Calculus

Problem 17901

Find the Elasticity of Demand for $D(p)=\sqrt{350-2p}$ at $p=\$161$ . Is it Unitary, Elastic, or Inelastic? What to do for revenue?

See Solution

Problem 17902

Find the general antiderivative of the piecewise function:
$f^{\prime}(x)=\begin{cases} 4 x^{2}+5, & x<4 \\ \frac{1}{9 x^{2}}, & x>4 \end{cases}$
with $f(x)=\begin{cases} +c_{1}, & x<4 \\ +c_{2}, & x>4 \end{cases}$

See Solution

Problem 17903

A circle's area grows at $5 \mathrm{~cm}^2/\mathrm{s}$ . Find the radius change rate when radius is $4 \mathrm{~cm}$ and circumference is $6 \mathrm{~cm}$ .

See Solution

Problem 17904

Prove that $G(x)=x \sin x+\cos x+C$ is the antiderivative of $g(x)=x \cos x$ . Show $\frac{d}{dx}(G(x))=x \cos x$ .

See Solution

Problem 17905

Approximate the integral $\int_{0}^{5}(-x^{2}+19x-3)dx$ using a Riemann Sum with 5 equal subintervals: (a) left endpoints, (b) right endpoints, (c) midpoints.

See Solution

Problem 17906

Given $f^{\prime \prime \prime}(x)=\cos (x)+\sin (x)$ with $f^{\prime \prime \prime}(0)=-1$ , $f^{\prime}(0)=-1$ , $f(0)=9$ , find:
(a) $f^{\prime \prime}(x)$ , (b) $f^{\prime}(x)$ , (c) $f(x)$ .

See Solution

Problem 17907

Find a function $f$ such that the limit represents the area under $f$ from $0$ to $1$ : $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{2 n}{n^{2}+i^{2}}; f(x)=\square$

See Solution

Problem 17908

Find a function $f$ such that the limit represents the area under $f$ from $0$ to $a$ : $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{2 n}{n^{2}+i^{2}} = 1$ , with $f(x)=\square$ .

See Solution

Problem 17909

For the function $f(x)=-2 x^{2}-4$ , find the average value on $[0,4]$ and graph it with a calculator. The average value is $\square$ .

See Solution

Problem 17910

Find the average value of $f(x)=-6 x^{2}-3$ on $[0,4]$ . Also, graph $f(x)$ and its average value on this interval.

See Solution

Problem 17911

Find the initial temperature and the temperature after 15 minutes for $T(x)=-8+26 e^{-0.03 x}$ . Round to the nearest degree.

See Solution

Problem 17912

Find the behavior of the sequence $a=\left(\frac{n(n+1)}{n^{2}+4}\right)$ as $n \to \infty$ and check if $\sum_{n=1}^{\infty} a_{n}$ converges or diverges.

See Solution

Problem 17913

Estimez $\tan(0.17 \pi)$ en utilisant la fonction $f(x)=\tan(\pi x)$ .
a) Calculez $f^{\prime}(1/6)$ .
b) Trouvez l'approximation linéaire $L(x)$ en $x=1/6$ .
c) Utilisez $L(x)$ pour estimer $\tan(0.17\pi)$ . Arrondissez à six décimales.

See Solution

Problem 17914

Estimez $\tan(0.17 \pi)$ avec l'approximation linéaire de $f(x)=\tan(\pi x)$ en $x=1/6$ .
a) Calculez $f'(1/6)$ .
b) Donnez $L(x)$ .
c) Estimez $\tan(0.17 \pi)$ avec $L(x)$ . Arrondissez à six décimales.

See Solution

Problem 17915

Find the limit: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{2 n}{n^{2}+i^{2}} ; a=1$

See Solution

Problem 17916

Trouvez le polynôme de Taylor d'ordre 4 pour $f(x)=\ln(x^{2})$ en calculant les dérivées jusqu'à l'ordre 4 et en les évaluant en $a=1$ .

See Solution

Problem 17917

Approximate the integral $\int_{0}^{5}(-x^{2}+19x-2)dx$ using Riemann Sums with 5 equal subintervals: (a) left endpoints, (b) right endpoints, (c) midpoints.

See Solution

Problem 17918

Researchers study an arctic ecosystem. Answer these:
1. What does $H^{\prime}(t)>0$ and $H^{\prime \prime}(t)<0$ indicate about snowshoe hares?
2. Given $L(t) \approx 35 t e^{-0.2 t}+200$ , find: (a) Year with max lemmings (2000-2022) and count. (b) Year with min lemmings (2000-2022) and count.

See Solution

Problem 17919

Find and list the critical numbers of the function $f(z)=\frac{7 z+7}{7 z^{2}+7 z+7}$ in increasing order. Use $\mathrm{N}$ if not needed.

See Solution

Problem 17920

Find the critical numbers of the function $f(x)=2x^{3}-3x^{2}-72x$ .

See Solution

Problem 17921

Find $\int_{-4}^{-1} f(x) d x$ and $\int_{-4}^{-1}(6 f(x)-5) d x$ given $\int_{-7}^{2} f(x) d x=6$ , $\int_{-7}^{-4} f(x) d x=5$ , and $\int_{-1}^{2} f(x) d x=1$ .

See Solution

Problem 17922

Determine the temperature change for a 0.2 mile elevation increase, using the given temperature change rate.

See Solution

Problem 17923

Find the average slope of the function $f(x)=2x^{3}-15x^{2}-36x+3$ on $[-4,9]$ and values of $c$ where $f'(c)=11$ .

See Solution

Problem 17924

A flu epidemic starts with 5 cases. Rate of growth is $r(t)=22 e^{0.04 t}$ . Find total cases $F(t)$ and $F(25)$ .

See Solution

Problem 17925

Which two series diverge according to the ratio test? Choose from the following:
1. $\sum_{n=1}^{\infty} \frac{(n^{2})!}{(2n)!}$
2. $\sum_{n=1}^{\infty} \frac{(n+1)^{2n}}{(n+1)!}$
3. $\frac{1}{3}+\frac{13}{3 \cdot 5}+\cdots$
4. $\frac{3}{1}+\frac{3.5}{1.8}+\cdots$
5. $\sum_{n=1}^{\infty} \frac{n!}{2^{n!}}$
6. $\sum_{n=1}^{\infty} \frac{(n!)^{2}}{(2n)!}$

See Solution

Problem 17926

Find the marginal cost function $C'(x)$ , compute $C'(100)$ , and determine the cost of the 101st pair of jeans.

See Solution

Problem 17927

Estimate when puffin population decreases fastest using $P(t) \approx \frac{-80 t^{2}}{t^{2}+192}+150$ . How many puffins then? For snowy owls, find linear model $L$ with 47 owls in 2023 and $S^{\prime}(23)=-4.5$ . When will it drop below 20? Will it drop below 20 sooner or later if decrease isn't linear? Explain.

See Solution

Problem 17928

Find the balance of \$ 2,500 invested at 12% interest, compounded continuously, after 3 years. Options: a) \$ 3,683.46 b) \$ 3,583.32 c) \$ 3,512.32 d) \$ 3,489.58

See Solution

Problem 17929

What is the balance after 10 years for an investment of \$ 5,500 at 9\% interest compounded continuously? Options: a.) \$ 12,298.02 b.) \$ 14,757.62 c.) \$ 13,527.82 d.) \$ 15,987.42

See Solution

Problem 17930

A rocket accelerates at $4.2 \mathrm{~m} / \mathrm{s}^2$ for $55 \mathrm{~s}$ . Find: a) max velocity, b) max height, c) time to hit ground.

See Solution

Problem 17931

Trouver la limite suivante :
$\lim _{x \rightarrow \infty}\left(x e^{4 / x}-x\right)$
a) Quelle est la forme indéterminée ?
b) Calculez la limite avec la règle de l'Hôpital.

See Solution

Problem 17932

Calculez la limite suivante : $\lim _{y \rightarrow \infty}\left(1+\frac{2}{y}\right)^{2 y}$ et identifiez sa forme indéterminée.

See Solution

Problem 17933

Find the critical numbers of the function $f(z)=\frac{7 z+7}{7 z^{2}+7 z+7}$ in increasing order. Use $\mathrm{N}$ for unused blanks.

See Solution

Problem 17934

Déterminez la forme indéterminée de la limite $\lim _{y \rightarrow \infty}\left(1+\frac{2}{y}\right)^{2 y}$ et évaluez-la.

See Solution

Problem 17935

Calculez la limite suivante : $\lim _{y \rightarrow \infty}\left(1+\frac{2}{y}\right)^{2 y}$ . Quelle est sa forme indéterminée ? Évaluez-la avec la règle de l'Hôpital.

See Solution

Problem 17936

Find fixed points of $2.5 N_{t}-\frac{N_{t}^{2}}{200}$ , check stability, and compute $\int 2.5 N_{t}-\frac{N_{t}^{2}}{200} dN_t$ .

See Solution

Problem 17937

Find the derivative of the function $f(x)=\ln(1+x)$ .

See Solution

Problem 17938

Find the area between the curve $f(x)=x^{3}-4x$ and the $x$ -axis from $x=-2$ to $x=3$ . Sketch the graph.

See Solution

Problem 17939

Evaluate $g^{\prime}(1)$ for $g(x)=(f(x))^{3}$ where $f$ is graphed on $[0,8]$ .

See Solution

Problem 17940

Find the derivative of $\ln(f(x))$ if $f$ is differentiable and $f(x) > 0$ .

See Solution

Problem 17941

If $f$ is differentiable and $f(x)>0$ , what is $\frac{d}{d x}(\ln (f(x)))$ ? Choose from: 1) $\frac{1}{f(x)}$ 2) $\frac{f^{\prime}(x)}{f(x)}$ 3) $\frac{f(x)}{f^{\prime}(x)}$

See Solution

Problem 17942

Which two series converge according to the root test? Choose from:
1. $\sum_{n=1}^{\infty}\left(\frac{2 \arctan n}{\pi}\right)^{n}$
2. $\sum_{n=1}^{\infty}\left(\frac{2 n}{2 n+1}\right)^{2 n}$
3. $\sum_{n=1}^{\infty}\left(\frac{n^{4} \ln ^{2}\left(\cos \frac{1}{n}\right)}{n+1}\right)^{n}$
4. $\sum_{n=1}^{\infty}\left(\frac{n+1}{2 n-1}\right)^{n}$
5. $\sum_{n=1}^{\infty}\left(\frac{n}{\ln n}\right)^{n / 2}$
6. $\sum_{n=1}^{\infty}\left(\frac{3 n}{3 n-1}\right)^{n}$

See Solution

Problem 17943

Which series converge by the root test? Pick 2 from:
1. $\sum_{n=1}^{\infty}\left(\frac{2 \arctan n}{\pi}\right)^{n}$
2. $\sum_{n=1}^{\infty}\left(\frac{2 n}{2 n+1}\right)^{2 n}$
3. $\sum_{n=1}^{\infty}\left(\frac{n^{4} \ln^{2}\left(\cos \frac{1}{n}\right)}{n+1}\right)^{n}$
4. $\sum_{n=1}^{\infty}\left(\frac{n+1}{2 n-1}\right)^{n}$
5. $\sum_{n=1}^{\infty}\left(\frac{n}{\ln n}\right)^{n/2}$
6. $\sum_{n=1}^{\infty}\left(\frac{3 n}{3 n-1}\right)^{n^{n}}$

See Solution

Problem 17944

Find the tangent line equation for $h(x)=f(3x)$ at $x=1$ , given $f'(1)=1$ and $f'(3)=-1$ .

See Solution

Problem 17945

Find the derivative of $k(x)=f(x-1) \cdot \sqrt{x^{2}+5}$ .

See Solution

Problem 17946

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{e^{x}-x-1}{\cos (x)-1}$ . State indeterminate forms and use L'Hospital's Rule if needed.

See Solution

Problem 17947

Two statements about the series $\sum_{n=1}^{\infty} \frac{(\ln n)^{2}}{n}$ are true. Which ones?

See Solution

Problem 17948

Find $k^{\prime}(2)$ for the function $k(x)=f(x-1) \cdot \sqrt{x^{2}+5}$ .

See Solution

Problem 17949

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

See Solution

Problem 17950

Given $h(x)=f(g(x))$ , find $h^{\prime}(1)$ using the values from the tables for $f$ and $g$ .

See Solution

Problem 17951

Find the limit as $x \to \infty$ for $(1+3x)^{\frac{1}{x}}$ .

See Solution

Problem 17952

Find the slope of the tangent line to $k(x)=\arctan (x)+4 x^{2}$ at $x=2$ .

See Solution

Problem 17953

Given $g(m(x))=x$ on $[3,9]$ , find $m'(4)$ using the tables of $g(x)$ and $g'(x)$ .

See Solution

Problem 17954

Find the power series and determine the interval of convergence for $f(x)=\frac{4}{2x+3}$ .

See Solution

Problem 17955

Given functions $f(x)$ and $g(x)$ with derivatives, find $n''(3)$ for $n(x)=f(x^2)$ if $f''(9)=2$ .

See Solution

Problem 17956

Find the limit as $x$ approaches infinity for $\left(\frac{x}{x+1}\right)^{x}$ .

See Solution

Problem 17957

Find the power series for $f(x)=\frac{x+a}{x^{2}+a^{2}}$ , where $a>0$ .

See Solution

Problem 17958

Find the smallest interval of convergence for the series that converges at $x=0$ and diverges at $x=7$ . Choices: a) $(-3,4)$ b) $(0,7)$ c) $(0,4)$ d) $(2,3]$ e) $(-3,7)$ f) $[-3,4)$ g) $[0,4)$ h) $[-3,7)$ .

See Solution

Problem 17959

Find the max and min of $f(x)=x^{3}-6 x^{2}-63 x+4$ on these intervals: (a) $[-4,0]$ , (b) $[-1,8]$ , (c) $[-4,8]$ .

See Solution

Problem 17960

Find the 1st, 2nd, and 3rd degree Taylor polynomials $T_1(x)$ , $T_2(x)$ , and $T_3(x)$ for $f(x)=e^x$ at 0.

See Solution

Problem 17961

Approximate the integral $\int_{0.2}^{6.2} f(t) dt$ using a right Riemann sum with 4 subintervals from the given data.

See Solution

Problem 17962

Find the coordinates of the steepest point on the curve $y=\frac{70}{1+6 e^{-2 t}}$ for $t \geq 0$ .
$t=$ $y=$

See Solution

Problem 17963

Calculate the length of the curve given by $y=\int_{1}^{x} \sqrt{t^{3}-1} dt$ for $4 \leq x \leq 9$ .

See Solution

Problem 17964

Find the limit: $\lim _{x \rightarrow 0^{+}}\left(\frac{1}{x}-\frac{1}{\tan x}\right)$ .

See Solution

Problem 17965

Find the total revenue function from $R'(x)=4x(x^2+25,000)^{-\frac{2}{3}}$ given $R(130)=\$48,396$ . Also, find $x$ for $R(x) \geq \$43,000$ .

See Solution

Problem 17966

Determine the radius and interval of convergence for the series $\sum_{n=0}^{\infty} \frac{(x-2)^{n}}{n^{2}+1}$ .

See Solution

Problem 17967

The Tojolobal Mayan community's land use is modeled by $P=\frac{1}{1+3 e^{-0.0275 t}}$ .
(a) Find $P$ in 1935. (b) What is the long-run prediction for $P$ ? (c) When was half the land in use? (d) When is land use increasing most rapidly?

See Solution

Problem 17968

Calculate the integral from 0 to 5 for the function $-x^{2}+16x-8$ .

See Solution

Problem 17969

Find the limit $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{7 n}{n^{2}+i^{2}}$ and determine the function $f(x)$ .

See Solution

Problem 17970

Find the limit of $f(x)=\frac{x^{3}-1}{x-1}$ as $x$ approaches 1.

See Solution

Problem 17971

Find the function $f(x)$ given $f^{\prime \prime \prime}(x)=\cos (x)+\sin (x)$ and conditions: $f^{\prime \prime}(0)=-1$ , $f^{\prime}(0)=-1$ , $f(0)=4$ .

See Solution

Problem 17972

Is the function $f(x)=\left\{\begin{array}{ll} \frac{x^{2}-9}{x-3} & \text{if } x \neq 3 \\ 10 & \text{if } x=3 \end{array}\right.$ continuous at $a=3$ ? Justify using the checklist.

See Solution

Problem 17973

Find the total revenue function from $R'(x)=4x(x^2+25000)^{-\frac{2}{3}}$ with $R(130)=\$38,396$ . Also, find $x$ for $R(x) \geq \$30,000$ .

See Solution

Problem 17974

Approximate the integral $\int_{0}^{5}(-x^{2}+16x-8)dx$ using a Riemann Sum with 5 equal subintervals and specified samples: (a) left endpoints, (b) right endpoints, (c) midpoints. Enter exact numbers for each.

See Solution

Problem 17975

Approximate the integral $\int_{0}^{5}(-x^{2}+16x-8)dx$ using 5 equal subintervals. Use left, right, and midpoint sample points.

See Solution

Problem 17976

Sketch the graph of $f(x) = \begin{cases} x^2 + 3 & x \neq 1 \\ 0 & x = 1 \end{cases}$ . Find $f(1)$ and the limits as $x \to 1^{-}$ , $x \to 1^{+}$ , and $x \to 1$ .

See Solution

Problem 17977

Find $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ for $f(x)=\frac{2 x+4}{3 x^{5}+4}$ and identify horizontal asymptotes.

See Solution

Problem 17978

Explain $\lim h(x)=K$ as $x \rightarrow a^{-}$ : A) from right, B) from left, C) either side approaches $K$ .

See Solution

Problem 17979

Check if the function $y=\frac{8 x-4}{x^{2}-12 x+32}$ is continuous at $a=4$ . Justify your answer using the checklist.

See Solution

Problem 17980

For $f(x)=19 x^{3}-x$ , find slopes of secant lines for intervals $[1,2]$ , $[1,1.5]$ , $[1,1.1]$ , $[1,1.01]$ , $[1,1.001]$ and conjecture tangent slope at $x=1$ .

See Solution

Problem 17981

Find the limit as x approaches infinity: $\lim _{x \rightarrow \infty} \frac{1+2 x+2 x^{2}}{x^{2}}$

See Solution

Problem 17982

Given the function $f(x)$ , show it's not continuous at 1, determine left/right continuity, and state intervals of continuity.
$f(x)=\left\{\begin{array}{ll} x^{2}+3 x & \text { if } x \geq 1 \\ 3 x & \text { if } x<1 \end{array}\right.$

See Solution

Problem 17983

Show why $\lim _{x \rightarrow 3} \frac{x^{2}-8 x+15}{x-3}=\lim _{x \rightarrow 3}(x-5)$ , then find $\lim _{x \rightarrow 3} \frac{x^{2}-8 x+15}{x-3}$ .

See Solution

Problem 17984

Check if the function $y=\frac{8 x-4}{x^{2}-12 x+32}$ is continuous at $a=4$ . Justify your answer.

See Solution

Problem 17985

Show why $\lim _{x \rightarrow 3} \frac{x^{2}-8 x+15}{x-3}=\lim _{x \rightarrow 3}(x-5)$ and find the limit value.

See Solution

Problem 17986

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{3 \sin 9 x}{7 x}$ using l'Hôpital's Rule.

See Solution

Problem 17987

Calculate the average rate of change for $f(x)=2(3)^{x}-4$ from $x=1$ to $x=4$ . Options: 31.2, 39, 52, 78.

See Solution

Problem 17988

Find the derivative of the function $s(t)=\ln(4 \tan^{-1}(t))$ .

See Solution

Problem 17989

Find the tangent line equation to $y=x \ln (x-7)$ at the point $(8,0)$ .

See Solution

Problem 17990

Rewrite $\int_{2}^{6}\left(5 x^{3}-6 x\right) d x$ as two separate integrals.

See Solution

Problem 17991

Find the average rate of change (velocity) of a bowling ball with height $h(t)=-16 t^{2}+62 t+317$ over $[1,2.6]$ . Round to two decimal places.

See Solution

Problem 17992

Sketch $f(x)=12-3x$ on [0,8]. Find net area using left, right, and midpoint Riemann sums with $n=4$ . Identify positive/negative intervals.

See Solution

Problem 17993

Find the derivative of $s(t)=\ln(e^{6t}+e^{-6t})$ .

See Solution

Problem 17994

A bacteria culture starts with 100 and grows to 300 in 3 hours. Find $P(t)$ , population after 7 hours, and time to reach 1290.

See Solution

Problem 17995

Find the maximum profit using $P=9-2x$ and $TC=x^{3}-3x^{2}+4x+1$ by analyzing stationary points.

See Solution

Problem 17996

A bacteria culture starts with 460 and grows proportionally. After 3 hours, there are 1380.
(a) Find the function $P(t)$ for population after $t$ hours.
(b) What is the population after 7 hours?
(c) When will it reach 1390? Give your answer to 2 decimal places.

See Solution

Problem 17997

Bestimmen Sie die erste Ableitung von $f(x)=3 x^{17}-5 x^{12}+4 x^{3}-6$ .

See Solution

Problem 17998

Find the derivative of $h(x) = x^{n} + \sqrt[3]{x^{3}}$ .

See Solution

Problem 17999

Calculate the derivative of the function $g(x)=\frac{1}{x}+\sqrt{x}$ .

See Solution

Problem 18000

Établir les équations du mouvement d'un proton dans un champ électrique avec $E=\frac{U}{d}$ , données $U=250\mathrm{~V}$ , $d=5,0\mathrm{~cm}$ , $v_{0}=70000\mathrm{~km/h}$ , $\alpha=5^{\circ}$ .

See Solution

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Calculus

Problem 17901

Find the Elasticity of Demand for D(p)=350−2pD(p)=\sqrt{350-2p}D(p)=350−2p​ at p=$161p=\$161p=$161. Is it Unitary, Elastic, or Inelastic? What to do for revenue?

Problem 17902

Problem 17903

A circle's area grows at 5 cm2/s5 \mathrm{~cm}^2/\mathrm{s}5 cm2/s. Find the radius change rate when radius is 4 cm4 \mathrm{~cm}4 cm and circumference is 6 cm6 \mathrm{~cm}6 cm.

Problem 17904

Prove that G(x)=xsin⁡x+cos⁡x+CG(x)=x \sin x+\cos x+CG(x)=xsinx+cosx+C is the antiderivative of g(x)=xcos⁡xg(x)=x \cos xg(x)=xcosx. Show ddx(G(x))=xcos⁡x\frac{d}{dx}(G(x))=x \cos xdxd​(G(x))=xcosx.

Problem 17905

Approximate the integral ∫05(−x2+19x−3)dx\int_{0}^{5}(-x^{2}+19x-3)dx∫05​(−x2+19x−3)dx using a Riemann Sum with 5 equal subintervals: (a) left endpoints, (b) right endpoints, (c) midpoints.

Problem 17906

Problem 17907

Find a function fff such that the limit represents the area under fff from 000 to 111: lim⁡n→∞∑i=1n2nn2+i2;f(x)=□\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{2 n}{n^{2}+i^{2}}; f(x)=\squaren→∞lim​i=1∑n​n2+i22n​;f(x)=□

Problem 17908

Find a function fff such that the limit represents the area under fff from 000 to aaa: lim⁡n→∞∑i=1n2nn2+i2=1\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{2 n}{n^{2}+i^{2}} = 1n→∞lim​i=1∑n​n2+i22n​=1, with f(x)=□f(x)=\squaref(x)=□.

Problem 17909

For the function f(x)=−2x2−4f(x)=-2 x^{2}-4f(x)=−2x2−4, find the average value on [0,4][0,4][0,4] and graph it with a calculator. The average value is □\square□.

Problem 17910

Find the average value of f(x)=−6x2−3f(x)=-6 x^{2}-3f(x)=−6x2−3 on [0,4][0,4][0,4]. Also, graph f(x)f(x)f(x) and its average value on this interval.

Problem 17911

Find the initial temperature and the temperature after 15 minutes for T(x)=−8+26e−0.03xT(x)=-8+26 e^{-0.03 x}T(x)=−8+26e−0.03x. Round to the nearest degree.

Problem 17912

Find the behavior of the sequence a=(n(n+1)n2+4)a=\left(\frac{n(n+1)}{n^{2}+4}\right)a=(n2+4n(n+1)​) as n→∞n \to \inftyn→∞ and check if ∑n=1∞an\sum_{n=1}^{\infty} a_{n}∑n=1∞​an​ converges or diverges.

Problem 17913

Problem 17914

Problem 17915

Find the limit: lim⁡n→∞∑i=1n2nn2+i2;a=1\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{2 n}{n^{2}+i^{2}} ; a=1limn→∞​∑i=1n​n2+i22n​;a=1

Problem 17916

Trouvez le polynôme de Taylor d'ordre 4 pour f(x)=ln⁡(x2)f(x)=\ln(x^{2})f(x)=ln(x2) en calculant les dérivées jusqu'à l'ordre 4 et en les évaluant en a=1a=1a=1.

Problem 17917

Approximate the integral ∫05(−x2+19x−2)dx\int_{0}^{5}(-x^{2}+19x-2)dx∫05​(−x2+19x−2)dx using Riemann Sums with 5 equal subintervals: (a) left endpoints, (b) right endpoints, (c) midpoints.

Problem 17918

Problem 17919

Find and list the critical numbers of the function f(z)=7z+77z2+7z+7f(z)=\frac{7 z+7}{7 z^{2}+7 z+7}f(z)=7z2+7z+77z+7​ in increasing order. Use N\mathrm{N}N if not needed.

Problem 17920

Find the critical numbers of the function f(x)=2x3−3x2−72xf(x)=2x^{3}-3x^{2}-72xf(x)=2x3−3x2−72x.

Problem 17921

Problem 17922

Determine the temperature change for a 0.2 mile elevation increase, using the given temperature change rate.

Problem 17923

Find the average slope of the function f(x)=2x3−15x2−36x+3f(x)=2x^{3}-15x^{2}-36x+3f(x)=2x3−15x2−36x+3 on [−4,9][-4,9][−4,9] and values of ccc where f′(c)=11f'(c)=11f′(c)=11.

Problem 17924

A flu epidemic starts with 5 cases. Rate of growth is r(t)=22e0.04tr(t)=22 e^{0.04 t}r(t)=22e0.04t. Find total cases F(t)F(t)F(t) and F(25)F(25)F(25).

Problem 17925

Problem 17926

Find the marginal cost function C′(x)C'(x)C′(x), compute C′(100)C'(100)C′(100), and determine the cost of the 101st pair of jeans.

Problem 17927

Problem 17928

Find the balance of \$ 2,500 invested at 12% interest, compounded continuously, after 3 years. Options: a) \$ 3,683.46 b) \$ 3,583.32 c) \$ 3,512.32 d) \$ 3,489.58

Problem 17929

What is the balance after 10 years for an investment of \$ 5,500 at 9\% interest compounded continuously? Options: a.) \$ 12,298.02 b.) \$ 14,757.62 c.) \$ 13,527.82 d.) \$ 15,987.42

Problem 17930

A rocket accelerates at 4.2 m/s24.2 \mathrm{~m} / \mathrm{s}^24.2 m/s2 for 55 s55 \mathrm{~s}55 s. Find: a) max velocity, b) max height, c) time to hit ground.

Problem 17931

Trouver la limite suivante : lim⁡x→∞(xe4/x−x) \lim _{x \rightarrow \infty}\left(x e^{4 / x}-x\right) x→∞lim​(xe4/x−x) a) Quelle est la forme indéterminée ? b) Calculez la limite avec la règle de l'Hôpital.

Problem 17932

Calculez la limite suivante : lim⁡y→∞(1+2y)2y\lim _{y \rightarrow \infty}\left(1+\frac{2}{y}\right)^{2 y}y→∞lim​(1+y2​)2y et identifiez sa forme indéterminée.

Problem 17933

Find the critical numbers of the function f(z)=7z+77z2+7z+7f(z)=\frac{7 z+7}{7 z^{2}+7 z+7}f(z)=7z2+7z+77z+7​ in increasing order. Use N\mathrm{N}N for unused blanks.

Problem 17934

Déterminez la forme indéterminée de la limite lim⁡y→∞(1+2y)2y\lim _{y \rightarrow \infty}\left(1+\frac{2}{y}\right)^{2 y}limy→∞​(1+y2​)2y et évaluez-la.

Problem 17935

Calculez la limite suivante : lim⁡y→∞(1+2y)2y\lim _{y \rightarrow \infty}\left(1+\frac{2}{y}\right)^{2 y}y→∞lim​(1+y2​)2y. Quelle est sa forme indéterminée ? Évaluez-la avec la règle de l'Hôpital.

Problem 17936

Find fixed points of 2.5Nt−Nt22002.5 N_{t}-\frac{N_{t}^{2}}{200}2.5Nt​−200Nt2​​, check stability, and compute ∫2.5Nt−Nt2200dNt\int 2.5 N_{t}-\frac{N_{t}^{2}}{200} dN_t∫2.5Nt​−200Nt2​​dNt​.

Problem 17937

Find the derivative of the function f(x)=ln⁡(1+x)f(x)=\ln(1+x)f(x)=ln(1+x).

Problem 17938

Find the area between the curve f(x)=x3−4xf(x)=x^{3}-4xf(x)=x3−4x and the xxx-axis from x=−2x=-2x=−2 to x=3x=3x=3. Sketch the graph.

Problem 17939

Evaluate g′(1)g^{\prime}(1)g′(1) for g(x)=(f(x))3g(x)=(f(x))^{3}g(x)=(f(x))3 where fff is graphed on [0,8][0,8][0,8].

Problem 17940

Find the derivative of ln⁡(f(x))\ln(f(x))ln(f(x)) if fff is differentiable and f(x)>0f(x) > 0f(x)>0.

Problem 17941

If fff is differentiable and f(x)>0f(x)>0f(x)>0, what is ddx(ln⁡(f(x)))\frac{d}{d x}(\ln (f(x)))dxd​(ln(f(x)))? Choose from: 1) 1f(x)\frac{1}{f(x)}f(x)1​ 2) f′(x)f(x)\frac{f^{\prime}(x)}{f(x)}f(x)f′(x)​ 3) f(x)f′(x)\frac{f(x)}{f^{\prime}(x)}f′(x)f(x)​

Problem 17942

Problem 17943

Problem 17944

Find the tangent line equation for h(x)=f(3x)h(x)=f(3x)h(x)=f(3x) at x=1x=1x=1, given f′(1)=1f'(1)=1f′(1)=1 and f′(3)=−1f'(3)=-1f′(3)=−1.

Problem 17945

Find the Elasticity of Demand for $D(p)=\sqrt{350-2p}$ at $p=\$161$ . Is it Unitary, Elastic, or Inelastic? What to do for revenue?

A circle's area grows at $5 \mathrm{~cm}^2/\mathrm{s}$ . Find the radius change rate when radius is $4 \mathrm{~cm}$ and circumference is $6 \mathrm{~cm}$ .

Prove that $G(x)=x \sin x+\cos x+C$ is the antiderivative of $g(x)=x \cos x$ . Show $\frac{d}{dx}(G(x))=x \cos x$ .

Approximate the integral $\int_{0}^{5}(-x^{2}+19x-3)dx$ using a Riemann Sum with 5 equal subintervals: (a) left endpoints, (b) right endpoints, (c) midpoints.

Find a function $f$ such that the limit represents the area under $f$ from $0$ to $1$ : $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{2 n}{n^{2}+i^{2}}; f(x)=\square$

Find a function $f$ such that the limit represents the area under $f$ from $0$ to $a$ : $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{2 n}{n^{2}+i^{2}} = 1$ , with $f(x)=\square$ .

For the function $f(x)=-2 x^{2}-4$ , find the average value on $[0,4]$ and graph it with a calculator. The average value is $\square$ .

Find the average value of $f(x)=-6 x^{2}-3$ on $[0,4]$ . Also, graph $f(x)$ and its average value on this interval.

Find the initial temperature and the temperature after 15 minutes for $T(x)=-8+26 e^{-0.03 x}$ . Round to the nearest degree.

Find the behavior of the sequence $a=\left(\frac{n(n+1)}{n^{2}+4}\right)$ as $n \to \infty$ and check if $\sum_{n=1}^{\infty} a_{n}$ converges or diverges.

Find the limit: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{2 n}{n^{2}+i^{2}} ; a=1$

Trouvez le polynôme de Taylor d'ordre 4 pour $f(x)=\ln(x^{2})$ en calculant les dérivées jusqu'à l'ordre 4 et en les évaluant en $a=1$ .

Approximate the integral $\int_{0}^{5}(-x^{2}+19x-2)dx$ using Riemann Sums with 5 equal subintervals: (a) left endpoints, (b) right endpoints, (c) midpoints.

Find and list the critical numbers of the function $f(z)=\frac{7 z+7}{7 z^{2}+7 z+7}$ in increasing order. Use $\mathrm{N}$ if not needed.

Find the critical numbers of the function $f(x)=2x^{3}-3x^{2}-72x$ .

Find the average slope of the function $f(x)=2x^{3}-15x^{2}-36x+3$ on $[-4,9]$ and values of $c$ where $f'(c)=11$ .

A flu epidemic starts with 5 cases. Rate of growth is $r(t)=22 e^{0.04 t}$ . Find total cases $F(t)$ and $F(25)$ .

Find the marginal cost function $C'(x)$ , compute $C'(100)$ , and determine the cost of the 101st pair of jeans.

A rocket accelerates at $4.2 \mathrm{~m} / \mathrm{s}^2$ for $55 \mathrm{~s}$ . Find: a) max velocity, b) max height, c) time to hit ground.

Trouver la limite suivante :
$\lim _{x \rightarrow \infty}\left(x e^{4 / x}-x\right)$
a) Quelle est la forme indéterminée ?
b) Calculez la limite avec la règle de l'Hôpital.

Calculez la limite suivante : $\lim _{y \rightarrow \infty}\left(1+\frac{2}{y}\right)^{2 y}$ et identifiez sa forme indéterminée.

Find the critical numbers of the function $f(z)=\frac{7 z+7}{7 z^{2}+7 z+7}$ in increasing order. Use $\mathrm{N}$ for unused blanks.

Déterminez la forme indéterminée de la limite $\lim _{y \rightarrow \infty}\left(1+\frac{2}{y}\right)^{2 y}$ et évaluez-la.

Calculez la limite suivante : $\lim _{y \rightarrow \infty}\left(1+\frac{2}{y}\right)^{2 y}$ . Quelle est sa forme indéterminée ? Évaluez-la avec la règle de l'Hôpital.

Find fixed points of $2.5 N_{t}-\frac{N_{t}^{2}}{200}$ , check stability, and compute $\int 2.5 N_{t}-\frac{N_{t}^{2}}{200} dN_t$ .

Find the derivative of the function $f(x)=\ln(1+x)$ .

Find the area between the curve $f(x)=x^{3}-4x$ and the $x$ -axis from $x=-2$ to $x=3$ . Sketch the graph.

Evaluate $g^{\prime}(1)$ for $g(x)=(f(x))^{3}$ where $f$ is graphed on $[0,8]$ .

Find the derivative of $\ln(f(x))$ if $f$ is differentiable and $f(x) > 0$ .

If $f$ is differentiable and $f(x)>0$ , what is $\frac{d}{d x}(\ln (f(x)))$ ? Choose from: 1) $\frac{1}{f(x)}$ 2) $\frac{f^{\prime}(x)}{f(x)}$ 3) $\frac{f(x)}{f^{\prime}(x)}$

Find the tangent line equation for $h(x)=f(3x)$ at $x=1$ , given $f'(1)=1$ and $f'(3)=-1$ .

Find the derivative of $k(x)=f(x-1) \cdot \sqrt{x^{2}+5}$ .

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{e^{x}-x-1}{\cos (x)-1}$ . State indeterminate forms and use L'Hospital's Rule if needed.

Two statements about the series $\sum_{n=1}^{\infty} \frac{(\ln n)^{2}}{n}$ are true. Which ones?

Find $k^{\prime}(2)$ for the function $k(x)=f(x-1) \cdot \sqrt{x^{2}+5}$ .

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

Given $h(x)=f(g(x))$ , find $h^{\prime}(1)$ using the values from the tables for $f$ and $g$ .

Find the limit as $x \to \infty$ for $(1+3x)^{\frac{1}{x}}$ .

Find the slope of the tangent line to $k(x)=\arctan (x)+4 x^{2}$ at $x=2$ .

Given $g(m(x))=x$ on $[3,9]$ , find $m'(4)$ using the tables of $g(x)$ and $g'(x)$ .

Find the power series and determine the interval of convergence for $f(x)=\frac{4}{2x+3}$ .

Given functions $f(x)$ and $g(x)$ with derivatives, find $n''(3)$ for $n(x)=f(x^2)$ if $f''(9)=2$ .

Find the limit as $x$ approaches infinity for $\left(\frac{x}{x+1}\right)^{x}$ .

Find the power series for $f(x)=\frac{x+a}{x^{2}+a^{2}}$ , where $a>0$ .

Find the max and min of $f(x)=x^{3}-6 x^{2}-63 x+4$ on these intervals: (a) $[-4,0]$ , (b) $[-1,8]$ , (c) $[-4,8]$ .

Find the 1st, 2nd, and 3rd degree Taylor polynomials $T_1(x)$ , $T_2(x)$ , and $T_3(x)$ for $f(x)=e^x$ at 0.

Approximate the integral $\int_{0.2}^{6.2} f(t) dt$ using a right Riemann sum with 4 subintervals from the given data.

Find the coordinates of the steepest point on the curve $y=\frac{70}{1+6 e^{-2 t}}$ for $t \geq 0$ .
$t=$ $y=$

Calculate the length of the curve given by $y=\int_{1}^{x} \sqrt{t^{3}-1} dt$ for $4 \leq x \leq 9$ .

Find the limit: $\lim _{x \rightarrow 0^{+}}\left(\frac{1}{x}-\frac{1}{\tan x}\right)$ .

Find the total revenue function from $R'(x)=4x(x^2+25,000)^{-\frac{2}{3}}$ given $R(130)=\$48,396$ . Also, find $x$ for $R(x) \geq \$43,000$ .

Determine the radius and interval of convergence for the series $\sum_{n=0}^{\infty} \frac{(x-2)^{n}}{n^{2}+1}$ .

The Tojolobal Mayan community's land use is modeled by $P=\frac{1}{1+3 e^{-0.0275 t}}$ .
(a) Find $P$ in 1935. (b) What is the long-run prediction for $P$ ? (c) When was half the land in use? (d) When is land use increasing most rapidly?

Calculate the integral from 0 to 5 for the function $-x^{2}+16x-8$ .

Find the limit $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{7 n}{n^{2}+i^{2}}$ and determine the function $f(x)$ .

Find the limit of $f(x)=\frac{x^{3}-1}{x-1}$ as $x$ approaches 1.

Find the function $f(x)$ given $f^{\prime \prime \prime}(x)=\cos (x)+\sin (x)$ and conditions: $f^{\prime \prime}(0)=-1$ , $f^{\prime}(0)=-1$ , $f(0)=4$ .

Is the function $f(x)=\left\{\begin{array}{ll} \frac{x^{2}-9}{x-3} & \text{if } x \neq 3 \\ 10 & \text{if } x=3 \end{array}\right.$ continuous at $a=3$ ? Justify using the checklist.

Find the total revenue function from $R'(x)=4x(x^2+25000)^{-\frac{2}{3}}$ with $R(130)=\$38,396$ . Also, find $x$ for $R(x) \geq \$30,000$ .

Approximate the integral $\int_{0}^{5}(-x^{2}+16x-8)dx$ using a Riemann Sum with 5 equal subintervals and specified samples: (a) left endpoints, (b) right endpoints, (c) midpoints. Enter exact numbers for each.