Calculus

Problem 13901

Find local maxima/minima and intervals of increase/decrease for $f^{\prime}(x)=x^{2}(x-4)(x+4)$ .

See Solution

Problem 13902

Evaluate the integrals: A. $\int_{5}^{1} 6x(8-x) dx$ and B. $\int_{1}^{5} x(8-x) dx$ . Provide exact answers.

See Solution

Problem 13903

For the function $q(x)=\frac{x}{x^{2}+49}$ , find critical points, classify them, determine increasing/decreasing intervals, find inflection points, and sketch the graph.

See Solution

Problem 13904

Given the function $f(x)=2 \cos ^{2}(x)-4 \sin (x)$ for $0 \leq x \leq 2 \pi$ , find:
(a) intervals of increase and decrease. (b) local min and max values. (c) inflection points and concavity intervals.

See Solution

Problem 13905

Find the limit: $\lim _{x \rightarrow 0} \frac{\tan (5 x)}{\sin (8 x)}$ using I'Hospital's Rule if needed.

See Solution

Problem 13906

Find the limit using I'Hospital's Rule if needed: $\lim _{x \rightarrow 4} \frac{x^{2}-2 x-8}{x-4}$

See Solution

Problem 13907

Find the tangent line equation to $y^{x}=x^{y}$ at $(2,4)$ . Use implicit differentiation with $\ln()$ .

See Solution

Problem 13908

Evaluate the integral $\int \frac{a x^{4}+b}{x^{3}-4 x^{2}+4 x} d x$ , with $a, b \neq 0$ .

See Solution

Problem 13909

Express $I=\int_{-\infty}^{5} \frac{1}{x^{2}(x-5)} d x$ as limits of proper integrals without evaluating.

See Solution

Problem 13910

Find critical points and use the First Derivative Test for $f(x)=2 x^{3}+3 x^{2}-120 x+7$ on $[-5,8]$ . Identify local and absolute extrema.

See Solution

Problem 13911

Find local extrema as ordered pairs for: a. $f(x)=x^{3}-6 x^{2}+5$ b. $g(x)=x \sqrt{9-x^{2}}$ c. $h(x)=x e^{-\frac{x^{2}}{9}}$ d. $j(x)=\ln(x^{4}+27)$ e. $k(x)=x \sqrt{6-x}$ f. $l(x)=\sqrt[3]{x^{2}-1}$

See Solution

Problem 13912

Evaluate the integral $I=\int_{-3}^{3} \frac{3 x+5}{(x+2)^{2}} d x$ and explain why it is improper.

See Solution

Problem 13913

Sketch the function $f(x)=12-3x$ on $[0,8]$ and find net area using left, right, and midpoint Riemann sums with $n=4$ .

See Solution

Problem 13914

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

See Solution

Problem 13915

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

See Solution

Problem 13916

Find $\lim _{x \rightarrow 1}(f(x) g(x))$ given $\lim _{x \rightarrow 1} f(x)=16$ and $\lim _{x \rightarrow 1} g(x)=5$ . State the limit law used.

See Solution

Problem 13917

Calculate the integral of the function: $\int \frac{4 x-10}{x^{2}-6 x+8} d x$ using partial fractions.

See Solution

Problem 13918

Calcula $\frac{\partial^{3} z}{\partial x^{3}}$ si $z=x^{2} y^{2}-y^{3}+3 x^{4}+5$ .

See Solution

Problem 13919

Differentiate the function $f(x) = x^{3} \log_{4}(3x + 5)$ .

See Solution

Problem 13920

Encuentra la derivada parcial $f_{x}$ de la función $f(x, y)=\cos \left(x y^{2}\right)$ .

See Solution

Problem 13921

Differentiate the function $u=\sqrt[3]{t^{2}}-2\sqrt{t^{3}}$ . Find $u'$ .

See Solution

Problem 13922

Find the limit using I'Hospital's Rule: $\lim _{x \rightarrow 0^{+}}\left(\frac{7}{x}-\frac{7}{\tan (x)}\right)$

See Solution

Problem 13923

Differentiate the function $f(x)=\frac{x \sqrt{x^{3}+3 x}}{1+x}$ using logarithmic differentiation.

See Solution

Problem 13924

Find the horizontal asymptote of the average cost function $\overline{\mathrm{C}}(x)$ for a bike company with given costs.

See Solution

Problem 13925

Calcula $\frac{\partial^{2} z}{\partial x^{2}}$ si $z=x^{2} y^{2}-y^{3}+3 x^{4}+5$ .

See Solution

Problem 13926

Given the function $f(x)=2 x^{3}-15 x^{2}+36 x-5$ , find its derivative $f'(x)$ , critical numbers, and intervals of increase/decrease. Also, determine local min/max values.

See Solution

Problem 13927

To find the volume of revolution, should you slice perpendicular or parallel to the axis to evaluate the radius of rotation?

See Solution

Problem 13928

Calculate the volume of the solid formed by rotating the triangle with vertices $(1,0)$ , $(2,1)$ , and $(1,1)$ around $x=1$ .

See Solution

Problem 13929

Find the volume of the solid formed by revolving the triangle with vertices $(1,0)$ , $(2,1)$ , $(1,1)$ around $x=1$ . Choose the correct integral:
1. $\int_{0}^{1} \pi(y+1)^{2} dy$
2. $\int_{0}^{1} \pi(x+1)^{2} dx$
3. $\int_{0}^{1} \pi y^{2} dy$
4. $\int_{0}^{1} \pi(x-1)^{2} dx$
5. $\int_{0}^{1} \pi x^{2} dx$

See Solution

Problem 13930

Find the average speed of an arrow shot upward at $69.2 \mathrm{~m/s}$ during the third second of its flight.

See Solution

Problem 13931

Find the time (in seconds) when fireworks, launched at $49.8 \mathrm{~m/s}$ , reach $76.0$ meters high.

See Solution

Problem 13932

Evaluate the integral from 1 to 16 of $(x-7)/\sqrt{x}$ dx.

See Solution

Problem 13933

Find the volume of the solid formed by rotating the triangle with vertices $(1,0),(2,1),(1,1)$ around $x=1$ .

See Solution

Problem 13934

Find the volume of the solid formed by rotating the triangle with vertices $(1,0),(2,1),(1,1)$ around $x=1$ .

See Solution

Problem 13935

Evaluate the integral from 1 to 4 of $\frac{1}{2x}$ with respect to $x$ .

See Solution

Problem 13936

Evaluate the integral: $\int_{0}^{\pi / 4} 9 \sec \theta \tan \theta \, d \theta$

See Solution

Problem 13937

Find the limit of $N(t)=\frac{600}{1+49 e^{-0.7 t}}$ as $t$ approaches infinity and determine the initial rumor starters.

See Solution

Problem 13938

Find $g^{\prime}(4)$ if $f(4)=1$ , $f^{\prime}(4)=-1$ , and $g(x)=\frac{f(x)}{x}$ .

See Solution

Problem 13939

Evaluate the integral from 0 to π/4: $\int_{0}^{\pi / 4} 3 \sec ^{2} t \, dt$

See Solution

Problem 13940

Find $g^{\prime}(3)$ if $f(3)=-2$ , $f^{\prime}(3)=4$ , and $g(x)=\frac{2x+1}{f(x)}$ .

See Solution

Problem 13941

Evaluate the integral from 0 to 5: $\int_{0}^{5}\left(5 e^{x}+8 \cos x\right) d x$ .

See Solution

Problem 13942

Evaluate the integral from -2 to 3: $\int_{-2}^{3} e^{u+2} d u$

See Solution

Problem 13943

Given functions $f(2)=-3$ , $g(2)=-5$ , $h(2)=-2$ , and derivatives $f'(2)=-5$ , $g'(2)=-5$ , $h'(2)=4$ , find:
1. $(3 f+4 h)'(2)$
2. $(2 f+5 g-h)'(2)$
3. $(f h)'(2)$
4. $\left(\frac{f}{g}\right)'(2)$

See Solution

Problem 13944

Find the sum of the series if it converges: $\sum_{n=1}^{\infty} \frac{3}{n(n+2)}$ .

See Solution

Problem 13945

Given $f'(2)=-6$ and $g'(2)=-1$ , find $h'(2)$ for each: (A) $h(x)=10f(x)$ , (B) $h(x)=-13g(x)$ , (C) $h(x)=6f(x)+3g(x)$ , (D) $h(x)=12g(x)-11f(x)$ , (E) $h(x)=2f(x)+11g(x)-12$ , (F) $h(x)=-12g(x)-5f(x)+12x$ .

See Solution

Problem 13946

A stone creates ripples in a lake at $10 \mathrm{~cm/s}$ . Find the area increase rate after (a) 1s, (b) 3s, (c) 5s.

See Solution

Problem 13947

Determine if the following geometric series converge. If they do, find the sum; if not, write "DNE".
(a) $\sum_{n=1}^{\infty} \frac{5^{n}}{4^{n}}=\mathrm{DNE}$ . (b) $\sum_{n=2}^{\infty} \frac{1}{3^{n}}=\frac{1}{6}$ . (c) $\sum_{n=0}^{\infty} \frac{3^{n}}{7^{2 n+1}}=$ . (d) $\sum_{n=5}^{\infty} \frac{4^{n}}{5^{n}}=$ . (e) $\sum_{n=1}^{\infty} \frac{9^{n}}{9^{n+4}}=$ . (f) $\sum_{n=1}^{\infty} \frac{4^{n}+3^{n}}{5^{n}}=$ .

See Solution

Problem 13948

Evaluate the integral from 0 to 1 of $\frac{10}{t^{2}+1}$ with respect to $t$ .

See Solution

Problem 13949

Evaluate the integral: $\int_{0}^{\pi / 3} \frac{9 \sin (\theta)+9 \sin (\theta) \tan ^{2}(\theta)}{\sec ^{2}(\theta)} d \theta$

See Solution

Problem 13950

Evaluate the integral: $\int_{1}^{2} \frac{v^{5}+3 v^{6}}{v^{4}} d v$

See Solution

Problem 13951

A company has a fixed cost of \ $200,000 and \$300 per bike. What is the average cost$ \overline{\mathrm{C}}(4000)$ and its horizontal asymptote?

See Solution

Problem 13952

Find the derivative of $h(x)=(2x-7)^3$ .

See Solution

Problem 13953

Find the derivative of $y=\sqrt[3]{1+4x}$ .

See Solution

Problem 13954

Calculate the limit: $\lim _{x \rightarrow+\infty} \frac{x^{2}-x}{1-|x|}$ .

See Solution

Problem 13955

Das Profil einer Skisprungschanze wird durch $h(x)=0,003(x-150)^{2}+4,5$ beschrieben.
a) Höhe bei $x=100$ ? b) Steigung beim Start und bei $x=70$ ?

See Solution

Problem 13956

Die Funktion $w(t)=10+\frac{5}{t+1}$ beschreibt die Wassermenge in einem Becken. Bestimmen Sie die Änderungsrate bei $t_{1}=1$ und $t_{2}=8$ und interpretieren Sie. Welche Witterungseinflüsse könnten vorgelegen haben?

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Problem 13957

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

See Solution

Problem 13958

Find the derivative of $y=(2x^{3}+5)^{4}$ .

See Solution

Problem 13959

Calculate the decay rate constant for Technetium-99, which has a half-life of $6.0 \mathrm{~h}$ . Options: $8.66 \mathrm{~h}^{-1}$ , $0.12 h^{-1}$ , $0.693 h^{-1}$ , $6.0 \mathrm{~h}^{-1}$ .

See Solution

Problem 13960

Find the derivative of $f(x)=\frac{3 x}{\sqrt{x^{2}+1}}$ . Choose from the options provided.

See Solution

Problem 13961

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

See Solution

Problem 13962

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

See Solution

Problem 13963

Calculate the integral $\int_{1}^{4} 2 x \, dx$ .

See Solution

Problem 13964

Find the limit as $x$ approaches 1 from the left: $\lim _{x \rightarrow 1^{-}} \frac{x^{2}-x}{1-|x|}$ .

See Solution

Problem 13965

Finde die Stammfunktionen für: 1. $f(x)=3x$ , 2. $F(x)=x^{3}$ , 3. $F(x)=3x^{2}$ , 4. $F(x)=2x^{3}$ , 5. $F(x)=1.5x^{2}$ , 6. $F(x)=2x^{1.5}$ .

See Solution

Problem 13966

Bestimmen Sie das Vorzeichen der Ableitungen von $f(x)=-x^{2}+5$ , $g(x)=(x-3)^{2}-1$ , $h(x)=x^{3}$ an den Stellen: a) $x_{0}=3$ , b) $x_{0}=-5$ , c) $x_{0}=100$ , d) $x_{0}=0$ .

See Solution

Problem 13967

Bestimme Verdopplungs- und Halbwertszeiten für: (1) $B_{0}=100$ , 20\% Zuwachs; (2) $B_{0}=50000$ , Faktor 0,95; (3) $B_{0}=810$ Mio.t, 0,7\% Rückgang. Erkläre, warum diese Zeiten unabhängig von $B_{0}$ sind.

See Solution

Problem 13968

Bestimmen Sie die 1. Ableitung von $f(x)=\frac{1}{2 x}$ . Welche der folgenden ist korrekt? $f^{\prime}(x)=\frac{1}{2 x^{2}}, -\frac{1}{x^{2}}, -\frac{1}{2 x^{2}}, \frac{1}{x^{2}}, \frac{1}{2}$ .

See Solution

Problem 13969

Calculer les limites: $\lim _{x \rightarrow 2^{+}} \frac{\sqrt[3]{x^{3}-2 x^{2}}}{x-2}$ et $\lim _{x \rightarrow 2^{-}} \frac{\frac{2}{\pi} \operatorname{Arctan}\left(\frac{1}{\sqrt{2-x}}\right)-1}{x-2}$ .

See Solution

Problem 13970

15. Verkehrszählung:
a) Bestimmen Sie die Uhrzeit mit der höchsten Verkehrsdichte aus $f(x)=\frac{1}{10^{10}} x^{5}-\frac{1}{130000} x^{3}+\frac{1}{6} x+6$ .
b) Berechnen Sie die Fahrzeuge zwischen 6 und 9 Uhr.
c) Ermitteln Sie den durchschnittlichen Fahrzeugstrom zwischen 6 und 8 Uhr.

See Solution

Problem 13971

A skier starts at $2.0 \mathrm{~m/s}$ from a height of $40 \mathrm{~m}$ . Find speed at $25 \mathrm{~m}$ : $v=17 \mathrm{~m/s}$ .

See Solution

Problem 13972

Find the derivative of $f(x)=8 \cdot \sqrt[4]{x^{3}}-\frac{2}{3 x^{2}}+4$ .

See Solution

Problem 13973

Find the growth rate $\frac{\mathrm{dL}}{\mathrm{dt}}$ at $t=10, 20, 25$ weeks for $L=-37.61+3.68t-6.26\times10^{-4}t^{3}$ .

See Solution

Problem 13974

Find points on the curve $y=x^{3}+4 x+4$ where the tangent is parallel to $5 x-y=-2$ . Are there multiple points?

See Solution

Problem 13975

(a) What does $\frac{d H}{d x}$ mean? (b) Prove $\frac{d H}{d x}$ is constant. Choose the correct interpretation of $\frac{d H}{d x}$ .

See Solution

Problem 13976

Find the derivative of $f(x)=\frac{x^{5}-4 x^{2}}{x^{5}-3 x}$ .

See Solution

Problem 13977

Find the local extremum of $f(x)=\frac{4}{3} x^{3}-2 x^{2}-120 x+6$ . What are the local minimum/minima?

See Solution

Problem 13978

Find the derivative of $g(x)=\frac{\csc(x)}{e x}$ .

See Solution

Problem 13979

Find the intervals where the function $f(x)=2 x^{3}-3 x^{2}-120 x+18$ is increasing or decreasing.

See Solution

Problem 13980

Find the local extremum of the function $f(x)=\frac{4}{3} x^{3}-2 x^{2}-8 x+10$ . What are the local minimum/minima?

See Solution

Problem 13981

Find the local extrema of the function $f(x)=\frac{4}{3} x^{3}-2 x^{2}-8 x+10$ . Identify local min/max and their locations.

See Solution

Problem 13982

Find the derivative of $f(x)=\sqrt[4]{(7-3x^{2})^{12}}$ .

See Solution

Problem 13983

Find the local extrema of $f(x)=\frac{4}{3} x^{3}-2 x^{2}-120 x+6$ . What are the local min/max values?

See Solution

Problem 13984

Find the year of maximum profit and the maximum profit from the function $f(x)=-29 x^{2}+812 x-3528$ . Derivative: $f^{\prime}(x)=\square$ .

See Solution

Problem 13985

Find the year with the local minimum stock price for $f(x)=5.3 x^{2}-121 x+764$ . First, find $f^{\prime}(x)$ . $f^{\prime}(x)=\square$

See Solution

Problem 13986

Find the training session length $t$ that maximizes the rating $R(t)=\frac{14 t}{t^{2}+49}$ .

See Solution

Problem 13987

Find the derivative of $f(x)=e^{3x^{2}-4x-5}$ .

See Solution

Problem 13988

Find the average rate of change of $f(x)$ from $x_{1}=-7$ to $x_{2}=-1$ . Round to the nearest hundredth. $f(x)=\sqrt{-8 x-5}$

See Solution

Problem 13989

Find the derivative of $f(x)=\log(1+x^{10})$ .

See Solution

Problem 13990

Find the water temperature $x$ (in °C) that maximizes salmon swimming upstream, given $S(x)=-x^{3}+3x^{2}+189x+5181$ for $3 \leq x \leq 19$ . Round to the nearest degree.

See Solution

Problem 13991

Find the derivative of $f(x)=\frac{\cot(x^{2})}{2x}$ .

See Solution

Problem 13992

Find the local minimum percentage of the function $f(x)=\frac{x^{2}+121}{2 x}$ for $1 \leq x \leq 12$ . Calculate $f^{\prime}(x)$ .

See Solution

Problem 13993

Find the local maximum number of salmon swimming upstream for $S(x)=-x^{3}+3 x^{2}+189 x+5181$ in the range $3 \leq x \leq 19$ . What is the water temperature in degrees Celsius?

See Solution

Problem 13994

Find the local minimum percentage of $f(x)=\frac{x^{2}+121}{2 x}$ for $1 \leq x \leq 12$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 13995

Find the tangent line to $f(x)=x^{3}-2 x^{2}+x$ at $x=1$ . What is its equation?

See Solution

Problem 13996

Find the limit of the sequence $a_{n}=\sqrt{n+1}-\sqrt{n}$ . If it doesn't exist, explain why.

See Solution

Problem 13997

Evaluate the integral $\int\left(\sqrt{x}-\frac{2}{\sqrt{x}}\right) d x$ .

See Solution

Problem 13998

If a stock's price is falling faster, are $P^{\prime}(t)$ and $P^{\prime \prime}(t)$ positive or negative? Explain.

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Problem 13999

Find the second derivative $f^{\prime \prime}(x)$ for $f(x)=-e^{x}$ and calculate $f^{\prime \prime}(0)$ , $f^{\prime \prime}(4)$ , and $f^{\prime \prime}(-2)$ .

See Solution

Problem 14000

Calculate the limit of the sequence $a_{n}=\frac{n(\sqrt{3 n}-7)^{2}}{(\sqrt{n+2}+2)^{4}}$ or explain why it doesn't exist.

See Solution

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Calculus

Problem 13901

Find local maxima/minima and intervals of increase/decrease for f′(x)=x2(x−4)(x+4)f^{\prime}(x)=x^{2}(x-4)(x+4)f′(x)=x2(x−4)(x+4).

Problem 13902

Evaluate the integrals: A. ∫516x(8−x)dx\int_{5}^{1} 6x(8-x) dx∫51​6x(8−x)dx and B. ∫15x(8−x)dx\int_{1}^{5} x(8-x) dx∫15​x(8−x)dx. Provide exact answers.

Problem 13903

For the function q(x)=xx2+49q(x)=\frac{x}{x^{2}+49}q(x)=x2+49x​, find critical points, classify them, determine increasing/decreasing intervals, find inflection points, and sketch the graph.

Problem 13904

Given the function f(x)=2cos⁡2(x)−4sin⁡(x)f(x)=2 \cos ^{2}(x)-4 \sin (x)f(x)=2cos2(x)−4sin(x) for 0≤x≤2π0 \leq x \leq 2 \pi0≤x≤2π, find: (a) intervals of increase and decrease. (b) local min and max values. (c) inflection points and concavity intervals.

Problem 13905

Find the limit: lim⁡x→0tan⁡(5x)sin⁡(8x)\lim _{x \rightarrow 0} \frac{\tan (5 x)}{\sin (8 x)}limx→0​sin(8x)tan(5x)​ using I'Hospital's Rule if needed.

Problem 13906

Find the limit using I'Hospital's Rule if needed: lim⁡x→4x2−2x−8x−4\lim _{x \rightarrow 4} \frac{x^{2}-2 x-8}{x-4}x→4lim​x−4x2−2x−8​

Problem 13907

Find the tangent line equation to yx=xyy^{x}=x^{y}yx=xy at (2,4)(2,4)(2,4). Use implicit differentiation with ln⁡()\ln()ln().

Problem 13908

Evaluate the integral ∫ax4+bx3−4x2+4xdx\int \frac{a x^{4}+b}{x^{3}-4 x^{2}+4 x} d x∫x3−4x2+4xax4+b​dx, with a,b≠0a, b \neq 0a,b=0.

Problem 13909

Express I=∫−∞51x2(x−5)dxI=\int_{-\infty}^{5} \frac{1}{x^{2}(x-5)} d xI=∫−∞5​x2(x−5)1​dx as limits of proper integrals without evaluating.

Problem 13910

Find critical points and use the First Derivative Test for f(x)=2x3+3x2−120x+7f(x)=2 x^{3}+3 x^{2}-120 x+7f(x)=2x3+3x2−120x+7 on [−5,8][-5,8][−5,8]. Identify local and absolute extrema.

Problem 13911

Problem 13912

Evaluate the integral I=∫−333x+5(x+2)2dxI=\int_{-3}^{3} \frac{3 x+5}{(x+2)^{2}} d xI=∫−33​(x+2)23x+5​dx and explain why it is improper.

Problem 13913

Sketch the function f(x)=12−3xf(x)=12-3xf(x)=12−3x on [0,8][0,8][0,8] and find net area using left, right, and midpoint Riemann sums with n=4n=4n=4.

Problem 13914

Check if the function f(x)={x2−9x−3 if x≠310 if x=3f(x)=\left\{\begin{array}{ll}\frac{x^{2}-9}{x-3} & \text { if } x \neq 3 \\ 10 & \text { if } x=3\end{array}\right.f(x)={x−3x2−9​10​ if x=3 if x=3​ is continuous at a=3a=3a=3.

Problem 13915

Sketch the graph of f(x)={x2+3,x≠10,x=1f(x)=\begin{cases} x^{2}+3, & x \neq 1 \\ 0, & x=1 \end{cases}f(x)={x2+3,0,​x=1x=1​. Conjecture f(1)f(1)f(1) and lim⁡x→1+f(x)\lim_{x \to 1^{+}} f(x)limx→1+​f(x).

Problem 13916

Find lim⁡x→1(f(x)g(x))\lim _{x \rightarrow 1}(f(x) g(x))limx→1​(f(x)g(x)) given lim⁡x→1f(x)=16\lim _{x \rightarrow 1} f(x)=16limx→1​f(x)=16 and lim⁡x→1g(x)=5\lim _{x \rightarrow 1} g(x)=5limx→1​g(x)=5. State the limit law used.

Problem 13917

Calculate the integral of the function: ∫4x−10x2−6x+8dx\int \frac{4 x-10}{x^{2}-6 x+8} d x∫x2−6x+84x−10​dx using partial fractions.

Problem 13918

Calcula ∂3z∂x3\frac{\partial^{3} z}{\partial x^{3}}∂x3∂3z​ si z=x2y2−y3+3x4+5z=x^{2} y^{2}-y^{3}+3 x^{4}+5z=x2y2−y3+3x4+5.

Problem 13919

Differentiate the function f(x)=x3log⁡4(3x+5)f(x) = x^{3} \log_{4}(3x + 5)f(x)=x3log4​(3x+5).

Problem 13920

Encuentra la derivada parcial fxf_{x}fx​ de la función f(x,y)=cos⁡(xy2)f(x, y)=\cos \left(x y^{2}\right)f(x,y)=cos(xy2).

Problem 13921

Differentiate the function u=t23−2t3u=\sqrt[3]{t^{2}}-2\sqrt{t^{3}}u=3t2​−2t3​. Find u′u'u′.

Problem 13922

Find the limit using I'Hospital's Rule: lim⁡x→0+(7x−7tan⁡(x))\lim _{x \rightarrow 0^{+}}\left(\frac{7}{x}-\frac{7}{\tan (x)}\right)x→0+lim​(x7​−tan(x)7​)

Problem 13923

Differentiate the function f(x)=xx3+3x1+xf(x)=\frac{x \sqrt{x^{3}+3 x}}{1+x}f(x)=1+xxx3+3x​​ using logarithmic differentiation.

Problem 13924

Find the horizontal asymptote of the average cost function C‾(x)\overline{\mathrm{C}}(x)C(x) for a bike company with given costs.

Problem 13925

Calcula ∂2z∂x2\frac{\partial^{2} z}{\partial x^{2}}∂x2∂2z​ si z=x2y2−y3+3x4+5z=x^{2} y^{2}-y^{3}+3 x^{4}+5z=x2y2−y3+3x4+5.

Problem 13926

Given the function f(x)=2x3−15x2+36x−5f(x)=2 x^{3}-15 x^{2}+36 x-5f(x)=2x3−15x2+36x−5, find its derivative f′(x)f'(x)f′(x), critical numbers, and intervals of increase/decrease. Also, determine local min/max values.

Problem 13927

To find the volume of revolution, should you slice perpendicular or parallel to the axis to evaluate the radius of rotation?

Problem 13928

Calculate the volume of the solid formed by rotating the triangle with vertices (1,0)(1,0)(1,0), (2,1)(2,1)(2,1), and (1,1)(1,1)(1,1) around x=1x=1x=1.

Problem 13929

Problem 13930

Find the average speed of an arrow shot upward at 69.2 m/s69.2 \mathrm{~m/s}69.2 m/s during the third second of its flight.

Problem 13931

Find the time (in seconds) when fireworks, launched at 49.8 m/s49.8 \mathrm{~m/s}49.8 m/s, reach 76.076.076.0 meters high.

Problem 13932

Evaluate the integral from 1 to 16 of (x−7)/x(x-7)/\sqrt{x}(x−7)/x​ dx.

Problem 13933

Find the volume of the solid formed by rotating the triangle with vertices (1,0),(2,1),(1,1)(1,0),(2,1),(1,1)(1,0),(2,1),(1,1) around x=1x=1x=1.

Problem 13934

Find the volume of the solid formed by rotating the triangle with vertices (1,0),(2,1),(1,1)(1,0),(2,1),(1,1)(1,0),(2,1),(1,1) around x=1x=1x=1.

Problem 13935

Evaluate the integral from 1 to 4 of 12x\frac{1}{2x}2x1​ with respect to xxx.

Problem 13936

Evaluate the integral: ∫0π/49sec⁡θtan⁡θ dθ\int_{0}^{\pi / 4} 9 \sec \theta \tan \theta \, d \theta∫0π/4​9secθtanθdθ

Problem 13937

Find the limit of N(t)=6001+49e−0.7tN(t)=\frac{600}{1+49 e^{-0.7 t}}N(t)=1+49e−0.7t600​ as ttt approaches infinity and determine the initial rumor starters.

Problem 13938

Find g′(4)g^{\prime}(4)g′(4) if f(4)=1f(4)=1f(4)=1, f′(4)=−1f^{\prime}(4)=-1f′(4)=−1, and g(x)=f(x)xg(x)=\frac{f(x)}{x}g(x)=xf(x)​.

Problem 13939

Evaluate the integral from 0 to π/4: ∫0π/43sec⁡2t dt\int_{0}^{\pi / 4} 3 \sec ^{2} t \, dt∫0π/4​3sec2tdt

Problem 13940

Find g′(3)g^{\prime}(3)g′(3) if f(3)=−2f(3)=-2f(3)=−2, f′(3)=4f^{\prime}(3)=4f′(3)=4, and g(x)=2x+1f(x)g(x)=\frac{2x+1}{f(x)}g(x)=f(x)2x+1​.

Problem 13941

Find local maxima/minima and intervals of increase/decrease for $f^{\prime}(x)=x^{2}(x-4)(x+4)$ .

Evaluate the integrals: A. $\int_{5}^{1} 6x(8-x) dx$ and B. $\int_{1}^{5} x(8-x) dx$ . Provide exact answers.

For the function $q(x)=\frac{x}{x^{2}+49}$ , find critical points, classify them, determine increasing/decreasing intervals, find inflection points, and sketch the graph.

Given the function $f(x)=2 \cos ^{2}(x)-4 \sin (x)$ for $0 \leq x \leq 2 \pi$ , find:
(a) intervals of increase and decrease. (b) local min and max values. (c) inflection points and concavity intervals.

Find the limit: $\lim _{x \rightarrow 0} \frac{\tan (5 x)}{\sin (8 x)}$ using I'Hospital's Rule if needed.

Find the limit using I'Hospital's Rule if needed: $\lim _{x \rightarrow 4} \frac{x^{2}-2 x-8}{x-4}$

Find the tangent line equation to $y^{x}=x^{y}$ at $(2,4)$ . Use implicit differentiation with $\ln()$ .

Evaluate the integral $\int \frac{a x^{4}+b}{x^{3}-4 x^{2}+4 x} d x$ , with $a, b \neq 0$ .

Express $I=\int_{-\infty}^{5} \frac{1}{x^{2}(x-5)} d x$ as limits of proper integrals without evaluating.

Find critical points and use the First Derivative Test for $f(x)=2 x^{3}+3 x^{2}-120 x+7$ on $[-5,8]$ . Identify local and absolute extrema.

Evaluate the integral $I=\int_{-3}^{3} \frac{3 x+5}{(x+2)^{2}} d x$ and explain why it is improper.

Sketch the function $f(x)=12-3x$ on $[0,8]$ and find net area using left, right, and midpoint Riemann sums with $n=4$ .

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

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

Find $\lim _{x \rightarrow 1}(f(x) g(x))$ given $\lim _{x \rightarrow 1} f(x)=16$ and $\lim _{x \rightarrow 1} g(x)=5$ . State the limit law used.

Calculate the integral of the function: $\int \frac{4 x-10}{x^{2}-6 x+8} d x$ using partial fractions.

Calcula $\frac{\partial^{3} z}{\partial x^{3}}$ si $z=x^{2} y^{2}-y^{3}+3 x^{4}+5$ .

Differentiate the function $f(x) = x^{3} \log_{4}(3x + 5)$ .

Encuentra la derivada parcial $f_{x}$ de la función $f(x, y)=\cos \left(x y^{2}\right)$ .

Differentiate the function $u=\sqrt[3]{t^{2}}-2\sqrt{t^{3}}$ . Find $u'$ .

Find the limit using I'Hospital's Rule: $\lim _{x \rightarrow 0^{+}}\left(\frac{7}{x}-\frac{7}{\tan (x)}\right)$

Differentiate the function $f(x)=\frac{x \sqrt{x^{3}+3 x}}{1+x}$ using logarithmic differentiation.

Find the horizontal asymptote of the average cost function $\overline{\mathrm{C}}(x)$ for a bike company with given costs.

Calcula $\frac{\partial^{2} z}{\partial x^{2}}$ si $z=x^{2} y^{2}-y^{3}+3 x^{4}+5$ .

Given the function $f(x)=2 x^{3}-15 x^{2}+36 x-5$ , find its derivative $f'(x)$ , critical numbers, and intervals of increase/decrease. Also, determine local min/max values.

Calculate the volume of the solid formed by rotating the triangle with vertices $(1,0)$ , $(2,1)$ , and $(1,1)$ around $x=1$ .

Find the average speed of an arrow shot upward at $69.2 \mathrm{~m/s}$ during the third second of its flight.

Find the time (in seconds) when fireworks, launched at $49.8 \mathrm{~m/s}$ , reach $76.0$ meters high.

Evaluate the integral from 1 to 16 of $(x-7)/\sqrt{x}$ dx.

Find the volume of the solid formed by rotating the triangle with vertices $(1,0),(2,1),(1,1)$ around $x=1$ .

Find the volume of the solid formed by rotating the triangle with vertices $(1,0),(2,1),(1,1)$ around $x=1$ .

Evaluate the integral from 1 to 4 of $\frac{1}{2x}$ with respect to $x$ .

Evaluate the integral: $\int_{0}^{\pi / 4} 9 \sec \theta \tan \theta \, d \theta$

Find the limit of $N(t)=\frac{600}{1+49 e^{-0.7 t}}$ as $t$ approaches infinity and determine the initial rumor starters.

Find $g^{\prime}(4)$ if $f(4)=1$ , $f^{\prime}(4)=-1$ , and $g(x)=\frac{f(x)}{x}$ .

Evaluate the integral from 0 to π/4: $\int_{0}^{\pi / 4} 3 \sec ^{2} t \, dt$

Find $g^{\prime}(3)$ if $f(3)=-2$ , $f^{\prime}(3)=4$ , and $g(x)=\frac{2x+1}{f(x)}$ .

Evaluate the integral from 0 to 5: $\int_{0}^{5}\left(5 e^{x}+8 \cos x\right) d x$ .

Evaluate the integral from -2 to 3: $\int_{-2}^{3} e^{u+2} d u$

Find the sum of the series if it converges: $\sum_{n=1}^{\infty} \frac{3}{n(n+2)}$ .

A stone creates ripples in a lake at $10 \mathrm{~cm/s}$ . Find the area increase rate after (a) 1s, (b) 3s, (c) 5s.

Evaluate the integral from 0 to 1 of $\frac{10}{t^{2}+1}$ with respect to $t$ .

Evaluate the integral: $\int_{0}^{\pi / 3} \frac{9 \sin (\theta)+9 \sin (\theta) \tan ^{2}(\theta)}{\sec ^{2}(\theta)} d \theta$

Evaluate the integral: $\int_{1}^{2} \frac{v^{5}+3 v^{6}}{v^{4}} d v$

A company has a fixed cost of \ $200,000 and \$300 per bike. What is the average cost$ \overline{\mathrm{C}}(4000)$ and its horizontal asymptote?

Find the derivative of $h(x)=(2x-7)^3$ .

Find the derivative of $y=\sqrt[3]{1+4x}$ .

Calculate the limit: $\lim _{x \rightarrow+\infty} \frac{x^{2}-x}{1-|x|}$ .

Das Profil einer Skisprungschanze wird durch $h(x)=0,003(x-150)^{2}+4,5$ beschrieben.
a) Höhe bei $x=100$ ? b) Steigung beim Start und bei $x=70$ ?

Die Funktion $w(t)=10+\frac{5}{t+1}$ beschreibt die Wassermenge in einem Becken. Bestimmen Sie die Änderungsrate bei $t_{1}=1$ und $t_{2}=8$ und interpretieren Sie. Welche Witterungseinflüsse könnten vorgelegen haben?

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

Find the derivative of $y=(2x^{3}+5)^{4}$ .

Calculate the decay rate constant for Technetium-99, which has a half-life of $6.0 \mathrm{~h}$ . Options: $8.66 \mathrm{~h}^{-1}$ , $0.12 h^{-1}$ , $0.693 h^{-1}$ , $6.0 \mathrm{~h}^{-1}$ .

Find the derivative of $f(x)=\frac{3 x}{\sqrt{x^{2}+1}}$ . Choose from the options provided.

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

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

Calculate the integral $\int_{1}^{4} 2 x \, dx$ .

Find the limit as $x$ approaches 1 from the left: $\lim _{x \rightarrow 1^{-}} \frac{x^{2}-x}{1-|x|}$ .

Finde die Stammfunktionen für: 1. $f(x)=3x$ , 2. $F(x)=x^{3}$ , 3. $F(x)=3x^{2}$ , 4. $F(x)=2x^{3}$ , 5. $F(x)=1.5x^{2}$ , 6. $F(x)=2x^{1.5}$ .

Bestimmen Sie das Vorzeichen der Ableitungen von $f(x)=-x^{2}+5$ , $g(x)=(x-3)^{2}-1$ , $h(x)=x^{3}$ an den Stellen: a) $x_{0}=3$ , b) $x_{0}=-5$ , c) $x_{0}=100$ , d) $x_{0}=0$ .

Bestimme Verdopplungs- und Halbwertszeiten für: (1) $B_{0}=100$ , 20\% Zuwachs; (2) $B_{0}=50000$ , Faktor 0,95; (3) $B_{0}=810$ Mio.t, 0,7\% Rückgang. Erkläre, warum diese Zeiten unabhängig von $B_{0}$ sind.

Bestimmen Sie die 1. Ableitung von $f(x)=\frac{1}{2 x}$ . Welche der folgenden ist korrekt? $f^{\prime}(x)=\frac{1}{2 x^{2}}, -\frac{1}{x^{2}}, -\frac{1}{2 x^{2}}, \frac{1}{x^{2}}, \frac{1}{2}$ .

A skier starts at $2.0 \mathrm{~m/s}$ from a height of $40 \mathrm{~m}$ . Find speed at $25 \mathrm{~m}$ : $v=17 \mathrm{~m/s}$ .

Find the derivative of $f(x)=8 \cdot \sqrt[4]{x^{3}}-\frac{2}{3 x^{2}}+4$ .

Find the growth rate $\frac{\mathrm{dL}}{\mathrm{dt}}$ at $t=10, 20, 25$ weeks for $L=-37.61+3.68t-6.26\times10^{-4}t^{3}$ .

Find points on the curve $y=x^{3}+4 x+4$ where the tangent is parallel to $5 x-y=-2$ . Are there multiple points?