Calculus

Problem 19901

Berechne den Flächeninhalt A zwischen $f(x)=4-x^{2}$ und $g(x)=\frac{1}{2} x+4$ im Intervall $[1 ; 2]$ .

See Solution

Problem 19902

Find the function $y$ for $x>0$ that satisfies the differential equation $y'=\frac{y^2 \ln (x)}{x y^2 + x y}$ with $y(1)=1$ .

See Solution

Problem 19903

Solve the equation $x^{3} y^{\prime}=x^{3}-3 x^{2} y$ with the condition $y(1)=1$ .

See Solution

Problem 19904

Solve the differential equation: $(2x + 4y^2)dx + (8xy + 4y^2)dy = 0$ .

See Solution

Problem 19905

Solve the DE $x^{3} y' = x^{3} - 3 x^{2} y$ with initial condition $y(1) = 1$ .

See Solution

Problem 19906

Ein Körper wird mit $6 \mathrm{~m/s}$ aus $140 \mathrm{~m}$ Höhe bei $30^{\circ}$ abgeworfen. Finde die Zeit $T$ bis zum Aufprall.

See Solution

Problem 19907

Find $F^{\prime}(\pi)$ if $F(x)=-\int_{0}^{x} \cos (t) dt$ .

See Solution

Problem 19908

Calculate the integral $\int_{0}^{4} \frac{3 \sqrt{x}}{4} d x$ . What is its value?

See Solution

Problem 19909

Find the value of $\int_{0}^{6} f(x) dx$ for the piecewise function $f(x) = -2x$ if $x<1$ and $f(x) = 3$ if $x \geq 1$ .

See Solution

Problem 19910

Find $F(\pi)$ where $F(x)=\int_{\pi}^{x} t^{4} \sin^{2}(t) dt$ .

See Solution

Problem 19911

If $-\int(\cos (\theta))^{2} \sin (\theta) d \theta=\int u^{2} d u$ , find $u$ . Choices: a. $\sin (\theta)$ , b. $\cos (t)$ , c. $-\cos (\theta)$ , d. $-\cos (t)$ , e. $\cos (\theta)$ , f. $\sin (\pi)$ .

See Solution

Problem 19912

Find the area between $f(x)=x^{e}+e^{x}$ and the $x$ -axis for $x \in[0,1]$ . What is it equal to?

See Solution

Problem 19913

Find the limit: $\lim _{t \rightarrow \infty}(t-7)(t-8)$ . Choose one: a. -8 b. limit does not exist. c. 7 d. 8 e. 0

See Solution

Problem 19914

Find the value of $\lim _{x \rightarrow \infty} \frac{5 \ln (x)}{4 x}$ . Choices: a. DNE, b. $5/4$ , c. -1, d. $4/5$ , e. 0, f. 1.

See Solution

Problem 19915

If $\int 3 t^{2} \cos(t^{3}) dt = \int \cos(u) du$ , find $u$ .
Options: a. $t$ b. $t^{3}$ c. $3 t^{2}$ d. $\cos(t)$ e. $3 t$ f. $\sin(t)$

See Solution

Problem 19916

Bewege einen Körper mit $v(t)=40-t^{2}$ in $\mathrm{m}/\mathrm{s}$ . a) Weg in 4s als Integral. b) Berechne Weg mit Summen.

See Solution

Problem 19917

Berechnen Sie die Reihenwerte für $x>0$ : (a) $\sum_{n=3}^{\infty}\left(\frac{1}{3}\right)^{\frac{n}{3}}$ , (b) $\sum_{k=0}^{\infty}\left(\frac{x-2}{x+2}\right)^{k}$ , (c) $\sum_{k=0}^{\infty} \frac{1}{\left((-1)^{k}+4\right)^{k}}$ .

See Solution

Problem 19918

Ein Radfahrer fährt bergauf mit der Geschwindigkeit $v(t)=-\frac{t^{3}}{1350000}+\frac{7 t^{2}}{150000}-\frac{3 t}{625}+\frac{1142}{125}$ . a) Geschwindigkeit in $\mathrm{km/h}$ zu Beginn? b) Dauer der Bergfahrt? Berechne die Strecke durch ein Integral.

See Solution

Problem 19919

Berechnen Sie die Reihenwerte für $x>0$ : (a) $\sum_{n=3}^{\infty}\left(\frac{1}{3}\right)^{\frac{n}{3}}$ , (b) $\sum_{k=0}^{\infty}\left(\frac{x-2}{x+2}\right)^{k}$ , (c) $\sum_{k=0}^{\infty} \frac{1}{\left((-1)^{k}+4\right)^{k}}$

See Solution

Problem 19920

Bestimmen Sie die Grenzwerte der Folgen: (a) $\sqrt[n]{3^{n}+4^{n}}$ , (b) $\frac{n^{2}+\cos(n)}{n^{2}-\sin(n)}$ , (c) $\left(1-\frac{1}{n^{2}-2n+5}\right)^{n}$ , (d) $\sum_{k=1}^{n} \frac{k}{2n^{2}+4k}$ .

See Solution

Problem 19921

Find $h^{\prime}(2)$ for $h(x)=f(g(x))$ given $g(2)=3$ , $g^{\prime}(2)=5$ , and $f^{\prime}(3)=-1$ .

See Solution

Problem 19922

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{\frac{-3}{\sqrt{x+\Delta x}}+\frac{3}{\sqrt{x}}}{\Delta x}$ . Options: A) -3 B) $\frac{-3}{x}$ C) $\frac{3}{2 \sqrt{x}}$ D) $\frac{3}{2 x \sqrt{x}}$

See Solution

Problem 19923

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{f(3+\Delta x)-f(3)}{\Delta x}$ for $f(x)=\frac{x+1}{x-1}$ . Options: A) $-\frac{1}{2}$ B) $\frac{1}{2}$ C) $-\frac{1}{4}$ D) $\frac{1}{4}$

See Solution

Problem 19924

Find the hole in $g(x)=\frac{3 x^{2}-8 x-3}{x^{2}-9}$ , its domain, and calculate these limits:
1. $\lim _{x \rightarrow-\infty} g(x)$
2. $\lim _{x \rightarrow \infty} g(x)$
3. $\lim _{x \rightarrow-3^{-}} g(x)$
4. $\lim _{x \rightarrow-3^{+}} g(x)$
5. $\lim _{x \rightarrow 3^{-}} g(x)$
6. $\lim _{x \rightarrow 3^{+}} g(x)$

See Solution

Problem 19925

Determine where $f(x)=\frac{1}{x^{2}+7}$ is concave up/down and find points of inflection. Select correct answers.

See Solution

Problem 19926

Vergleichen Sie die Differenzenquotienten der Intervalle $[-100, -1]$ , $[-10, -1]$ , $[-1, 1]$ und $[-1.01, -1]$ .

See Solution

Problem 19927

Gegeben ist die Funktion $f_{a}(x)=2 a x^{3}+(2-4 a) x$ .
a) Zeigen Sie, dass alle Graphen von $f_{a}$ durch den Hochpunkt von $f_{-0,25}$ gehen. b) Bestimmen Sie $a$ , so dass die Steigung von $f_{a}$ am Hochpunkt 6 ist. c) Gibt es ein $a$ , sodass $f_{a}$ keine lokalen Extrema hat? d) Finde die Gleichung der Wendenormale von $f_{a}$ und den Parameter $a$ für die Steigung 0,5.

See Solution

Problem 19928

Find the general antiderivative of $f(u)=\frac{-10 u^{4}-1 \sqrt{u}}{u^{2}}$ . Use upper-case "C" for constants. $F(u)=$

See Solution

Problem 19929

Find the satellite's orbital period in seconds given Earth's gravity is $6 \, m/s^{2}$ and radius is $6,371,000 \, m$ .

See Solution

Problem 19930

Find the antiderivative $F$ of $f(x)=4-3(1+x^{2})^{-1}$ with $F(1)=-1$ .

See Solution

Problem 19931

Given the function $f(x)=\frac{1}{x^{2}+8 x+20}$ , find:
a) The domain in interval notation. b) The critical numbers of $f$ . c) Intervals where $f$ is increasing and decreasing. d) Use the First Derivative Test for relative maxima and minima.

See Solution

Problem 19932

Find the general antiderivative of $f(u)=\frac{-3 u^{4}-4 \sqrt{u}}{u^{2}}$ . Use upper-case " $\mathrm{C}$ " for constants.

See Solution

Problem 19933

Find the derivative of $f(x)=\frac{x^{2}-2 x}{x^{2}+5 x}$ using the quotient rule.

See Solution

Problem 19934

Find the function $g(x)$ such that $g'(x) = -4096 - x^3$ and the maximum value of $g(x)$ is $-5$ .

See Solution

Problem 19935

Find the function $f(x)$ given that $f^{\prime \prime}(x)=3 x+6 \sin (x)$ , $f(0)=4$ , and $f^{\prime}(0)=3$ .

See Solution

Problem 19936

Simplify the expression: $\frac{d}{d x} \int_{4}^{x}(7 t^{2}+t+9) d t = \square$

See Solution

Problem 19937

Find the limit: $\lim _{x \rightarrow-1} \frac{5-5}{4 x^{2}+4 x}$ .

See Solution

Problem 19938

Approximate local and global minima and maxima for $f(x)=2 x^{3}-5 x-3$ . Round to two decimal places.

See Solution

Problem 19939

Bestimmen Sie die Tangentengleichung für die Punkte und Ableitungen: a) $B(2 \mid 4), f^{\prime}(2)=1$ b) $B(3 \mid 1), f^{\prime}(3)=2$ c) $B(0 \mid -2), f^{\prime}(0)=\frac{1}{2}$ d) $B(-2 \mid -3), f^{\prime}(-2)=\frac{3}{2}$ e) $B(-4 \mid 6), f^{\prime}(-4)=-2$ f) $B\left(-\frac{1}{2} \mid 3\right), f^{\prime}\left(-\frac{1}{2}\right)=0$

See Solution

Problem 19940

Bestimmen Sie die Tangentengleichungen für die Funktionen an den Punkten: a) $f(x)=x^{2}, b=2$ ; b) $f(x)=2 x^{2}-3 x, b=1$ ; c) $f(x)=\frac{8}{3} x^{3}, b=\frac{1}{2}$ ; d) $f(x)=x^{3}-2 x^{2}, b=2$ ; e) $f(x)=3 x^{4}+4 x, b=-1$ ; f) $f(x)=\frac{1}{x^{2}}, b=-2$ .

See Solution

Problem 19941

Bestimme den Schnittpunkt der Tangente an $f$ bei $B$ mit der $x$ -Achse für die Funktionen a) $f(x)=0,5 x^{2}$ , b) $f(x)=\frac{1}{3} x^{3}+2 x^{2}$ , c) $f(x)=\frac{3}{x^{\prime}}$ .

See Solution

Problem 19942

Determine if the integral $\int^{\infty} t^{-p} d t$ converges or diverges.

See Solution

Problem 19943

Analyze the function $y=3x^{4}+4x^{3}$ : find intercepts, extrema, inflection points, and asymptotes.

See Solution

Problem 19944

Evaluate the integral: $\int \frac{d x}{2 x+9}$ . Use absolute values and $C$ for the constant of integration.

See Solution

Problem 19945

Analyze the function $y=\frac{x^{2}+1}{x^{2}-25}$ : find intercepts, extrema, inflection points, asymptotes, domain, and range.

See Solution

Problem 19946

Evaluate the integral $\int \frac{x^{4}}{x^{5}-2} d x$ using the substitution $u=x^{5}-2$ . Find $d u$ and rewrite the integral in terms of $u$ .

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Problem 19947

Find where the function $f(x)=(x^{2}-4)e^{x}$ is concave up or down and identify the inflection points.

See Solution

Problem 19948

Considérez la fonction $f(x)=\sqrt{x^{4}+19}-\frac{x^{4}}{20}$ . Trouvez les points critiques, minima et maxima locaux.
(a) Points critiques: $x=$ (b) Minima locaux: $x=$ (c) Maxima locaux: $x=$

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Problem 19949

Calculate the integral $\int_{1}^{2} x \ln (8 x) d x$ .

See Solution

Problem 19950

Find $k$ so that $f(x)=\frac{k}{(x+1)^{2}}$ is a probability density function for $0 \leq x \leq 1$ .

See Solution

Problem 19951

Find the integral of the function $x e^{x}$ with respect to $x$ .

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Problem 19952

Evaluate the integral from 0 to 1 of $(x+1)e^{x} \, dx$ .

See Solution

Problem 19953

Evaluate the integral $\int_{1}^{\infty} t^{-2} e^{t} d t$ .

See Solution

Problem 19954

Calculate the area between the curve $y=7 x \ln (x)$ , the $x$ -axis, and the lines $x=1$ and $x=e$ .

See Solution

Problem 19955

Find the limit as $t$ approaches infinity for the expression $-\frac{t}{e^{t}} - \frac{1}{e^{t}}$ .

See Solution

Problem 19956

A substance grows at $15\%$ daily. If it starts at 85 grams, find its mass after 5 days. Round to the nearest tenth.

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Problem 19957

Check for a typo in $f(4)=4$ and $f(4)=1$ . Then find $\lim _{x \rightarrow 4}\left(\frac{2-\sqrt{f x}}{2-\sqrt{x}}\right)$ . Options: (A) 0 (B) 1 (C) -1 (D) none.

See Solution

Problem 19958

Calculez la limite $\lim _{x \rightarrow \infty} x \sin \left(\frac{2}{x}\right)$ . Quelle est sa forme indéterminée? Évaluez-la.

See Solution

Problem 19959

Find the point $(x, 0)$ on the $x$ -axis to minimize cable length $f(x)=\sqrt{(8-x)^2+3^2}+\sqrt{(8-x)^2+(-3)^2}$ .

See Solution

Problem 19960

Sketch the graph of $y=2x^{3}-3x^{2}$ and find local maxima, minima, inflection points, concavity, and increase/decrease intervals.

See Solution

Problem 19961

Find the point $(x, 0)$ on the $x$ -axis that minimizes cable length from Centerville $(9,0)$ to Springfield $(0,6)$ and Shelbyville $(0,-6)$ . Calculate the minimum cable length and justify your answer.

See Solution

Problem 19962

Evaluate the integral from 0 to 1: $\int_{0}^{1}\left(3 x-3^{x}\right) d x$ .

See Solution

Problem 19963

Gegeben ist die Funktion $f_{a}(x)=2 a x^{3}+(2-4 a) x$ .
a) Zeigen, dass alle Graphen durch $H$ der Kurve $f-0.25$ gehen. b) Bestimmen Sie $a$ , damit die Steigung in $H$ gleich 6 ist. c) Gibt es ein $a$ , sodass $f_{a}$ keine lokalen Extrema hat? d) Finden Sie die Wendenormale und den $a$ , für den die Steigung 0,5 ist.

See Solution

Problem 19964

Calculate the indefinite integral and include the constant $C$ for integration: $\int\left(\cos (x)+\frac{1}{3} x\right) d x$

See Solution

Problem 19965

Use Stokes' Theorem to find the circulation of $\vec{F}=6 y \vec{i}+4 z \vec{j}+6 x \vec{k}$ around the triangle from $(6,0,0)$ to $(6,0,3)$ to $(6,2,3)$ back to $(6,0,0)$ .

See Solution

Problem 19966

Gegeben ist die Funktion $f_{a}(x)=2 a x^{3}+(2-4 a) x$ für $a \in \mathbb{R}$ und $a \neq 0$ .
a) Zeigen Sie, dass alle Graphen durch den Hochpunkt $H$ von $f_{-0,25}$ gehen.
b) Finden Sie $a$ , sodass $f_{a}$ im Punkt $H$ die Steigung $6$ hat.
c) Gibt es $a$ , sodass $f_{a}$ keine lokalen Extrema hat?
d) Bestimmen Sie die Wendenormale von $f_{a}$ und den Parameter $a$ , bei dem die Steigung $0,5$ ist.

See Solution

Problem 19967

Find the indefinite integral: $\int(u+2)(5u+3) \, du$ (Include constant C).

See Solution

Problem 19968

Evaluate the integral using the definition: $\int_{-1}^{1}(1+4 x) d x$ .

See Solution

Problem 19969

A boy walks towards a 20 m pole at 4 km/h. Find the rate (m/s) at which distance to the top changes when 5 m away. Round to 3 decimal places.

See Solution

Problem 19970

Find the flow rate out of the sphere $x^{2}+y^{2}+z^{2}=9$ for a fluid with density $4 \mathrm{~kg} / \mathrm{m}^{3}$ and velocity $\mathbf{v}=-y \mathbf{i}+x \mathbf{j}+5 z \mathbf{k}$ .

See Solution

Problem 19971

Find the flux of the vector field $\mathbf{F}=4 \mathbf{i}+3 \mathbf{j}+1 \mathbf{k}$ across the surface $S$ defined by $4x+5y+z=4$ in the first octant.

See Solution

Problem 19972

Water is poured into a cone at $3 \mathrm{ft}^3/\mathrm{min}$ . Find the water level rise rate when it's $9 \mathrm{ft}$ deep.

See Solution

Problem 19973

Find the limit: $\lim _{x \rightarrow 0^{+}} \sqrt{x} \ln x$

See Solution

Problem 19974

Find the electric flux across the closed surface $M$ consisting of a hemisphere and its base with $\mathbf{E}=\langle 16 x, 16 y, 16 z\rangle$ . Use spherical coordinates for the integral over the hemisphere.

See Solution

Problem 19975

Calculate the surface integral $\iint_{\mathcal{S}} z \, d S$ for $y=3-z^{2}$ , $0 \leq x, z \leq 6$ .

See Solution

Problem 19976

Find the flux of $\vec{F}=3 y \vec{i}+3 \vec{j}-3 x z \vec{k}$ through the surface $S: y=x^{2}+z^{2}$ , $x^{2}+z^{2} \leq 1$ .

See Solution

Problem 19977

Find the flux of $\vec{F}=x \vec{i}+y \vec{j}+z \vec{k}$ through the curved surface of the cylinder $x^{2}+y^{2}=16$ between $x+y+z=3$ and $x+y+z=5$ .

See Solution

Problem 19978

Find the flux of $\vec{F}=x \vec{i}+y \vec{j}+z \vec{k}$ through the curved surface of the cylinder $x^{2}+y^{2}=16$ between the planes $x+y+z=3$ and $x+y+z=5$ .

See Solution

Problem 19979

A snowball's surface area decreases at $9 \mathrm{~cm}^{2} / \mathrm{min}$ . Find the diameter decrease rate when it's $12 \mathrm{~cm}$ . Round to three decimal places.

See Solution

Problem 19980

Ein Körper wird beschleunigt mit $a(t)=1,5 t$ . Finde $v_{1}(t)$ und $v_{2}(t)$ und bestimme a und v für $t=0$ und $t=10$ .

See Solution

Problem 19981

Find the derivative of $g(x)=e^{x^{6}-7x}$ . What is $g'(x)$ ?

See Solution

Problem 19982

Differentiate the function $g(t)=\frac{9-5 t}{7 t+6}$ . Find $g^{\prime}(t)=$ .

See Solution

Problem 19983

Differentiate the function F(t) = t^{9} + e^{9}. What is F'(t)?

See Solution

Problem 19984

Calculate the average value of $f(x)=2x+1$ over the interval $[0,6]$ .

See Solution

Problem 19985

A snowball's surface area decreases at $9 \mathrm{~cm}^{2} / \mathrm{min}$ . Find the diameter's decrease rate when it's $12 \mathrm{~cm}$ . Round to three decimal places.

See Solution

Problem 19986

Bestimmen Sie die Folgenglieder $a_{n}$ , Häufungspunkte, $\varlimsup_{n \in \mathbb{N}} a_{n}$ , $\sup _{n \in \mathbb{N}} a_{n}$ , $\varliminf_{n \in \mathbb{N}} a_{n}$ und das Konvergenzverhalten.

See Solution

Problem 19987

Find the linearization $L(x)$ of $g(x)=\sqrt[7]{1+x}$ at $a=0$ . First, compute $g^{\prime}(x)$ and then $g^{\prime}(0)$ .

See Solution

Problem 19988

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=x^{2}-3x+5$ , simplifying the result.

See Solution

Problem 19989

Berechnen Sie die Grenzwerte der Funktion $f(x)=\frac{x^{2}-3x}{x^{2}-x-6}$ .

See Solution

Problem 19990

Calculate the average value of $f(x)=x^{2}$ on the interval $[0,3]$ .

See Solution

Problem 19991

Calculate the average value of $f(x)=e^{x / 8}$ over the interval $[0,8]$ .

See Solution

Problem 19992

Find the average value of $f(x)=4x+1$ over the interval $[0,6]$ .

See Solution

Problem 19993

Calculate the average value of the function $f(x)=25-x^{2}$ over the interval $[-4,4]$ .

See Solution

Problem 19994

Evaluate the integral using areas: $\int_{-9}^{9} \sqrt{81-x^{2}} d x$ .

See Solution

Problem 19995

Calculate the average value of $f(x)=3 \sqrt{x}$ over the interval $[0,9]$ .

See Solution

Problem 19996

Auto A bremst von $100 \mathrm{~km/h}$ auf $0 \mathrm{~km/h}$ in $39,08 \mathrm{~m}$ mit $-9,87 \mathrm{~m/s^2}$ . Finde $v(t)$ und $s(t)$ , Bremsdauer, Wege in den ersten zwei Sekunden und Geschwindigkeit nach $35,9 \mathrm{~m}$ .

See Solution

Problem 19997

Calculez les valeurs de $A$ , $B$ et $C$ pour l'intégrale $\int \frac{x^{2}+3}{x^{3}+9 x^{2}} d x$ avec $\frac{x^{2}+3}{x^{3}+9 x^{2}}=\frac{A}{x}+\frac{B}{x^{2}}+\frac{C}{x+9}.$

See Solution

Problem 19998

Find the integral: $\int x^{2} e^{4 x} d x$

See Solution

Problem 19999

Calculez l'intégrale indéfinie $\int \frac{x^{2}+3}{x^{3}+9 x^{2}} d x$ en trouvant $A, B, C$ avec $\frac{x^{2}+3}{x^{3}+9 x^{2}}=\frac{A}{x}+\frac{B}{x^{2}}+\frac{C}{x+9}$ .

See Solution

Problem 20000

Find the derivative of the function $f(x)=\int_{-19}^{x}(\sin^2(t)-t^4) dt$ . What is $f'(x)$ ?

See Solution

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Calculus

Problem 19901

Berechne den Flächeninhalt A zwischen f(x)=4−x2f(x)=4-x^{2}f(x)=4−x2 und g(x)=12x+4g(x)=\frac{1}{2} x+4g(x)=21​x+4 im Intervall [1;2][1 ; 2][1;2].

Problem 19902

Find the function yyy for x>0x>0x>0 that satisfies the differential equation y′=y2ln⁡(x)xy2+xyy'=\frac{y^2 \ln (x)}{x y^2 + x y}y′=xy2+xyy2ln(x)​ with y(1)=1y(1)=1y(1)=1.

Problem 19903

Solve the equation x3y′=x3−3x2yx^{3} y^{\prime}=x^{3}-3 x^{2} yx3y′=x3−3x2y with the condition y(1)=1y(1)=1y(1)=1.

Problem 19904

Solve the differential equation: (2x+4y2)dx+(8xy+4y2)dy=0(2x + 4y^2)dx + (8xy + 4y^2)dy = 0(2x+4y2)dx+(8xy+4y2)dy=0.

Problem 19905

Solve the DE x3y′=x3−3x2yx^{3} y' = x^{3} - 3 x^{2} yx3y′=x3−3x2y with initial condition y(1)=1y(1) = 1y(1)=1.

Problem 19906

Ein Körper wird mit 6 m/s6 \mathrm{~m/s}6 m/s aus 140 m140 \mathrm{~m}140 m Höhe bei 30∘30^{\circ}30∘ abgeworfen. Finde die Zeit TTT bis zum Aufprall.

Problem 19907

Find F′(π)F^{\prime}(\pi)F′(π) if F(x)=−∫0xcos⁡(t)dtF(x)=-\int_{0}^{x} \cos (t) dtF(x)=−∫0x​cos(t)dt.

Problem 19908

Calculate the integral ∫043x4dx\int_{0}^{4} \frac{3 \sqrt{x}}{4} d x∫04​43x​​dx. What is its value?

Problem 19909

Find the value of ∫06f(x)dx\int_{0}^{6} f(x) dx∫06​f(x)dx for the piecewise function f(x)=−2xf(x) = -2xf(x)=−2x if x<1x<1x<1 and f(x)=3f(x) = 3f(x)=3 if x≥1x \geq 1x≥1.

Problem 19910

Find F(π)F(\pi)F(π) where F(x)=∫πxt4sin⁡2(t)dtF(x)=\int_{\pi}^{x} t^{4} \sin^{2}(t) dtF(x)=∫πx​t4sin2(t)dt.

Problem 19911

Problem 19912

Find the area between f(x)=xe+exf(x)=x^{e}+e^{x}f(x)=xe+ex and the xxx-axis for x∈[0,1]x \in[0,1]x∈[0,1]. What is it equal to?

Problem 19913

Find the limit: lim⁡t→∞(t−7)(t−8)\lim _{t \rightarrow \infty}(t-7)(t-8)limt→∞​(t−7)(t−8). Choose one: a. -8 b. limit does not exist. c. 7 d. 8 e. 0

Problem 19914

Find the value of lim⁡x→∞5ln⁡(x)4x\lim _{x \rightarrow \infty} \frac{5 \ln (x)}{4 x}limx→∞​4x5ln(x)​. Choices: a. DNE, b. 5/45/45/4, c. -1, d. 4/54/54/5, e. 0, f. 1.

Problem 19915

If ∫3t2cos⁡(t3)dt=∫cos⁡(u)du\int 3 t^{2} \cos(t^{3}) dt = \int \cos(u) du∫3t2cos(t3)dt=∫cos(u)du, find uuu. Options: a. ttt b. t3t^{3}t3 c. 3t23 t^{2}3t2 d. cos⁡(t)\cos(t)cos(t) e. 3t3 t3t f. sin⁡(t)\sin(t)sin(t)

Problem 19916

Bewege einen Körper mit v(t)=40−t2v(t)=40-t^{2}v(t)=40−t2 in m/s\mathrm{m}/\mathrm{s}m/s. a) Weg in 4s als Integral. b) Berechne Weg mit Summen.

Problem 19917

Problem 19918

Problem 19919

Problem 19920

Problem 19921

Find h′(2)h^{\prime}(2)h′(2) for h(x)=f(g(x))h(x)=f(g(x))h(x)=f(g(x)) given g(2)=3g(2)=3g(2)=3, g′(2)=5g^{\prime}(2)=5g′(2)=5, and f′(3)=−1f^{\prime}(3)=-1f′(3)=−1.

Problem 19922

Problem 19923

Problem 19924

Problem 19925

Determine where f(x)=1x2+7f(x)=\frac{1}{x^{2}+7}f(x)=x2+71​ is concave up/down and find points of inflection. Select correct answers.

Problem 19926

Vergleichen Sie die Differenzenquotienten der Intervalle [−100,−1][-100, -1][−100,−1], [−10,−1][-10, -1][−10,−1], [−1,1][-1, 1][−1,1] und [−1.01,−1][-1.01, -1][−1.01,−1].

Problem 19927

Problem 19928

Find the general antiderivative of f(u)=−10u4−1uu2f(u)=\frac{-10 u^{4}-1 \sqrt{u}}{u^{2}}f(u)=u2−10u4−1u​​. Use upper-case "C" for constants. F(u)=F(u)= F(u)=

Problem 19929

Find the satellite's orbital period in seconds given Earth's gravity is 6 m/s26 \, m/s^{2}6m/s2 and radius is 6,371,000 m6,371,000 \, m6,371,000m.

Problem 19930

Find the antiderivative FFF of f(x)=4−3(1+x2)−1f(x)=4-3(1+x^{2})^{-1}f(x)=4−3(1+x2)−1 with F(1)=−1F(1)=-1F(1)=−1.

Problem 19931

Given the function f(x)=1x2+8x+20f(x)=\frac{1}{x^{2}+8 x+20}f(x)=x2+8x+201​, find:a) The domain in interval notation. b) The critical numbers of fff. c) Intervals where fff is increasing and decreasing. d) Use the First Derivative Test for relative maxima and minima.

Problem 19932

Find the general antiderivative of f(u)=−3u4−4uu2f(u)=\frac{-3 u^{4}-4 \sqrt{u}}{u^{2}}f(u)=u2−3u4−4u​​. Use upper-case " C\mathrm{C}C " for constants.

Problem 19933

Find the derivative of f(x)=x2−2xx2+5xf(x)=\frac{x^{2}-2 x}{x^{2}+5 x}f(x)=x2+5xx2−2x​ using the quotient rule.

Problem 19934

Find the function g(x)g(x)g(x) such that g′(x)=−4096−x3g'(x) = -4096 - x^3g′(x)=−4096−x3 and the maximum value of g(x)g(x)g(x) is −5-5−5.

Problem 19935

Find the function f(x)f(x)f(x) given that f′′(x)=3x+6sin⁡(x)f^{\prime \prime}(x)=3 x+6 \sin (x)f′′(x)=3x+6sin(x), f(0)=4f(0)=4f(0)=4, and f′(0)=3f^{\prime}(0)=3f′(0)=3.

Problem 19936

Simplify the expression: ddx∫4x(7t2+t+9)dt=□\frac{d}{d x} \int_{4}^{x}(7 t^{2}+t+9) d t = \squaredxd​∫4x​(7t2+t+9)dt=□

Problem 19937

Find the limit: lim⁡x→−15−54x2+4x\lim _{x \rightarrow-1} \frac{5-5}{4 x^{2}+4 x}limx→−1​4x2+4x5−5​.

Problem 19938

Approximate local and global minima and maxima for f(x)=2x3−5x−3f(x)=2 x^{3}-5 x-3f(x)=2x3−5x−3. Round to two decimal places.

Problem 19939

Problem 19940

Problem 19941

Bestimme den Schnittpunkt der Tangente an fff bei BBB mit der xxx-Achse für die Funktionen a) f(x)=0,5x2f(x)=0,5 x^{2}f(x)=0,5x2, b) f(x)=13x3+2x2f(x)=\frac{1}{3} x^{3}+2 x^{2}f(x)=31​x3+2x2, c) f(x)=3x′f(x)=\frac{3}{x^{\prime}}f(x)=x′3​.

Problem 19942

Determine if the integral ∫∞t−pdt\int^{\infty} t^{-p} d t∫∞t−pdt converges or diverges.

Problem 19943

Analyze the function y=3x4+4x3y=3x^{4}+4x^{3}y=3x4+4x3: find intercepts, extrema, inflection points, and asymptotes.

Problem 19944

Evaluate the integral: ∫dx2x+9\int \frac{d x}{2 x+9}∫2x+9dx​. Use absolute values and CCC for the constant of integration.

Problem 19945

Analyze the function y=x2+1x2−25y=\frac{x^{2}+1}{x^{2}-25}y=x2−25x2+1​: find intercepts, extrema, inflection points, asymptotes, domain, and range.

Berechne den Flächeninhalt A zwischen $f(x)=4-x^{2}$ und $g(x)=\frac{1}{2} x+4$ im Intervall $[1 ; 2]$ .

Find the function $y$ for $x>0$ that satisfies the differential equation $y'=\frac{y^2 \ln (x)}{x y^2 + x y}$ with $y(1)=1$ .

Solve the equation $x^{3} y^{\prime}=x^{3}-3 x^{2} y$ with the condition $y(1)=1$ .

Solve the differential equation: $(2x + 4y^2)dx + (8xy + 4y^2)dy = 0$ .

Solve the DE $x^{3} y' = x^{3} - 3 x^{2} y$ with initial condition $y(1) = 1$ .

Ein Körper wird mit $6 \mathrm{~m/s}$ aus $140 \mathrm{~m}$ Höhe bei $30^{\circ}$ abgeworfen. Finde die Zeit $T$ bis zum Aufprall.

Find $F^{\prime}(\pi)$ if $F(x)=-\int_{0}^{x} \cos (t) dt$ .

Calculate the integral $\int_{0}^{4} \frac{3 \sqrt{x}}{4} d x$ . What is its value?

Find the value of $\int_{0}^{6} f(x) dx$ for the piecewise function $f(x) = -2x$ if $x<1$ and $f(x) = 3$ if $x \geq 1$ .

Find $F(\pi)$ where $F(x)=\int_{\pi}^{x} t^{4} \sin^{2}(t) dt$ .

Find the area between $f(x)=x^{e}+e^{x}$ and the $x$ -axis for $x \in[0,1]$ . What is it equal to?

Find the limit: $\lim _{t \rightarrow \infty}(t-7)(t-8)$ . Choose one: a. -8 b. limit does not exist. c. 7 d. 8 e. 0

Find the value of $\lim _{x \rightarrow \infty} \frac{5 \ln (x)}{4 x}$ . Choices: a. DNE, b. $5/4$ , c. -1, d. $4/5$ , e. 0, f. 1.

If $\int 3 t^{2} \cos(t^{3}) dt = \int \cos(u) du$ , find $u$ .
Options: a. $t$ b. $t^{3}$ c. $3 t^{2}$ d. $\cos(t)$ e. $3 t$ f. $\sin(t)$

Bewege einen Körper mit $v(t)=40-t^{2}$ in $\mathrm{m}/\mathrm{s}$ . a) Weg in 4s als Integral. b) Berechne Weg mit Summen.

Find $h^{\prime}(2)$ for $h(x)=f(g(x))$ given $g(2)=3$ , $g^{\prime}(2)=5$ , and $f^{\prime}(3)=-1$ .

Determine where $f(x)=\frac{1}{x^{2}+7}$ is concave up/down and find points of inflection. Select correct answers.

Vergleichen Sie die Differenzenquotienten der Intervalle $[-100, -1]$ , $[-10, -1]$ , $[-1, 1]$ und $[-1.01, -1]$ .

Find the general antiderivative of $f(u)=\frac{-10 u^{4}-1 \sqrt{u}}{u^{2}}$ . Use upper-case "C" for constants. $F(u)=$

Find the satellite's orbital period in seconds given Earth's gravity is $6 \, m/s^{2}$ and radius is $6,371,000 \, m$ .

Find the antiderivative $F$ of $f(x)=4-3(1+x^{2})^{-1}$ with $F(1)=-1$ .

Given the function $f(x)=\frac{1}{x^{2}+8 x+20}$ , find:
a) The domain in interval notation. b) The critical numbers of $f$ . c) Intervals where $f$ is increasing and decreasing. d) Use the First Derivative Test for relative maxima and minima.

Find the general antiderivative of $f(u)=\frac{-3 u^{4}-4 \sqrt{u}}{u^{2}}$ . Use upper-case " $\mathrm{C}$ " for constants.

Find the derivative of $f(x)=\frac{x^{2}-2 x}{x^{2}+5 x}$ using the quotient rule.

Find the function $g(x)$ such that $g'(x) = -4096 - x^3$ and the maximum value of $g(x)$ is $-5$ .

Find the function $f(x)$ given that $f^{\prime \prime}(x)=3 x+6 \sin (x)$ , $f(0)=4$ , and $f^{\prime}(0)=3$ .

Simplify the expression: $\frac{d}{d x} \int_{4}^{x}(7 t^{2}+t+9) d t = \square$

Find the limit: $\lim _{x \rightarrow-1} \frac{5-5}{4 x^{2}+4 x}$ .

Approximate local and global minima and maxima for $f(x)=2 x^{3}-5 x-3$ . Round to two decimal places.

Bestimme den Schnittpunkt der Tangente an $f$ bei $B$ mit der $x$ -Achse für die Funktionen a) $f(x)=0,5 x^{2}$ , b) $f(x)=\frac{1}{3} x^{3}+2 x^{2}$ , c) $f(x)=\frac{3}{x^{\prime}}$ .

Determine if the integral $\int^{\infty} t^{-p} d t$ converges or diverges.

Analyze the function $y=3x^{4}+4x^{3}$ : find intercepts, extrema, inflection points, and asymptotes.

Evaluate the integral: $\int \frac{d x}{2 x+9}$ . Use absolute values and $C$ for the constant of integration.

Analyze the function $y=\frac{x^{2}+1}{x^{2}-25}$ : find intercepts, extrema, inflection points, asymptotes, domain, and range.

Evaluate the integral $\int \frac{x^{4}}{x^{5}-2} d x$ using the substitution $u=x^{5}-2$ . Find $d u$ and rewrite the integral in terms of $u$ .

Find where the function $f(x)=(x^{2}-4)e^{x}$ is concave up or down and identify the inflection points.

Considérez la fonction $f(x)=\sqrt{x^{4}+19}-\frac{x^{4}}{20}$ . Trouvez les points critiques, minima et maxima locaux.
(a) Points critiques: $x=$ (b) Minima locaux: $x=$ (c) Maxima locaux: $x=$

Calculate the integral $\int_{1}^{2} x \ln (8 x) d x$ .

Find $k$ so that $f(x)=\frac{k}{(x+1)^{2}}$ is a probability density function for $0 \leq x \leq 1$ .

Find the integral of the function $x e^{x}$ with respect to $x$ .

Evaluate the integral from 0 to 1 of $(x+1)e^{x} \, dx$ .

Evaluate the integral $\int_{1}^{\infty} t^{-2} e^{t} d t$ .

Calculate the area between the curve $y=7 x \ln (x)$ , the $x$ -axis, and the lines $x=1$ and $x=e$ .

Find the limit as $t$ approaches infinity for the expression $-\frac{t}{e^{t}} - \frac{1}{e^{t}}$ .

A substance grows at $15\%$ daily. If it starts at 85 grams, find its mass after 5 days. Round to the nearest tenth.

Check for a typo in $f(4)=4$ and $f(4)=1$ . Then find $\lim _{x \rightarrow 4}\left(\frac{2-\sqrt{f x}}{2-\sqrt{x}}\right)$ . Options: (A) 0 (B) 1 (C) -1 (D) none.

Calculez la limite $\lim _{x \rightarrow \infty} x \sin \left(\frac{2}{x}\right)$ . Quelle est sa forme indéterminée? Évaluez-la.

Find the point $(x, 0)$ on the $x$ -axis to minimize cable length $f(x)=\sqrt{(8-x)^2+3^2}+\sqrt{(8-x)^2+(-3)^2}$ .

Sketch the graph of $y=2x^{3}-3x^{2}$ and find local maxima, minima, inflection points, concavity, and increase/decrease intervals.

Find the point $(x, 0)$ on the $x$ -axis that minimizes cable length from Centerville $(9,0)$ to Springfield $(0,6)$ and Shelbyville $(0,-6)$ . Calculate the minimum cable length and justify your answer.

Evaluate the integral from 0 to 1: $\int_{0}^{1}\left(3 x-3^{x}\right) d x$ .

Calculate the indefinite integral and include the constant $C$ for integration: $\int\left(\cos (x)+\frac{1}{3} x\right) d x$

Use Stokes' Theorem to find the circulation of $\vec{F}=6 y \vec{i}+4 z \vec{j}+6 x \vec{k}$ around the triangle from $(6,0,0)$ to $(6,0,3)$ to $(6,2,3)$ back to $(6,0,0)$ .

Find the indefinite integral: $\int(u+2)(5u+3) \, du$ (Include constant C).

Evaluate the integral using the definition: $\int_{-1}^{1}(1+4 x) d x$ .

Find the flow rate out of the sphere $x^{2}+y^{2}+z^{2}=9$ for a fluid with density $4 \mathrm{~kg} / \mathrm{m}^{3}$ and velocity $\mathbf{v}=-y \mathbf{i}+x \mathbf{j}+5 z \mathbf{k}$ .

Find the flux of the vector field $\mathbf{F}=4 \mathbf{i}+3 \mathbf{j}+1 \mathbf{k}$ across the surface $S$ defined by $4x+5y+z=4$ in the first octant.

Water is poured into a cone at $3 \mathrm{ft}^3/\mathrm{min}$ . Find the water level rise rate when it's $9 \mathrm{ft}$ deep.

Find the limit: $\lim _{x \rightarrow 0^{+}} \sqrt{x} \ln x$

Find the electric flux across the closed surface $M$ consisting of a hemisphere and its base with $\mathbf{E}=\langle 16 x, 16 y, 16 z\rangle$ . Use spherical coordinates for the integral over the hemisphere.