Calculus

Problem 18901

Selenium-83 has a half-life of 25 min. How long for a 10 mg sample to decay to 1 mg? Use $k=\ln \left(\frac{A}{A_{0}}\right)$ and $t=$ .

See Solution

Problem 18902

Find the initial butterflies at $t=0$ , max butterflies as $t \to \infty$ , and butterflies at $t=20$ for $f(t)=\frac{360}{1+11.0 e^{-0.12 t}}$ .

See Solution

Problem 18903

Function $A=A_0 e^{-0.01155 t}$ models radioactive decay.
a) Find remaining pounds after 50 years with $A_0=300$ . b) Determine years until only 50 pounds remain using $t=\frac{\ln(A/A_0)}{k}$ .

See Solution

Problem 18904

Find the derivative $\frac{d y}{d x}$ for the function $y=\sqrt[3]{x}-\cos x+7 x^{5}$ .

See Solution

Problem 18905

Find the average rate of change of $p(t)=\frac{(t^{2}-5)(t+1)}{t^{2}+2}$ on the interval $[1,3]$ .

See Solution

Problem 18906

Calculate the average rate of change of $g(x)=2x^{2}-2$ over the interval $[x, x+h]$ .

See Solution

Problem 18907

Find the derivatives using the Fundamental Theorem of Calculus: a) $f(x)=\int_{0}^{x} \sqrt{t+t^{3}} dt$ b) $f(x)=\int_{x}^{0} \sqrt{1+\sec(t)} dt$

See Solution

Problem 18908

Find the 1st, 2nd, and 3rd derivatives of $f(x)=x \cdot e^{2-x}$ .

See Solution

Problem 18909

Find the derivatives using the Fundamental Theorem of Calculus Part I for: a) $f(x)=\int_{0}^{x} \sqrt{t+t^{3}} dt$ b) $f(x)=\int_{x}^{0} \sqrt{1+\sec(t)} dt$ c) $f(x)=\int_{1}^{3x+2} \frac{t}{1+t^{3}} dt$

See Solution

Problem 18910

Find the derivative of the function $f(x)=e^{x}$ . Show that $\frac{d}{dx}(e^{x})=e^{x}$ .

See Solution

Problem 18911

Choose the integral for the area between $y=x^{2}$ , $x=2$ , and the $x$ -axis: $\int_{0}^{2} x^{2} d x$ .

See Solution

Problem 18912

Find the area between the curve $y=8x^{3}-3x^{2}$ and the $x$ -axis from $x=1$ to $x=2$ .

See Solution

Problem 18913

Find the area between the curve $y=x^{2}-2x$ and the $x$ -axis from $x=0$ to $x=3$ . Enter the exact value.

See Solution

Problem 18914

Aufgabe 1: Finde die Extrempunkte von $f(x)=x^{3}-12x$ und bestimme ihre Art mit der zweiten Ableitung.
Aufgabe 2: Gegeben $f_{a}(x)=\frac{1}{3}x^{3}-a x^{2}+a^{2}$ , bestimme $f_{2}(x)$ , die Steigung bei $x=3$ und den Wert von $a$ für Steigung 15.
Aufgabe 3: Beurteile: a) Ist die zweite Ableitung einer nach unten geöffneten Parabel negativ? b) Hat $f^{\prime}(3)=0$ , $f^{\prime \prime}(3)=-2$ einen Hochpunkt bei $\mathrm{H}(3 \mid-2)$ ?

See Solution

Problem 18915

Two flies on a balloon inflate at 5 cm³/s. Find their distance over time and measure it in two ways. Steps: A. Draw; B. Equation; C. Speed after 3s, 0.1s, 0.01s, and initially.

See Solution

Problem 18916

Berechnen Sie die Steigung von $f$ bei $A(2 \mid f(2))$ für a) $f(x)=\frac{3}{2} x^{2}$ , b) $f(t)=\frac{1}{4} t^{4}-\frac{5}{3} t^{3}$ , c) $f(z)=\frac{3}{4} z^{2}-3 z^{-1}$ , d) $f(x)=-x-2 x^{-4}$ . Leiten Sie a) $s(t)=t^{4}$ und b) $h(y)=y 12$ ab.

See Solution

Problem 18917

Leiten Sie die folgenden Funktionen ab und analysieren Sie die Raketenhöhe:
1) $s(t)=t^{4}$ , $h(y)=y^{12}$ , $m(a)=a^{-3}$ , $g(z)=\frac{1}{z^{2}}$ . 2) Für $h(t)=t^{3}$ : Höhe und Geschwindigkeit nach 1,2 s? Wann ist die Geschwindigkeit $12 \frac{\mathrm{m}}{\mathrm{s}}$ ?

See Solution

Problem 18918

Find $h(x)$ given that $g(x)=e^{f(x)}$ and $g^{\prime \prime}(x)=h(x)e^{f(x)}$ .

See Solution

Problem 18919

Gegeben sind $f_{a}(x)=\frac{1}{6} x^{3}-\frac{a^{2}}{4} x^{2}$ und $g_{a}(x)=-\frac{1}{a} x^{2}+\frac{3 a}{2} x$ . Zeigen Sie, dass Nullstellen von $f_{a}$ und $g_{a}$ für $a>0$ übereinstimmen. Bestimmen Sie Hochpunkte von $g_{a}$ in Abhängigkeit von $a$ und wann diese oberhalb von $g(x)=4,5$ liegen. Berechnen Sie Hoch- und Tiefpunkte von $f_{a}$ .

See Solution

Problem 18920

Evaluate the integral $\int_{1 / 5}^{5} \frac{1}{x^{2}} dx$ .

See Solution

Problem 18921

Welche Aussagen zur Funktion $f(x)=\ln(4x^4+5x^2+1)$ sind korrekt? Wähle aus: 1. Polynom 4. Grades, 2. Minimale Stelle bei $x^=0$ , 3. Nullstelle bei $x^=0$ , 4. Nach unten beschränkt.

See Solution

Problem 18922

Bestimmen Sie die erste und zweite Ableitung für die Funktionen: a) $f(x)=t x^{3}-4 x^{2}$ , b) $f(x)=3 x^{2}+2 x-b$ , c) $g(x)=3 x^{3}-2 a x-a$ , d) $f(x)=(t-x)^{2}+3 x$ .

See Solution

Problem 18923

Bestimmen Sie die Ableitungen bis $f^{(n)}(x)=0$ für die Funktionen: a) $f(x)=3 x^{5}-2 x^{3}+6 x$ b) $f(x)=-\frac{3}{8} x^{4}-\frac{1}{12} x^{3}+\frac{1}{14} x$ c) $f(x)=\sqrt{5} \cdot x^{6}-\pi x^{4}-3$ Bestimmen Sie die 1. und 2. Ableitung für: a) $f(x)=t x^{3}-4 x^{2}$ b) $f(x)=3 x^{2}+2 x-b$ c) $g(x)=3 x^{3}-2 a x-a$ d) $f(x)=(t-x)^{2}+3 x$

See Solution

Problem 18924

Solve $\frac{d y}{d x}=(1+\ln x) y$ with initial condition $y(1)=1$ . Find $y=$ A) $e^{\frac{x^{2}-1}{x^{2}}}$ B) $\ln x$ C) $e^{2 x+x \ln x-2}$ D) $e^{x \ln x}$ .

See Solution

Problem 18925

Find the second derivative $d^{2} y / d x^{2}$ at $y=1$ for $\tan^{-1}(y^{2})=3x$ . Choices: a) $-9/2$ , b) $-9$ , c) $9$ .

See Solution

Problem 18926

Find the derivative of $f_{a}(x)=x^{3}-3 a x^{2}$ .

See Solution

Problem 18927

Find the second derivative $d^{2} y / d x^{2}$ at $y=1$ for the equation $\tan^{-1}(y^{2})=3x$ .

See Solution

Problem 18928

Solve the equation $\frac{d y}{d x}=(1+\ln x) y$ with the condition $y(1)=1$ . Find $y$ .

See Solution

Problem 18929

Evaluate the integral: $\int \sin (t) \sqrt{1+\cos (t)} d t$ (use constant $C$ ).

See Solution

Problem 18930

Skizzieren Sie den Graphen von $f'$ für $f(x)=3 x^{2}-4 x$ und bestimmen Sie die Ableitungsfunktion.

See Solution

Problem 18931

Use the Mean Value Theorem to estimate $f(5) - f(3)$ given $-4 \leq f^{\prime}(x) \leq 4$ on $(3,5)$ .

See Solution

Problem 18932

Find the maximum possible value of $f(4)$ given $f(0)=-3$ and $f^{\prime}(x) \leq 4$ for all $x$ . Answer: $f(4) \leq$

See Solution

Problem 18933

Gegeben ist die Funktion $f_{a}(x)=x^{3}-3 a x^{2}$ . Bestimme Nullstellen und lokale Extrema in Abhängigkeit von $a$ .

See Solution

Problem 18934

Find the horizontal asymptotes by calculating these limits:
1. $\lim _{x \rightarrow \infty} \frac{-4 x}{13+2 x}$
2. $\lim _{x \rightarrow-\infty} \frac{15 x-11}{x^{3}+14 x-2}$
3. $\lim _{x \rightarrow \infty} \frac{x^{2}-10 x-2}{3-2 x^{2}}$
4. $\lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}+4 x}}{6-5 x}$
5. $\lim _{x \rightarrow-\infty} \frac{\sqrt{x^{2}+4 x}}{6-5 x}$

See Solution

Problem 18935

Evaluate the integral from 4 to 9 of the function $18 x^{2}-10 x+7$ .

See Solution

Problem 18936

Find the limit as $x$ approaches 4 for the expression $\frac{4-x}{\sqrt{x}-2}$ .

See Solution

Problem 18937

Find intervals where $f(x)=-x^{4}-8x^{3}+5x+7$ is concave up/down and $x$ -coordinates of inflection points.

See Solution

Problem 18938

Evaluate the integral from 2 to 3 of $(8 + e^{x})$ dx. Provide the answer exactly or rounded to two decimal places.

See Solution

Problem 18939

Find $\int_{-5.5}^{-3} f(x) d x$ and $\int_{-5.5}^{-3}(2 f(x)-10) d x=$ .

See Solution

Problem 18940

Given $\int_{-8}^{-0.5} f(x) d x=2$ , $\int_{-8}^{-5.5} f(x) d x=10$ , $\int_{-3}^{-0.5} f(x) d x=6$ , find $\int_{-5.5}^{-3} f(x) d x$ and $\int_{-5.5}^{-3}(2 f(x)-10) d x$ .

See Solution

Problem 18941

Evaluate the integral from 2 to 5 of (5x + 8) dx. Provide the answer exactly or rounded to two decimal places.

See Solution

Problem 18942

Evaluate the integral: $\int_{0}^{13} \frac{4}{x} d x$ . Provide the answer in exact form or to two decimal places.

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Problem 18943

Evaluate the integral from 10 to 19 of $\frac{4}{x}$ dx. Provide the answer exactly or rounded to two decimal places.

See Solution

Problem 18944

Evaluate the integral $\int_{-2}^{-1} 5 \sqrt[3]{x} \, dx$ . Provide the exact answer or round to two decimal places.

See Solution

Problem 18945

Evaluate the integral: $\int_{-4}^{-3}-9 x^{2} d x$ . Provide the exact or rounded answer.

See Solution

Problem 18946

Calculate the integral from 3 to 4 of $\frac{4}{x}$ dx. Provide the answer exactly or rounded to two decimal places.

See Solution

Problem 18947

Calculate the integral from 1 to 3 of $8 \sqrt{x} \, dx$ . Provide the answer exactly or rounded to two decimal places.

See Solution

Problem 18948

Evaluate the integral: $\int_{-3}^{2}\left(12 x\left(3 x^{2}-6\right)^{3}\right) d x$ . Provide the exact answer or round to two decimal places.

See Solution

Problem 18949

Use the Divergence Test to check if the series $\sum_{k=0}^{\infty} \frac{k}{2 k^{6}+1}$ diverges or is inconclusive.

See Solution

Problem 18950

Determine if the series $\sum_{k=0}^{\infty} \frac{k}{7 k+1}$ diverges using the Divergence Test.

See Solution

Problem 18951

Use the Divergence Test for the series $\sum_{k=0}^{\infty} \frac{k}{2 k^{6}+1}$ . What is $\lim_{k \to \infty} a_k$ ? Choose A, B, C, or D.

See Solution

Problem 18952

Bestimme $f^{\prime}(2)$ für $f(x)=3 x^{2}-2$ und $f^{\prime}(1)$ für $f(x)=5 x^{2}+4$ .

See Solution

Problem 18953

Evaluate the integral: $\int_{-5}^{-3}(3(3x+7)^{4}) \, dx$ . Provide the answer in exact form or rounded to two decimal places.

See Solution

Problem 18954

Berechnen Sie $f^{\prime}(2)$ für die Funktion $f(x)=3 x^{2}-2$ .

See Solution

Problem 18955

Use the Integral Test to check if $\sum_{k=0}^{\infty} \frac{9}{\sqrt{k+2}}$ converges. Which conditions apply to $f(x)=\frac{9}{\sqrt{x+2}}$ ?

See Solution

Problem 18956

Use the Divergence Test to check if the series $\sum_{k=2}^{\infty} \frac{3 \sqrt{k}}{\ln 10 k}$ diverges. Choose A, B, C, or D.

See Solution

Problem 18957

Find the sum of the series $S = \sum_{n=1}^{\infty}-7\left(\frac{3}{8}\right)^{n}$ .

See Solution

Problem 18958

Use the Integral Test to check if the series $\sum_{k=0}^{\infty} \frac{9}{\sqrt{k+2}}$ converges. Identify conditions satisfied by $f(x)=\frac{9}{\sqrt{x+2}}$ .

See Solution

Problem 18959

Bestimmen Sie die Ableitung $f^{\prime}(1)$ für die Funktion $f(x)=5 x^{2}+4$ .

See Solution

Problem 18960

Find the average value of the function $f(x)=3 x^{2}+1$ on the interval $[1,3]$ .

See Solution

Problem 18961

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

See Solution

Problem 18962

Approximate local and global minima/maxima of $f(x)=3 x^{3}-2 x+3$ using a calculator. Round to two decimal places.

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Problem 18963

A 90 kg man drops from 2.7 m and stops 0.82 s after hitting the water. What force does the water exert on him in N?

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Problem 18964

Solve the differential equation: $\frac{d y}{d t}=\frac{t e^{t}}{y \sqrt{8+y^{2}}}$

See Solution

Problem 18965

Find the positive $x$ -coordinate of the point of inflection for $f(x)=x \exp \left(-x^{2} / 2\right)$ . Round to three decimal places.

See Solution

Problem 18966

Bestimmen Sie die Extrempunkte der Funktion $f(x)=\frac{e^{x}}{x}$ .

See Solution

Problem 18967

Find the area of the surface formed by rotating the curve $x=\sqrt{a^{2}-y^{2}}$ , $0 \leq y \leq a / 8$ around the $y$ -axis.

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Problem 18968

Untersuchen Sie die Funktionen auf lokale Extrema und Wendepunkte: a) $f(x)=\frac{1}{3} x^{3}-2 x^{2}+3 x$ b) $f(x)=\frac{1}{6} x^{3}+x^{2}$ c) $f(x)=x^{4}-3 x^{2}+4$ d) $f(x)=1$

See Solution

Problem 18969

Calculate the integral $A=\int_{-5}^{-2}\left(-x^{2}-6 x-8\right)-(x+2) d x$ .

See Solution

Problem 18970

Solve the differential equation: $\frac{d u}{d t}=2+2 u+t+t u$

See Solution

Problem 18971

Find the max altitude of a hot air balloon with rate $R(t)=t^{3}-4 t^{2}+6$ for $0 \leq t \leq 4$ and when it rises fastest.

See Solution

Problem 18972

Find the limit: $\lim _{x \rightarrow \infty}\left(x \operatorname{arctg} \frac{x+1}{x-1}-\frac{\pi}{4} x\right)$ .

See Solution

Problem 18973

Find the integral of $x(2x+5)^{8}$ with respect to $x$ .

See Solution

Problem 18974

Find where the function $y=9 x^{5}-60 x^{3}$ is (a) increasing, (b) decreasing, and (c) where relative extrema occur.

See Solution

Problem 18975

Find where the function $y=(x^{2}-100)^{4}$ is (a) increasing, (b) decreasing, and (c) where relative extrema occur.

See Solution

Problem 18976

Find the total area between the curve $f(x)=x^{3}+x^{2}-2 x$ and the $x$ -axis, considering all relevant intervals.

See Solution

Problem 18977

Berechne die Ableitung der Funktion $f a(x)=-a x^{3}+4 a x$ .

See Solution

Problem 18978

Una carretera inclinada a 8° y velocidad de 20 pies. Si se bloquean las ruedas, ¿qué distancia recorren? $\mu_{k}=0.5$

See Solution

Problem 18979

Find where the function $y=2 e^{x}-e^{-x}$ is (a) increasing, (b) decreasing, and (c) where relative extrema occur.

See Solution

Problem 18980

Evaluate the integral $\int (9 \sqrt{x} - 3) \, dx$ .

See Solution

Problem 18981

Find the derivative $f^{\prime}(x)$ for the function $f(x)=e^{x}+2 x-5 \ln x$ .

See Solution

Problem 18982

Find where $F(x)=x^{3}+5x^{2}+3x-9$ is concave up/down, its turning and inflection points, and sketch the graph with labels.

See Solution

Problem 18983

Find the derivative of $f(x)=\frac{3}{(2x+7)^{2}}$ .

See Solution

Problem 18984

Calculate the integral: $\int(10 \sqrt{x^{3}}-8 x) \, dx$

See Solution

Problem 18985

Bestimmen Sie das unbestimmte Integral der Funktion $f(x)=\frac{3}{(2 x+7)^{2}}$ .

See Solution

Problem 18986

Find the area between $y=x^{2}+4x$ and $y=x-7$ from $x=-2$ to $x=2$ .

See Solution

Problem 18987

Find the derivative $f'(x)$ for the function $f(x) = 6x - 5 \ln(x^7)$ .

See Solution

Problem 18988

Given the price-demand equation $p + 0.02x = 20$ : a. Solve for demand $x$ in terms of price $p$ . b. Determine the elasticity of demand $E(p)$ . c. Calculate $E(4)$ .

See Solution

Problem 18989

Find the area between $f(x)=1-x^{2}$ and $g(x)=4-4x^{2}$ from $x=0$ to $x=1$ .

See Solution

Problem 18990

Given the price-demand equation $p+0.02 x=20$ :
a. Solve for demand $x$ in terms of price $p$ . b. Determine the elasticity of demand function $E(p)$ . c. Calculate $E(4)$ .

See Solution

Problem 18991

Find the condition for $f(x)=2-x-e^{x}$ on $[0,1]$ to satisfy the Intermediate Value Theorem (IVT).

See Solution

Problem 18992

Given the price-demand equation $15p + 0.06x = 300$ :
a. Solve for demand $x$ in terms of price $p$ . b. Determine the elasticity of demand function $E(p)$ . c. Calculate $E(1)$ .

See Solution

Problem 18993

Find the labor supply function from the utility $U(C, L)=C-(1-L)^{2}$ and derive the equilibrium wage and profits.

See Solution

Problem 18994

Nikola drives at 72 ft/s and decelerates at $24 \mathrm{ft} / \mathrm{s}^{2}$ . Find $d(0)$ , a formula for $d(t)$ , stop time, and braking distance.

See Solution

Problem 18995

Bestimmen Sie die Stammfunktion für die Funktionen a) bis l) und nennen Sie die verwendete Regel.

See Solution

Problem 18996

Find the volume of the solid with a base in the first quadrant bounded by $y=18-2 x^{2}$ using square cross-sections.

See Solution

Problem 18997

Given the function $L(t)=2.5 t+1.5 \cos \left(\frac{2 \pi t}{24}\right)$ , find (a) the growth rate $\frac{\mathrm{dL}}{\mathrm{dt}}$ and (b) the largest growth rate.

See Solution

Problem 18998

Find the velocity $v(t)$ and position $s(t)$ of a particle with $a(t)=6t+4$ , $v(0)=-2$ , $s(0)=1$ .

See Solution

Problem 18999

Find the general antiderivative of $g(x)=\sec ^{2}(x)-2 e^{x}-\frac{1}{x}$ , considering $x$ can be negative.

See Solution

Problem 19000

Find the general antiderivative of $f(t)=\frac{1}{x^{2}}+\cos (x)$ .

See Solution

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Calculus

Problem 18901

Selenium-83 has a half-life of 25 min. How long for a 10 mg sample to decay to 1 mg? Use k=ln⁡(AA0)k=\ln \left(\frac{A}{A_{0}}\right)k=ln(A0​A​) and t=t=t=.

Problem 18902

Find the initial butterflies at t=0t=0t=0, max butterflies as t→∞t \to \inftyt→∞, and butterflies at t=20t=20t=20 for f(t)=3601+11.0e−0.12tf(t)=\frac{360}{1+11.0 e^{-0.12 t}}f(t)=1+11.0e−0.12t360​.

Problem 18903

Function A=A0e−0.01155tA=A_0 e^{-0.01155 t}A=A0​e−0.01155t models radioactive decay. a) Find remaining pounds after 50 years with A0=300A_0=300A0​=300. b) Determine years until only 50 pounds remain using t=ln⁡(A/A0)kt=\frac{\ln(A/A_0)}{k}t=kln(A/A0​)​.

Problem 18904

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=x3−cos⁡x+7x5y=\sqrt[3]{x}-\cos x+7 x^{5}y=3x​−cosx+7x5.

Problem 18905

Find the average rate of change of p(t)=(t2−5)(t+1)t2+2p(t)=\frac{(t^{2}-5)(t+1)}{t^{2}+2}p(t)=t2+2(t2−5)(t+1)​ on the interval [1,3][1,3][1,3].

Problem 18906

Calculate the average rate of change of g(x)=2x2−2g(x)=2x^{2}-2g(x)=2x2−2 over the interval [x,x+h][x, x+h][x,x+h].

Problem 18907

Find the derivatives using the Fundamental Theorem of Calculus: a) f(x)=∫0xt+t3dtf(x)=\int_{0}^{x} \sqrt{t+t^{3}} dtf(x)=∫0x​t+t3​dt b) f(x)=∫x01+sec⁡(t)dtf(x)=\int_{x}^{0} \sqrt{1+\sec(t)} dtf(x)=∫x0​1+sec(t)​dt

Problem 18908

Find the 1st, 2nd, and 3rd derivatives of f(x)=x⋅e2−xf(x)=x \cdot e^{2-x}f(x)=x⋅e2−x.

Problem 18909

Problem 18910

Find the derivative of the function f(x)=exf(x)=e^{x}f(x)=ex. Show that ddx(ex)=ex\frac{d}{dx}(e^{x})=e^{x}dxd​(ex)=ex.

Problem 18911

Choose the integral for the area between y=x2y=x^{2}y=x2, x=2x=2x=2, and the xxx-axis: ∫02x2dx\int_{0}^{2} x^{2} d x∫02​x2dx.

Problem 18912

Find the area between the curve y=8x3−3x2y=8x^{3}-3x^{2}y=8x3−3x2 and the xxx-axis from x=1x=1x=1 to x=2x=2x=2.

Problem 18913

Find the area between the curve y=x2−2xy=x^{2}-2xy=x2−2x and the xxx-axis from x=0x=0x=0 to x=3x=3x=3. Enter the exact value.

Problem 18914

Problem 18915

Two flies on a balloon inflate at 5 cm³/s. Find their distance over time and measure it in two ways. Steps: A. Draw; B. Equation; C. Speed after 3s, 0.1s, 0.01s, and initially.

Problem 18916

Problem 18917

Problem 18918

Find h(x)h(x)h(x) given that g(x)=ef(x)g(x)=e^{f(x)}g(x)=ef(x) and g′′(x)=h(x)ef(x)g^{\prime \prime}(x)=h(x)e^{f(x)}g′′(x)=h(x)ef(x).

Problem 18919

Problem 18920

Evaluate the integral ∫1/551x2dx\int_{1 / 5}^{5} \frac{1}{x^{2}} dx∫1/55​x21​dx.

Problem 18921

Welche Aussagen zur Funktion f(x)=ln⁡(4x4+5x2+1)f(x)=\ln(4x^4+5x^2+1)f(x)=ln(4x4+5x2+1) sind korrekt? Wähle aus: 1. Polynom 4. Grades, 2. Minimale Stelle bei x∗=0x^*=0x∗=0, 3. Nullstelle bei x∗=0x^*=0x∗=0, 4. Nach unten beschränkt.

Problem 18922

Bestimmen Sie die erste und zweite Ableitung für die Funktionen: a) f(x)=tx3−4x2f(x)=t x^{3}-4 x^{2}f(x)=tx3−4x2, b) f(x)=3x2+2x−bf(x)=3 x^{2}+2 x-bf(x)=3x2+2x−b, c) g(x)=3x3−2ax−ag(x)=3 x^{3}-2 a x-ag(x)=3x3−2ax−a, d) f(x)=(t−x)2+3xf(x)=(t-x)^{2}+3 xf(x)=(t−x)2+3x.

Problem 18923

Problem 18924

Solve dydx=(1+ln⁡x)y\frac{d y}{d x}=(1+\ln x) ydxdy​=(1+lnx)y with initial condition y(1)=1y(1)=1y(1)=1. Find y=y=y= A) ex2−1x2e^{\frac{x^{2}-1}{x^{2}}}ex2x2−1​ B) ln⁡x\ln xlnx C) e2x+xln⁡x−2e^{2 x+x \ln x-2}e2x+xlnx−2 D) exln⁡xe^{x \ln x}exlnx.

Problem 18925

Find the second derivative d2y/dx2d^{2} y / d x^{2}d2y/dx2 at y=1y=1y=1 for tan⁡−1(y2)=3x\tan^{-1}(y^{2})=3xtan−1(y2)=3x. Choices: a) −9/2-9/2−9/2, b) −9-9−9, c) 999.

Problem 18926

Find the derivative of fa(x)=x3−3ax2f_{a}(x)=x^{3}-3 a x^{2}fa​(x)=x3−3ax2.

Problem 18927

Find the second derivative d2y/dx2d^{2} y / d x^{2}d2y/dx2 at y=1y=1y=1 for the equation tan⁡−1(y2)=3x\tan^{-1}(y^{2})=3xtan−1(y2)=3x.

Problem 18928

Solve the equation dydx=(1+ln⁡x)y\frac{d y}{d x}=(1+\ln x) ydxdy​=(1+lnx)y with the condition y(1)=1y(1)=1y(1)=1. Find yyy.

Problem 18929

Evaluate the integral: ∫sin⁡(t)1+cos⁡(t)dt\int \sin (t) \sqrt{1+\cos (t)} d t∫sin(t)1+cos(t)​dt (use constant CCC).

Problem 18930

Skizzieren Sie den Graphen von f′f'f′ für f(x)=3x2−4xf(x)=3 x^{2}-4 xf(x)=3x2−4x und bestimmen Sie die Ableitungsfunktion.

Problem 18931

Use the Mean Value Theorem to estimate f(5)−f(3)f(5) - f(3)f(5)−f(3) given −4≤f′(x)≤4-4 \leq f^{\prime}(x) \leq 4−4≤f′(x)≤4 on (3,5)(3,5)(3,5).

Problem 18932

Find the maximum possible value of f(4)f(4)f(4) given f(0)=−3f(0)=-3f(0)=−3 and f′(x)≤4f^{\prime}(x) \leq 4f′(x)≤4 for all xxx. Answer: f(4)≤f(4) \leqf(4)≤

Problem 18933

Gegeben ist die Funktion fa(x)=x3−3ax2f_{a}(x)=x^{3}-3 a x^{2}fa​(x)=x3−3ax2. Bestimme Nullstellen und lokale Extrema in Abhängigkeit von aaa.

Problem 18934

Problem 18935

Evaluate the integral from 4 to 9 of the function 18x2−10x+718 x^{2}-10 x+718x2−10x+7.

Problem 18936

Find the limit as xxx approaches 4 for the expression 4−xx−2\frac{4-x}{\sqrt{x}-2}x​−24−x​.

Problem 18937

Find intervals where f(x)=−x4−8x3+5x+7f(x)=-x^{4}-8x^{3}+5x+7f(x)=−x4−8x3+5x+7 is concave up/down and xxx-coordinates of inflection points.

Problem 18938

Evaluate the integral from 2 to 3 of (8+ex)(8 + e^{x})(8+ex) dx. Provide the answer exactly or rounded to two decimal places.

Problem 18939

Find ∫−5.5−3f(x)dx\int_{-5.5}^{-3} f(x) d x∫−5.5−3​f(x)dx and ∫−5.5−3(2f(x)−10)dx=\int_{-5.5}^{-3}(2 f(x)-10) d x=∫−5.5−3​(2f(x)−10)dx=.

Problem 18940

Problem 18941

Evaluate the integral from 2 to 5 of (5x + 8) dx. Provide the answer exactly or rounded to two decimal places.

Problem 18942

Evaluate the integral: ∫0134xdx\int_{0}^{13} \frac{4}{x} d x∫013​x4​dx. Provide the answer in exact form or to two decimal places.

Problem 18943

Evaluate the integral from 10 to 19 of 4x\frac{4}{x}x4​ dx. Provide the answer exactly or rounded to two decimal places.

Problem 18944

Selenium-83 has a half-life of 25 min. How long for a 10 mg sample to decay to 1 mg? Use $k=\ln \left(\frac{A}{A_{0}}\right)$ and $t=$ .

Find the initial butterflies at $t=0$ , max butterflies as $t \to \infty$ , and butterflies at $t=20$ for $f(t)=\frac{360}{1+11.0 e^{-0.12 t}}$ .

Function $A=A_0 e^{-0.01155 t}$ models radioactive decay.
a) Find remaining pounds after 50 years with $A_0=300$ . b) Determine years until only 50 pounds remain using $t=\frac{\ln(A/A_0)}{k}$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=\sqrt[3]{x}-\cos x+7 x^{5}$ .

Find the average rate of change of $p(t)=\frac{(t^{2}-5)(t+1)}{t^{2}+2}$ on the interval $[1,3]$ .

Calculate the average rate of change of $g(x)=2x^{2}-2$ over the interval $[x, x+h]$ .

Find the derivatives using the Fundamental Theorem of Calculus: a) $f(x)=\int_{0}^{x} \sqrt{t+t^{3}} dt$ b) $f(x)=\int_{x}^{0} \sqrt{1+\sec(t)} dt$

Find the 1st, 2nd, and 3rd derivatives of $f(x)=x \cdot e^{2-x}$ .

Find the derivative of the function $f(x)=e^{x}$ . Show that $\frac{d}{dx}(e^{x})=e^{x}$ .

Choose the integral for the area between $y=x^{2}$ , $x=2$ , and the $x$ -axis: $\int_{0}^{2} x^{2} d x$ .

Find the area between the curve $y=8x^{3}-3x^{2}$ and the $x$ -axis from $x=1$ to $x=2$ .

Find the area between the curve $y=x^{2}-2x$ and the $x$ -axis from $x=0$ to $x=3$ . Enter the exact value.

Find $h(x)$ given that $g(x)=e^{f(x)}$ and $g^{\prime \prime}(x)=h(x)e^{f(x)}$ .

Evaluate the integral $\int_{1 / 5}^{5} \frac{1}{x^{2}} dx$ .

Welche Aussagen zur Funktion $f(x)=\ln(4x^4+5x^2+1)$ sind korrekt? Wähle aus: 1. Polynom 4. Grades, 2. Minimale Stelle bei $x^=0$ , 3. Nullstelle bei $x^=0$ , 4. Nach unten beschränkt.

Bestimmen Sie die erste und zweite Ableitung für die Funktionen: a) $f(x)=t x^{3}-4 x^{2}$ , b) $f(x)=3 x^{2}+2 x-b$ , c) $g(x)=3 x^{3}-2 a x-a$ , d) $f(x)=(t-x)^{2}+3 x$ .

Solve $\frac{d y}{d x}=(1+\ln x) y$ with initial condition $y(1)=1$ . Find $y=$ A) $e^{\frac{x^{2}-1}{x^{2}}}$ B) $\ln x$ C) $e^{2 x+x \ln x-2}$ D) $e^{x \ln x}$ .

Find the second derivative $d^{2} y / d x^{2}$ at $y=1$ for $\tan^{-1}(y^{2})=3x$ . Choices: a) $-9/2$ , b) $-9$ , c) $9$ .

Find the derivative of $f_{a}(x)=x^{3}-3 a x^{2}$ .

Find the second derivative $d^{2} y / d x^{2}$ at $y=1$ for the equation $\tan^{-1}(y^{2})=3x$ .

Solve the equation $\frac{d y}{d x}=(1+\ln x) y$ with the condition $y(1)=1$ . Find $y$ .

Evaluate the integral: $\int \sin (t) \sqrt{1+\cos (t)} d t$ (use constant $C$ ).

Skizzieren Sie den Graphen von $f'$ für $f(x)=3 x^{2}-4 x$ und bestimmen Sie die Ableitungsfunktion.

Use the Mean Value Theorem to estimate $f(5) - f(3)$ given $-4 \leq f^{\prime}(x) \leq 4$ on $(3,5)$ .

Find the maximum possible value of $f(4)$ given $f(0)=-3$ and $f^{\prime}(x) \leq 4$ for all $x$ . Answer: $f(4) \leq$

Gegeben ist die Funktion $f_{a}(x)=x^{3}-3 a x^{2}$ . Bestimme Nullstellen und lokale Extrema in Abhängigkeit von $a$ .

Evaluate the integral from 4 to 9 of the function $18 x^{2}-10 x+7$ .

Find the limit as $x$ approaches 4 for the expression $\frac{4-x}{\sqrt{x}-2}$ .

Find intervals where $f(x)=-x^{4}-8x^{3}+5x+7$ is concave up/down and $x$ -coordinates of inflection points.

Evaluate the integral from 2 to 3 of $(8 + e^{x})$ dx. Provide the answer exactly or rounded to two decimal places.

Find $\int_{-5.5}^{-3} f(x) d x$ and $\int_{-5.5}^{-3}(2 f(x)-10) d x=$ .

Evaluate the integral: $\int_{0}^{13} \frac{4}{x} d x$ . Provide the answer in exact form or to two decimal places.

Evaluate the integral from 10 to 19 of $\frac{4}{x}$ dx. Provide the answer exactly or rounded to two decimal places.

Evaluate the integral $\int_{-2}^{-1} 5 \sqrt[3]{x} \, dx$ . Provide the exact answer or round to two decimal places.

Evaluate the integral: $\int_{-4}^{-3}-9 x^{2} d x$ . Provide the exact or rounded answer.

Calculate the integral from 3 to 4 of $\frac{4}{x}$ dx. Provide the answer exactly or rounded to two decimal places.

Calculate the integral from 1 to 3 of $8 \sqrt{x} \, dx$ . Provide the answer exactly or rounded to two decimal places.

Evaluate the integral: $\int_{-3}^{2}\left(12 x\left(3 x^{2}-6\right)^{3}\right) d x$ . Provide the exact answer or round to two decimal places.

Use the Divergence Test to check if the series $\sum_{k=0}^{\infty} \frac{k}{2 k^{6}+1}$ diverges or is inconclusive.

Determine if the series $\sum_{k=0}^{\infty} \frac{k}{7 k+1}$ diverges using the Divergence Test.

Use the Divergence Test for the series $\sum_{k=0}^{\infty} \frac{k}{2 k^{6}+1}$ . What is $\lim_{k \to \infty} a_k$ ? Choose A, B, C, or D.

Bestimme $f^{\prime}(2)$ für $f(x)=3 x^{2}-2$ und $f^{\prime}(1)$ für $f(x)=5 x^{2}+4$ .

Evaluate the integral: $\int_{-5}^{-3}(3(3x+7)^{4}) \, dx$ . Provide the answer in exact form or rounded to two decimal places.

Berechnen Sie $f^{\prime}(2)$ für die Funktion $f(x)=3 x^{2}-2$ .

Use the Integral Test to check if $\sum_{k=0}^{\infty} \frac{9}{\sqrt{k+2}}$ converges. Which conditions apply to $f(x)=\frac{9}{\sqrt{x+2}}$ ?

Use the Divergence Test to check if the series $\sum_{k=2}^{\infty} \frac{3 \sqrt{k}}{\ln 10 k}$ diverges. Choose A, B, C, or D.

Find the sum of the series $S = \sum_{n=1}^{\infty}-7\left(\frac{3}{8}\right)^{n}$ .

Use the Integral Test to check if the series $\sum_{k=0}^{\infty} \frac{9}{\sqrt{k+2}}$ converges. Identify conditions satisfied by $f(x)=\frac{9}{\sqrt{x+2}}$ .

Bestimmen Sie die Ableitung $f^{\prime}(1)$ für die Funktion $f(x)=5 x^{2}+4$ .

Find the average value of the function $f(x)=3 x^{2}+1$ on the interval $[1,3]$ .

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

Approximate local and global minima/maxima of $f(x)=3 x^{3}-2 x+3$ using a calculator. Round to two decimal places.

Solve the differential equation: $\frac{d y}{d t}=\frac{t e^{t}}{y \sqrt{8+y^{2}}}$

Find the positive $x$ -coordinate of the point of inflection for $f(x)=x \exp \left(-x^{2} / 2\right)$ . Round to three decimal places.

Bestimmen Sie die Extrempunkte der Funktion $f(x)=\frac{e^{x}}{x}$ .

Find the area of the surface formed by rotating the curve $x=\sqrt{a^{2}-y^{2}}$ , $0 \leq y \leq a / 8$ around the $y$ -axis.

Untersuchen Sie die Funktionen auf lokale Extrema und Wendepunkte: a) $f(x)=\frac{1}{3} x^{3}-2 x^{2}+3 x$ b) $f(x)=\frac{1}{6} x^{3}+x^{2}$ c) $f(x)=x^{4}-3 x^{2}+4$ d) $f(x)=1$

Calculate the integral $A=\int_{-5}^{-2}\left(-x^{2}-6 x-8\right)-(x+2) d x$ .

Solve the differential equation: $\frac{d u}{d t}=2+2 u+t+t u$

Find the max altitude of a hot air balloon with rate $R(t)=t^{3}-4 t^{2}+6$ for $0 \leq t \leq 4$ and when it rises fastest.

Find the limit: $\lim _{x \rightarrow \infty}\left(x \operatorname{arctg} \frac{x+1}{x-1}-\frac{\pi}{4} x\right)$ .

Find the integral of $x(2x+5)^{8}$ with respect to $x$ .

Find where the function $y=9 x^{5}-60 x^{3}$ is (a) increasing, (b) decreasing, and (c) where relative extrema occur.