-2}$ .

See Solution

Problem 16583

Find the derivatives:
1. For $z=w^{e}+e^{\pi}+\pi^{w}$ , find $\frac{d z}{d w}$ .
2. For $f(x)=e^{2}+\ln (6)$ , find $f^{\prime}(x)$ .
3. For $f(x)=\frac{\log _{9}(x)}{2}$ , find $f^{\prime}(x)$ .
4. For $g(s)=\frac{2^{s}}{\ln s}$ , find $g^{\prime}(s)$ .

See Solution

Problem 16584

Rewrite the functions $f(x)=e^{9x+8}$ and $g(x)=5^{-5x}$ in forms $C e^{kx}$ and $C \cdot b^x$ , then find their derivatives.

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Problem 16585

Find $f^{\prime}(x)$ for $f(x)=x^{3} \ln x$ and identify the critical points (separate with commas or write none).

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Problem 16586

Berechnen Sie die Grenzwerte: a) $\lim _{x \rightarrow 4} \frac{x^{2}-16}{x-4}$ , b) $\lim _{x \rightarrow-1} \frac{x^{3}-x}{x+1}$ , c) $\lim _{x \rightarrow 3} \frac{3-x}{2 x^{2}-6 x}$ , d) $\lim _{x \rightarrow 2} \frac{x^{4}-16}{x-2}$ .

See Solution

Problem 16587

Find the derivative of $y=\frac{\cos x}{1+\sin x}$ .

See Solution

Problem 16588

Bestimme die Tangentengleichung an den Graphen von $f$ für die Punkte: a) $P(1, 1)$ , b) $P(1,5)$ , c) $P(4, 16)$ , d) $P(1, 2)$ .

See Solution

Problem 16589

Find the $x$ -values where the function $f(x)=\frac{1}{6} x^{4}+4 x^{3}+36 x^{2}$ has an inflection point.

See Solution

Problem 16590

Find intervals where the function $f(x)=\frac{1}{6} x^{6}-x^{5}-5 x^{4}$ is concave up.

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Problem 16591

Find all $x$ -values for the function $f(x)=x^{3}-30 x^{2}$ where it has an inflection point.

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Problem 16592

Ein Metallstab kühlt von $95^{\circ} \mathrm{C}$ auf $5^{\circ} \mathrm{C}$ . Welche Funktion beschreibt den Abkühlungsprozess?

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Problem 16593

Find intervals where the function $f(x)=\frac{1}{6} x^{6}+x^{5}-5 x^{4}$ is concave up.

See Solution

Problem 16594

Find all $x$ -values where the function $f(x)=\frac{1}{4} x^{4}-x^{3}$ has an inflection point.

See Solution

Problem 16595

Find the intervals where the function $f(x)=\frac{2}{15} x^{6}-7 x^{4}$ is concave down.

See Solution

Problem 16596

Find the $x$ -values where the function $f(x)=\frac{1}{6} x^{4}+3 x^{3}$ has an inflection point.

See Solution

Problem 16597

Find all $x$ -values where the function $f(x)=\frac{1}{6} x^{4}+x^{3}-10 x^{2}$ has an inflection point.

See Solution

Problem 16598

Find intervals where the function $f(x)=\frac{1}{3} x^{6}+x^{5}$ is concave down.

See Solution

Problem 16599

Find all $x$ -values where the function $f(x)=\frac{3}{20} x^{5}-x^{4}$ has an inflection point.

See Solution

Problem 16600

Calculate the difference quotient for $g(x)=6 x^{2}-2 x+1$ .

See Solution

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Calculus

Problem 16501

Find F(x)=∫2x(t3+4t−4)dtF(x)=\int_{2}^{x}(t^{3}+4t-4) dtF(x)=∫2x​(t3+4t−4)dt and evaluate F(2)F(2)F(2), F(5)F(5)F(5), and F(8)F(8)F(8).

Problem 16502

Find ccc in [−π3,π3]\left[-\frac{\pi}{3}, \frac{\pi}{3}\right][−3π​,3π​] for f(x)=cos⁡xf(x)=\cos xf(x)=cosx satisfying f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a}f′(c)=b−af(b)−f(a)​.

Problem 16503

An object at 190∘F190^{\circ} \mathrm{F}190∘F is in water at 40∘F40^{\circ} \mathrm{F}40∘F. Find F(t)F(t)F(t) with cooling constant k=−0.6k=-0.6k=−0.6. F(t)= F(t)= F(t)=

Problem 16504

Find the derivative F′(x)F^{\prime}(x)F′(x) for F(x)=∫xx+2(6t+3)dtF(x)=\int_{x}^{x+2}(6 t+3) d tF(x)=∫xx+2​(6t+3)dt.

Problem 16505

Calculate the present value of a continuous income stream F(t)=30+7tF(t)=30+7tF(t)=30+7t over 20 years with a 1.9%1.9\%1.9% continuous interest rate.

Problem 16506

Find the derivative of the function y=∫7tan⁡(x)3t+t dty=\int_{7}^{\tan (x)} \sqrt{3 t+\sqrt{t}} \, dty=∫7tan(x)​3t+t​​dt. What is y′y'y′?

Problem 16507

Find the derivative F′(x)F^{\prime}(x)F′(x) where F(x)=∫0sin⁡(x)3tdtF(x)=\int_{0}^{\sin (x)} 3 \sqrt{t} dtF(x)=∫0sin(x)​3t​dt.

Problem 16508

Problem 16509

Find the derivative F′(x)F^{\prime}(x)F′(x) if F(x)=∫1x21t3dtF(x)=\int_{1}^{x^{2}} \frac{1}{t^{3}} dtF(x)=∫1x2​t31​dt.

Problem 16510

Problem 16511

Evaluate the integral ∫02(33x+2+2x2+4)dx\int_{0}^{2}\left(\frac{3}{3 x+2}+\frac{2}{x^{2}+4}\right) d x∫02​(3x+23​+x2+42​)dx.

Problem 16512

Problem 16513

Evaluate the integral: ∫sin⁡(48t)sec⁡2(cos⁡(48t))dt\int \sin (48 t) \sec ^{2}(\cos (48 t)) dt∫sin(48t)sec2(cos(48t))dt (use C for the constant).

Problem 16514

Evaluate the integral: ∫(t4+2t3+t2−t+5)e3tdt\int\left(t^{4}+2 t^{3}+t^{2}-t+5\right) e^{3 t} d t∫(t4+2t3+t2−t+5)e3tdt

Problem 16515

Evaluate the integral: ∫cos⁡(x)sin⁡6(x)dx\int \frac{\cos (x)}{\sin ^{6}(x)} dx∫sin6(x)cos(x)​dx and include the constant CCC.

Problem 16516

A radioactive sample starts with 618.8mg618.8 \mathrm{mg}618.8mg and has a half-life of 14 days. (a) Write the formula y=□eDy = \square e^{\mathbb{D}}y=□eD for amount yyy after ttt days. (b) Find the amount after 16 days, rounded to the nearest tenth in mg\mathrm{mg}mg.

Problem 16517

Problem 16518

Evaluate the integral using the substitution u=1/x9u=1/x^{9}u=1/x9. Find ∫sec⁡2(u)x10dx+C\int \frac{\sec^{2}(u)}{x^{10}} dx + C∫x10sec2(u)​dx+C.

Problem 16519

Evaluate the derivative y′y'y′ at the point P=(0,π2)P=(0, \frac{\pi}{2})P=(0,2π​) from the equation y′cos⁡(y+x+x2)−ysin⁡(y+x+x2)(y′+1+2x)=3x2y' \cos(y+x+x^2) - y \sin(y+x+x^2)(y'+1+2x) = 3x^2y′cos(y+x+x2)−ysin(y+x+x2)(y′+1+2x)=3x2.

Problem 16520

Find the area between the curves y=x2+3y=x^{2}+3y=x2+3 and y=6x−6y=6x-6y=6x−6 from x=−1x=-1x=−1 to x=2x=2x=2. Round to three decimal places.

Problem 16521

Find the area between the curves y=exy=e^{x}y=ex and y=−1xy=-\frac{1}{x}y=−x1​ from x=1x=1x=1 to x=2x=2x=2. Area = □\square□ square units.

Problem 16522

Find the area between y=e0.75xy=e^{0.75 x}y=e0.75x and y=−1xy=-\frac{1}{x}y=−x1​ from x=1x=1x=1 to x=2x=2x=2. The area is □\square□ square units.

Problem 16523

Find the instantaneous velocity using s=st2s=\frac{s}{t^{2}}s=t2s​ at t1=−2t_{1}=-2t1​=−2.

Problem 16524

Evaluate the limit: lim⁡n→+∞∑i=1nsin⁡(iπ/n)n\lim _{n \rightarrow+\infty} \sum_{i=1}^{n} \frac{\sin (i \pi / n)}{n}n→+∞lim​i=1∑n​nsin(iπ/n)​ using a definite integral on [0,1][0,1][0,1].

Problem 16525

Find the tangent line equation to cos⁡(x)+10y2=xy9+31\cos(x) + 10y^2 = xy^9 + 31cos(x)+10y2=xy9+31 at (0,3)(0, \sqrt{3})(0,3​).

Problem 16526

Problem 16527

Find the remaining amount QQQ of a \6milliongrantafter70days,given6 million grant after 70 days, given 6milliongrantafter70days,given\frac{dQ}{dt} \propto (100-t)^2$.

Problem 16528

Calculate the integral ∫sin⁡6θcos⁡3θ dθ\int \sin^{6} \theta \cos^{3} \theta \, d\theta∫sin6θcos3θdθ.

Problem 16529

Check if the series ∑n=2∞1nln⁡n\sum_{n=2}^{\infty} \frac{1}{n \sqrt{\ln n}}∑n=2∞​nlnn​1​ converges or diverges.

Problem 16530

Evaluate the integral: ∫1273x⋅x3dx\int_{1}^{27} \frac{3}{x \cdot \sqrt[3]{x}} dx∫127​x⋅3x​3​dx

Problem 16531

Kenny wants to withdraw \$33,000 annually for 25 years at 8.7% interest. How much must he have at age 65?

Problem 16532

Determine where f(x)f(x)f(x) is increasing, decreasing, and find local extrema for f(x)=−2x2−20x−21f(x)=-2 x^{2}-20 x-21f(x)=−2x2−20x−21.

Problem 16533

Determine where the function f(x)=−2x2−20x−20f(x) = -2x^2 - 20x - 20f(x)=−2x2−20x−20 is increasing, decreasing, and find local extrema.

Problem 16534

Given the sequence defined by a1=2a_{1}=2a1​=2 and an+1=12(an+6)a_{n+1}=\frac{1}{2}(a_{n}+6)an+1​=21​(an​+6), prove it's nondecreasing & bounded by 6, then find lim⁡n→∞an\lim_{n \to \infty} a_{n}limn→∞​an​.

Problem 16535

Evaluate the integral: ∫02πsin⁡2(θ)5+4cos⁡(θ)dθ\int_{0}^{2 \pi} \frac{\sin ^{2}(\theta)}{5+4 \cos (\theta)} d \theta∫02π​5+4cos(θ)sin2(θ)​dθ

Problem 16536

Find the absolute maximum and minimum of f(x)=2x3−6xf(x)=2 x^{3}-6 xf(x)=2x3−6x on the interval [0,3][0,3][0,3].

Problem 16537

Find the tangent line equation for y=f(x)y=f(x)y=f(x) at x=ex=ex=e, where f(x)=x−8ln⁡xf(x)=x-8 \ln xf(x)=x−8lnx.

Problem 16538

Find the tangent line equation to y=f(x)y=f(x)y=f(x) at x=ex=ex=e, where f(x)=x−8ln⁡xf(x)=x-8 \ln xf(x)=x−8lnx.

Problem 16539

Evaluate the integral ∫−12(−1x−3+2x+2)dx\int_{-1}^{2}\left(\frac{-1}{x-3}+\frac{2}{x+2}\right) d x∫−12​(x−3−1​+x+22​)dx.

Problem 16540

Find where the function f(x)=−3x2−12x−20f(x) = -3x^2 - 12x - 20f(x)=−3x2−12x−20 is increasing, decreasing, and its local extrema.

Problem 16541

Problem 16542

Determine where the function f(x)f(x)f(x) is increasing, decreasing, and find the local extrema for f(x)=−3x2−18x−27f(x)=-3 x^{2}-18 x-27f(x)=−3x2−18x−27.

Problem 16543

Find $F(x)=\int_{2}^{x}(t^{3}+4t-4) dt$ and evaluate $F(2)$ , $F(5)$ , and $F(8)$ .

Find $c$ in $\left[-\frac{\pi}{3}, \frac{\pi}{3}\right]$ for $f(x)=\cos x$ satisfying $f'(c) = \frac{f(b) - f(a)}{b - a}$ .

An object at $190^{\circ} \mathrm{F}$ is in water at $40^{\circ} \mathrm{F}$ . Find $F(t)$ with cooling constant $k=-0.6$ . $F(t)=$

Find the derivative $F^{\prime}(x)$ for $F(x)=\int_{x}^{x+2}(6 t+3) d t$ .

Calculate the present value of a continuous income stream $F(t)=30+7t$ over 20 years with a $1.9\%$ continuous interest rate.

Find the derivative of the function $y=\int_{7}^{\tan (x)} \sqrt{3 t+\sqrt{t}} \, dt$ . What is $y'$ ?

Find the derivative $F^{\prime}(x)$ where $F(x)=\int_{0}^{\sin (x)} 3 \sqrt{t} dt$ .

Find the derivative $F^{\prime}(x)$ if $F(x)=\int_{1}^{x^{2}} \frac{1}{t^{3}} dt$ .

Evaluate the integral $\int_{0}^{2}\left(\frac{3}{3 x+2}+\frac{2}{x^{2}+4}\right) d x$ .

Evaluate the integral: $\int \sin (48 t) \sec ^{2}(\cos (48 t)) dt$ (use C for the constant).

Evaluate the integral: $\int\left(t^{4}+2 t^{3}+t^{2}-t+5\right) e^{3 t} d t$

Evaluate the integral: $\int \frac{\cos (x)}{\sin ^{6}(x)} dx$ and include the constant $C$ .

A radioactive sample starts with $618.8 \mathrm{mg}$ and has a half-life of 14 days.
(a) Write the formula $y = \square e^{\mathbb{D}}$ for amount $y$ after $t$ days.
(b) Find the amount after 16 days, rounded to the nearest tenth in $\mathrm{mg}$ .

Evaluate the integral using the substitution $u=1/x^{9}$ . Find $\int \frac{\sec^{2}(u)}{x^{10}} dx + C$ .

Evaluate the derivative $y'$ at the point $P=(0, \frac{\pi}{2})$ from the equation $y' \cos(y+x+x^2) - y \sin(y+x+x^2)(y'+1+2x) = 3x^2$ .

Find the area between the curves $y=x^{2}+3$ and $y=6x-6$ from $x=-1$ to $x=2$ . Round to three decimal places.

Find the area between the curves $y=e^{x}$ and $y=-\frac{1}{x}$ from $x=1$ to $x=2$ . Area = $\square$ square units.

Find the area between $y=e^{0.75 x}$ and $y=-\frac{1}{x}$ from $x=1$ to $x=2$ . The area is $\square$ square units.

Find the instantaneous velocity using $s=\frac{s}{t^{2}}$ at $t_{1}=-2$ .

Evaluate the limit: $\lim _{n \rightarrow+\infty} \sum_{i=1}^{n} \frac{\sin (i \pi / n)}{n}$ using a definite integral on $[0,1]$ .

Find the tangent line equation to $\cos(x) + 10y^2 = xy^9 + 31$ at $(0, \sqrt{3})$ .

Find the remaining amount $Q$ of a \ $6 million grant after 70 days, given$ \frac{dQ}{dt} \propto (100-t)^2$.

Calculate the integral $\int \sin^{6} \theta \cos^{3} \theta \, d\theta$ .

Check if the series $\sum_{n=2}^{\infty} \frac{1}{n \sqrt{\ln n}}$ converges or diverges.

Evaluate the integral: $\int_{1}^{27} \frac{3}{x \cdot \sqrt[3]{x}} dx$

Determine where $f(x)$ is increasing, decreasing, and find local extrema for $f(x)=-2 x^{2}-20 x-21$ .

Determine where the function $f(x) = -2x^2 - 20x - 20$ is increasing, decreasing, and find local extrema.

Given the sequence defined by $a_{1}=2$ and $a_{n+1}=\frac{1}{2}(a_{n}+6)$ , prove it's nondecreasing & bounded by 6, then find $\lim_{n \to \infty} a_{n}$ .

Evaluate the integral: $\int_{0}^{2 \pi} \frac{\sin ^{2}(\theta)}{5+4 \cos (\theta)} d \theta$

Find the absolute maximum and minimum of $f(x)=2 x^{3}-6 x$ on the interval $[0,3]$ .

Find the tangent line equation for $y=f(x)$ at $x=e$ , where $f(x)=x-8 \ln x$ .

Find the tangent line equation to $y=f(x)$ at $x=e$ , where $f(x)=x-8 \ln x$ .

Evaluate the integral $\int_{-1}^{2}\left(\frac{-1}{x-3}+\frac{2}{x+2}\right) d x$ .

Find where the function $f(x) = -3x^2 - 12x - 20$ is increasing, decreasing, and its local extrema.

Determine where the function $f(x)$ is increasing, decreasing, and find the local extrema for $f(x)=-3 x^{2}-18 x-27$ .

Find where $f(x)$ is increasing, decreasing, and its local extrema for $f(x)=-3 x^{2}-18 x-20$ .

Evaluate the limit: $\lim _{x \rightarrow-1^{+}} \frac{5 \cos ^{-1}(x)-5 \pi}{14 \sin ^{-1}(x)+7 \pi}$ .

Find the dimensions of a Norman window with a perimeter of 90 feet that maximize its area. $(r, y)=(\square) \mathrm{ft}$

Find the critical points of the function $f(x)=x^{4} \ln x$ by determining where $f^{\prime}(x)=0$ .

Find the difference quotient for the function $f(x)=5 x^{2}+2 x-2$ : $\frac{f(x+h)-f(x)}{h}$ .

Maximize the area of a window with a semicircle and rectangle. Total height is 90 ft; width is $2r$ . Find $r$ and $y$ .

Find the maximum volume of a cylinder formed by rotating a rectangle around the $x$ -axis, bounded by $f(x)=\frac{x}{x^{2}+1}$ .

Rewrite $f(x)=e^{8 x+5}$ as $f(x)=C e^{k x}$ and $f(x)=C \cdot b^{x}$ . Find $f'(x)$ . Do the same for $g(x)=7^{-3 x}$ . Find $g'(x)$ .

Calculate the series sum: $\sum_{n=1}^{\infty}\left(\frac{5}{n(n+1)}-\frac{1}{2^{n}}\right)$ .

Find the area under the standard normal curve from $z=0$ to $z=0.89$ . Round to four decimal places.

Evaluate the integral and verify by differentiating: $\int \frac{9}{8+9 x} d x, x \neq -\frac{8}{9}$

Find the difference quotient, $\frac{f(x+h)-f(x)}{h}$ , for the function $f(x)=-3x-7$ .

Prove that if the series $\sum_{n \in \mathbb{N}} a_{n}$ and $\sum_{n \in \mathbb{N}} b_{n}$ converge, then $\sum_{n \in \mathbb{N}} a_{n} b_{n}$ converges.

Determine the local min and max of the polynomial $P(x)=\frac{4}{9} x^{3}-2 x^{2}$ .

In a town of 4000, the infected after $t$ days is $N(t)=\frac{4000}{1+15.1 e^{-0.6 t}}$ . Find $N(0)$ , $N(2)$ , $N(5)$ , $N(8)$ , $N(12)$ , $N(16)$ , and discuss if all 4000 will be infected.

Find the initial temperature and the temperature after 20 minutes of a soda cooling in a cooler, given $T(x)=-4+24 e^{-0.034 x}$ .

Find the population size $P(t)=\frac{800}{1+4 e^{-0.4 t}}$ after 6 and 8 years. Round to the nearest whole number.

Austin invested \ $92,000 at$ 5 \frac{7}{8} \% $daily, and Josiah at$ 5 \frac{3}{4} \%$ continuously. After 17 years, find the difference.

Invest \ $10,000 at$ 9\% $daily or$ 8.92\%$ continuously. Which gives more in 3 years?

Find the difference quotient for the function $f(x)=-3 x^{2}+x-8$ : $\frac{f(x+h)-f(x)}{h}$ .

Insect population $P(t)=500 e^{0.02 t}$ : (a) Find $P(0)$ . (b) Determine growth rate. (c) Find $P(10)$ . (d) When is $P=650$ ? (e) When does $P$ double?

Find the value of $\int_{3}^{3} f(x) d x$ given $\int_{0}^{1} f(x) d x=1$ , $\int_{0}^{2} f(x) d x=3$ , and $\int_{2}^{5} f(x) d x=2$ .

Find the mosquito population after 4 days if it grows from 1000 to 1900 in 1 day. Approximately $\square$ mosquitoes.

Find $g^{\prime}\left(\frac{5}{2}\right)$ given $g(x)-9 x \sin (g(x))=6 x^{2}-x-35$ and $g\left(\frac{5}{2}\right)=0$ .

Evaluate the integral: $\int_{-\infty}^{\infty} \frac{3 x^{2}}{(x^{2}+2 x+2)(x^{2}+1)^{2}} dx$

Temukan titik ekstrem dari $y=\frac{1}{3} x^{3}+8 x^{2}+60 x$ .