Calculus

Problem 25901

Find the displacement and distance traveled of a particle with velocity $v(t)=3 t^{2}-42 t+135$ over $[0,10]$ .

See Solution

Problem 25902

Find the critical points of the function $f(x)=\frac{e^{x}+e^{-x}}{6}$ .

See Solution

Problem 25903

Find the critical points of the function $f(x)=\frac{1}{x}-\ln x$ .

See Solution

Problem 25904

Approximate $\int_{2}^{6} f(x) dx$ using a Riemann sum with 2 rectangles and left endpoints. Values: 4, -2, -3.

See Solution

Problem 25905

Evaluate the integral: $\int \frac{48 x^{2}}{x^{4}-26 x^{2}+25} dx$ and find its partial fraction decomposition.

See Solution

Problem 25906

Find the absolute extreme values of $f(x)=24 x^{\frac{1}{2}}-x$ on the interval $[0,576]$ .

See Solution

Problem 25907

Find the min/max, domain, range, and intervals of increase/decrease for $f(x)=4x^2+16x-3$ and $g(x)=-x^2+5x+9$ .

See Solution

Problem 25908

Find the absolute extreme values of $f(x)=2 \sin ^{2} x$ on the interval $[0, \pi]$ .

See Solution

Problem 25909

Estimate total water drained in 3 minutes using average of left- and right-endpoint approximations from $v(t)$ values: $44, 40, 36, 32, 28, 24, 20$ .

See Solution

Problem 25910

Estimate the total extinct marine families from $t=514$ to $t=524$ using $M_{10}$ with $r(t)=\frac{3130}{t+262}$ .

See Solution

Problem 25911

Find $\lim _{x \rightarrow(-7)+} f(x)$ for the piecewise function: $f(x)=\left\{\begin{array}{ll}\left(x^{2}-1\right)/(x+1) & \text{if } x \leq-7 \\ \left(x^{2}+3x-28\right)/(x+7) & \text{if } x>-7\end{array}\right.$ . Choices: (A) -7, (B) -8, (C) 7, (D) -11, (E) does not exist.

See Solution

Problem 25912

Find the 4th derivative of $f(x) = \cos(\sin(x))$ at $x=0$ using Taylor series: $f^{(4)}(0)$ .

See Solution

Problem 25913

Estimate cardiac output $R$ using $A=4 \mathrm{mg}$ and $c(t)$ values from $t=0$ to $t=10$ . Calculate $R$ in $\mathrm{L} / \mathrm{min}$ .

See Solution

Problem 25914

Evaluate the integral $\int \frac{22 x^{2}}{x^{4}-61 x^{2}+900} d x$ and find its partial fraction decomposition.

See Solution

Problem 25915

Find where the function $h(x)=\frac{5+5 \sin x}{\cos x}$ is continuous and analyze the limits: $\lim _{x \rightarrow \pi / 2^{-}} h(x)$ and $\lim _{x \rightarrow 4 \pi / 3} h(x)$ .

See Solution

Problem 25916

Find the limit: $\lim _{x \rightarrow \pi / 2^{-}} \frac{5+5 \sin x}{\cos x}$ .

See Solution

Problem 25917

Calculate the first four approximations of the largest positive root of $x^{3}+x=17$ using Newton's Method, starting with $x_{0}=3$ . Round to six decimal places.

See Solution

Problem 25918

Evaluate the integral: $\int \frac{54 x^{2}}{x^{4}-45 x^{2}+324} d x$ . Find the result: $\int \frac{54 x^{2}}{x^{4}-45 x^{2}+324} d x=\square$ .

See Solution

Problem 25919

Evaluate the integral: $\int \frac{42 x^{2}}{x^{4}-29 x^{2}+100} d x=\square$

See Solution

Problem 25920

Find the limit: $\lim _{x \rightarrow 4 \pi / 3} \frac{5+5 \sin x}{\cos x}$ .

See Solution

Problem 25921

Evaluate the integral: $\int \frac{2 z^{3}-3 z^{2}-17 z+23}{z^{2}-z-12} d z$ .

See Solution

Problem 25922

Find the derivative of y with respect to θ for y = ln(8 cos^6(θ)).

See Solution

Problem 25923

Find the length $x$ that maximizes the area of a rectangular playground using 120 meters of fencing for three sides.

See Solution

Problem 25924

Find the absolute extreme values of $f(x)=2 x^{2}+2 \cos ^{-1} x$ on the interval $[-1,1]$ .

See Solution

Problem 25925

Find the absolute extreme values of $f(x)=5(4x)^{x}$ on the interval $[0.05,1]$ .

See Solution

Problem 25926

Determine if $f(x)=7 x^{4}-6 x^{2}-3$ has a real zero between 1 and 2 using the Intermediate Value Theorem.

See Solution

Problem 25927

Check if $f(x)=7 x^{4}-6 x^{2}-3$ has a real zero between 1 and 2 using the Intermediate Value Theorem.

See Solution

Problem 25928

Rolle's Theorem doesn't apply to $f(x)=|x|$ on $[-a, a]$ for any $a>0$ . Explain why this is the case.

See Solution

Problem 25929

Find the absolute extreme values of $f(x)=5 \csc x$ on the interval $\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right]$ .

See Solution

Problem 25930

Find the absolute extreme values of $f(x)=x^{3} e^{-x}$ on the interval $[-1,4]$ .

See Solution

Problem 25931

Find points $\mathrm{c}$ where the Mean Value Theorem applies for $f(x)=x^{3}$ on $[-13,13]$ .

See Solution

Problem 25932

Find points $c$ where the Mean Value Theorem applies for $f(x)=x^{3}$ on $[-4,4]$ .

See Solution

Problem 25933

Check if Rolle's theorem applies to $f(x)=x(x-3)^{2}$ on $[0,3]$ and find the guaranteed point(s).

See Solution

Problem 25934

Find the critical points of the function $f(x)=(x-2)^{5}$ .

See Solution

Problem 25935

Find $c$ such that $\frac{f(b)-f(a)}{b-a}=f^{\prime}(c)$ for $f(x)=\tan^{-1} x$ on $[-\sqrt{3}, \sqrt{3}]$ . Round to the nearest thousandth.

See Solution

Problem 25936

Determine the absolute max and min of $F(x)=\sqrt[3]{x}$ for $-27 \leq x \leq 27$ .

See Solution

Problem 25937

Evaluate the integral $\int (x^{4}-5 x^{3}-5) \, dx = \mathrm{C}$ .

See Solution

Problem 25938

Find $c$ for the Mean Value Theorem with $f(x) = \tan^{-1}x$ on $[-\sqrt{3}, \sqrt{3}]$ : $f'(c) = \frac{f(\sqrt{3}) - f(-\sqrt{3})}{2\sqrt{3}}$ .

See Solution

Problem 25939

Find values of $c$ for $f(x)=x^{2}+4x+1$ in $[2,3]$ satisfying $\frac{f(3)-f(2)}{3-2}=f^{\prime}(c)$ .

See Solution

Problem 25940

Calculate the total revenue from the investment stream $R(t)=400 e^{0.08 t}$ between 4 and 6 years. Round to the nearest cent.

See Solution

Problem 25941

Find the absolute extreme values of $h(x)=\frac{1}{3} x+5$ on the interval $-3 \leq x \leq 4$ .

See Solution

Problem 25942

Interpret $P(20)=80$ and $\left.\frac{d P}{d x}\right|_{x=20}=2$ for advertising profit in thousands.

See Solution

Problem 25943

Find the absolute max and min of $f(x)=5 x^{2 / 3}$ on the interval $[-27, 1]$ .

See Solution

Problem 25944

Find the derivative of $G(x)=(3x^{2}+4x-5)^{6}$ .

See Solution

Problem 25945

Find $c$ such that $\frac{f(7)-f(5)}{7-5}=f'(c)$ for $f(x)=\ln(x-4)$ on $[5,7]$ . Round to the nearest thousandth.

See Solution

Problem 25946

Evaluate the limits: $\lim _{x \rightarrow \infty} \frac{x+x^{3}+x^{5}}{1-x^{2}+2023 x^{5}}$ , $\lim _{x \rightarrow 5} \frac{2^{x}-32}{x-5}$ , $\lim _{x \rightarrow 0} \frac{x}{\tan x}$ , and $\lim _{x \rightarrow \infty} x^{\frac{1}{x}}$ .

See Solution

Problem 25947

Find the maximum and minimum values of the function $f(x)=x^{2}+x-4$ on the interval $[-4,4]$ .

See Solution

Problem 25948

Hydrazine, $\mathrm{N}_{2} \mathrm{H}_{4}$ , reacts with $\mathrm{O}_{2}$ to form $\mathrm{N}_{2}$ and $\mathrm{H}_{2} \mathrm{O}$ . Find the temperature for spontaneity given $\Delta S^{\circ} = -122.1 \mathrm{~J/K}$ and $\Delta H^{\circ} = 95.4 \mathrm{~kJ/mol}$ .

See Solution

Problem 25949

Find the area between the curves $S(t)=3.5t+9$ and $C(t)=5.2t$ for $t$ years, and interpret the savings in \$. Round answers accordingly.

See Solution

Problem 25950

Bestimme den Tag $x$ , an dem die Funktion $f(x)=(2 x+1) \cdot e^{-0,02 x}$ ihr Maximum erreicht.

See Solution

Problem 25951

Find the limit: $\lim _{x \rightarrow 0^{+}}(9 x+1)^{\cot (x)}$ . Use l'Hospital's Rule if needed.

See Solution

Problem 25952

Find the light intensity $I$ (in thousands of foot-candles) that maximizes the photosynthesis rate $P=\frac{90 I}{I^{2}+I+1}$ .

See Solution

Problem 25953

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{f(4+3 x)+f(4+6 x)}{x}$ given $f(4)=0$ and $f^{\prime}(4)=8$ .

See Solution

Problem 25954

Find the area between the curves $S(t)=3.8t+9$ and $C(t)=5.4t$ for $t$ in years. Round to 2 decimal and nearest integer.

See Solution

Problem 25955

Find the function $f$ if $f'(x) = 1 - 6x$ and $f(0) = 8$ .

See Solution

Problem 25956

Find the general antiderivative of $f(x)=\frac{9-x^{3}+2 x^{5}}{x^{6}}$ .

See Solution

Problem 25957

Find $f(4)$ given $f(1)=12$ , $f^{\prime}$ is continuous, and $\int_{1}^{4} f^{\prime}(x) dx=17$ .

See Solution

Problem 25958

Find $\int_{5}^{9} f(x) d x$ given that $\int_{3}^{9} f(x) d x=8$ and $\int_{3}^{5} f(x) d x=5$ .

See Solution

Problem 25959

Evaluate the integral $\int_{0}^{2} \frac{1}{\sqrt[3]{x-1}} d x$ .

See Solution

Problem 25960

Find the dimensions (in cm) of a box with a square base and volume $13,500 \mathrm{~cm}^{3}$ that minimizes material use.

See Solution

Problem 25961

Find the function $f$ such that $f^{\prime \prime}(x)=10+6x+36x^{2}$ with conditions $f(0)=2$ and $f(1)=14$ .

See Solution

Problem 25962

Find $F^{\prime}(x)$ if $F(x)=\int_{\sin x}^{\cos x}(t+1)^{2024} dt$ .

See Solution

Problem 25963

Bestimme die Maße und maximale Fläche eines rechteckigen Tors in der Seitenwand $f(x)=\frac{1}{25} x^{4}-\frac{2}{3} x^{2}+\frac{9}{5}$ für $[-1,84 ; 1,84]$ .

See Solution

Problem 25964

Find the function $f$ given that $f''(x)=-2+36x-12x^2$ , $f(0)=2$ , and $f'(0)=16$ .

See Solution

Problem 25965

Aufgabe 1: Finde die Nullstellen von $f(x)=x^{5}+7,25 x^{3}+2,25 x$ , $g(x)=x^{3}+10 x^{2}+7 x-18$ , $h(x)=(3 x-7)(x^{2}+5)(\sqrt{x})$ .
Aufgabe 2: Bestimme die Ableitung und die Steigung an $x_{0}$ für $f(x)=3 x^{4}$ , $g(x)=3$ , $h(x)=2 x^{6}-5 x^{3}+2 x$ bei $x_{0}=2$ , $5$ , $-2$ .

See Solution

Problem 25966

Evaluate the integral from -1 to 2: $\int_{-1}^{2}(x-|x|) d x$ .

See Solution

Problem 25967

Find the general antiderivative of $f(x)=(x+7)(2x-3)$ and check by differentiating. Use $C$ for the constant.

See Solution

Problem 25968

Find the general antiderivative of $f(x)=3 \sqrt{x}+9 \cos (x)$ and check by differentiating. Use $C$ .

See Solution

Problem 25969

Find a function $f$ where $f'(x)=3x^3$ and the line $81x+y=0$ is tangent to $f$ . What is $f(x)=?$

See Solution

Problem 25970

Find the function $f$ such that $f''(x)=x^{-2}$ for $x>0$ , with conditions $f(1)=0$ and $f(7)=0$ .

See Solution

Problem 25971

Determine if the sequence $a_{n}=\sqrt{n+1}-\sqrt{n}$ converges or diverges. If it converges, find the limit.

See Solution

Problem 25972

Find the position of a particle, $s(t)$ , given $a(t)=t^{2}-7t+5$ , $s(0)=0$ , and $s(1)=20$ .

See Solution

Problem 25973

Find $f$ given that $f(x)=f^{\prime \prime}(x)=\frac{8}{9} x^{8 / 9}$ . Use constants $C$ and $D$ .

See Solution

Problem 25974

Check if the series $\sum_{n=1}^{\infty} \frac{\ln n}{n^{4}}$ converges or diverges.

See Solution

Problem 25975

Find the position function $s(t)$ for a particle with $a(t)=2t+9$ , $s(0)=2$ , and $v(0)=-5$ .

See Solution

Problem 25976

Find $f$ if $f^{\prime \prime}(x)=4x+\sin(x)$ , using constants $C$ and $D$ . What is $f(x)$ ?

See Solution

Problem 25977

Calculate the integrals $\int x \sin(x^{2}+1) \, dx$ and $\int \sin^{2} x \, dx$ .

See Solution

Problem 25978

Find the function $f$ given that $f^{\prime}(x)=\frac{8}{\sqrt{1-x^{2}}}$ and $f\left(\frac{1}{2}\right)=7$ .

See Solution

Problem 25979

Bestimme die unbestimmten Integrale der Funktionen: a) $f(x)=-\frac{1}{2} x^{2}+3 x$ , b) $f(x)=x^{\frac{2}{3}}-x$ , c) $f(x)=-x^{3}-1$ , d) $f(x)=-3 x^{2}+4 x-7$ .

See Solution

Problem 25980

Bestimme die unbestimmten Integrale für die Funktionen: a) $f(x)=-\frac{1}{2} x^{2}+3 x$ , b) $f(x)=x^{\frac{2}{3}}-x$ , c) $f(x)=-x^{3}-1$ , d) $f(x)=-3 x^{2}+4 x-7$ .

See Solution

Problem 25981

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

See Solution

Problem 25982

Find the limit as $b$ approaches infinity for $A_b$ where $f(x)=\frac{4}{x^{2}}$ and $b>3$ .

See Solution

Problem 25983

Invest how much now at $6.5\%$ interest, compounded continuously, to have \$3000 in 2 years? Round to the nearest cent.

See Solution

Problem 25984

Find the tangent line equation for $f(x)=2 e^{-4 x}$ at the point $(0,2)$ .

See Solution

Problem 25985

Find the steepest tangent slope of $f(x)=\sqrt{x+2}$ on $[0,1]$ , the secant line through endpoints, and compare slopes.

See Solution

Problem 25986

Find the derivative of $y=\frac{\ln x}{x^{3}}$ .

See Solution

Problem 25987

Speicherung elektrischer Energie:
a) Zeichne den Graphen von $f(x)=x^{3}-38 x^{2}+295 x+750$ und interpretiere die Extrempunkte. b) Finde die Integralfunktion zu $f(x)$ und interpretiere sie. c) Berechne das Wasser im Becken nach 30 Tagen, wenn zu Beginn $10 \mathrm{Mill}^{\mathrm{m}} \mathrm{m}^{3}$ vorhanden waren.

See Solution

Problem 25988

Find the steepest slope of the tangent line for $f(x)=\sqrt{x+2}$ on the interval $[0,1]$ .

See Solution

Problem 25989

Finde die Hoch- und Tiefpunkte der Funktion $g(x) = 2x^3 - 9x^2 + 12x - 4$ .

See Solution

Problem 25990

Bestimme die Tangentengleichung von $f(x)=\frac{1}{3} x^{3}-3 x^{2}-5 x+6$ bei $x=0$ .

See Solution

Problem 25991

Find the blood flow speed in the plaque region and the pressure drop given diameter changes from $0.9 \mathrm{~cm}$ to $0.7 \mathrm{~cm}$ .

See Solution

Problem 25992

Given a differentiable function $g$ with $g^{\prime}(x)<0$ , find where $h(x)=g(x^{2}-2x-8)$ is increasing or decreasing.

See Solution

Problem 25993

Berechne die Tangentensteigung von $f(x)=x^{2}+5$ bei $x_{0}=1$ mit der h-Methode.

See Solution

Problem 25994

Entscheide, welche Aussagen zu den Verschiebungen und Streckungen der Funktion g und ihrer Ableitung zutreffen.

See Solution

Problem 25995

Two stones, $A$ (15 m/s) and $B$ (30 m/s), are thrown from a cliff. How does their fall time compare? a) same b) twice c) half d) four times

See Solution

Problem 25996

Finde den Punkt im Intervall $(-3, 0)$ , wo die Funktion $g(x) = \frac{1}{4} x^{4} + \frac{2}{3} x^{3} - \frac{3}{2} x^{2}$ die größte Steigung hat.

See Solution

Problem 25997

Vergleiche die Steigungen von $h(t)=-\frac{1}{3} t^{3}+3 t^{2}+150 t+500$ bei $t=1$ und $t=4$ .

See Solution

Problem 25998

Find the height of a projectile after 3 seconds using 6 rectangles and the midpoint Riemann sum for $v(t)=-15.3t+145$ .

See Solution

Problem 25999

Evaluate the limit: $\lim _{x \rightarrow 0}\left(\frac{2+\sqrt[5]{1+\sin x}}{3}\right)^{\frac{1}{x}}$

See Solution

Problem 26000

Find the marginal profit at a production level of 60 items for $C(q)=q^{3}-11 q^{2}+58 q+5000$ and $R(q)=-3 q^{2}+2400 q$ .

See Solution

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Calculus

Problem 25901

Find the displacement and distance traveled of a particle with velocity v(t)=3t2−42t+135v(t)=3 t^{2}-42 t+135v(t)=3t2−42t+135 over [0,10][0,10][0,10].

Problem 25902

Find the critical points of the function f(x)=ex+e−x6f(x)=\frac{e^{x}+e^{-x}}{6}f(x)=6ex+e−x​.

Problem 25903

Find the critical points of the function f(x)=1x−ln⁡xf(x)=\frac{1}{x}-\ln xf(x)=x1​−lnx.

Problem 25904

Approximate ∫26f(x)dx\int_{2}^{6} f(x) dx∫26​f(x)dx using a Riemann sum with 2 rectangles and left endpoints. Values: 4, -2, -3.

Problem 25905

Evaluate the integral: ∫48x2x4−26x2+25dx\int \frac{48 x^{2}}{x^{4}-26 x^{2}+25} dx∫x4−26x2+2548x2​dx and find its partial fraction decomposition.

Problem 25906

Find the absolute extreme values of f(x)=24x12−xf(x)=24 x^{\frac{1}{2}}-xf(x)=24x21​−x on the interval [0,576][0,576][0,576].

Problem 25907

Find the min/max, domain, range, and intervals of increase/decrease for f(x)=4x2+16x−3f(x)=4x^2+16x-3f(x)=4x2+16x−3 and g(x)=−x2+5x+9g(x)=-x^2+5x+9g(x)=−x2+5x+9.

Problem 25908

Find the absolute extreme values of f(x)=2sin⁡2xf(x)=2 \sin ^{2} xf(x)=2sin2x on the interval [0,π][0, \pi][0,π].

Problem 25909

Estimate total water drained in 3 minutes using average of left- and right-endpoint approximations from v(t)v(t)v(t) values: 44,40,36,32,28,24,2044, 40, 36, 32, 28, 24, 2044,40,36,32,28,24,20.

Problem 25910

Estimate the total extinct marine families from t=514t=514t=514 to t=524t=524t=524 using M10M_{10}M10​ with r(t)=3130t+262r(t)=\frac{3130}{t+262}r(t)=t+2623130​.

Problem 25911

Problem 25912

Find the 4th derivative of f(x)=cos⁡(sin⁡(x))f(x) = \cos(\sin(x))f(x)=cos(sin(x)) at x=0x=0x=0 using Taylor series: f(4)(0)f^{(4)}(0)f(4)(0).

Problem 25913

Estimate cardiac output RRR using A=4mgA=4 \mathrm{mg}A=4mg and c(t)c(t)c(t) values from t=0t=0t=0 to t=10t=10t=10. Calculate RRR in L/min\mathrm{L} / \mathrm{min}L/min.

Problem 25914

Evaluate the integral ∫22x2x4−61x2+900dx\int \frac{22 x^{2}}{x^{4}-61 x^{2}+900} d x∫x4−61x2+90022x2​dx and find its partial fraction decomposition.

Problem 25915

Find where the function h(x)=5+5sin⁡xcos⁡xh(x)=\frac{5+5 \sin x}{\cos x}h(x)=cosx5+5sinx​ is continuous and analyze the limits: lim⁡x→π/2−h(x)\lim _{x \rightarrow \pi / 2^{-}} h(x)limx→π/2−​h(x) and lim⁡x→4π/3h(x)\lim _{x \rightarrow 4 \pi / 3} h(x)limx→4π/3​h(x).

Problem 25916

Find the limit: lim⁡x→π/2−5+5sin⁡xcos⁡x\lim _{x \rightarrow \pi / 2^{-}} \frac{5+5 \sin x}{\cos x}limx→π/2−​cosx5+5sinx​.

Problem 25917

Calculate the first four approximations of the largest positive root of x3+x=17x^{3}+x=17x3+x=17 using Newton's Method, starting with x0=3x_{0}=3x0​=3. Round to six decimal places.

Problem 25918

Evaluate the integral: ∫54x2x4−45x2+324dx\int \frac{54 x^{2}}{x^{4}-45 x^{2}+324} d x∫x4−45x2+32454x2​dx. Find the result: ∫54x2x4−45x2+324dx=□\int \frac{54 x^{2}}{x^{4}-45 x^{2}+324} d x=\square∫x4−45x2+32454x2​dx=□.

Problem 25919

Evaluate the integral: ∫42x2x4−29x2+100dx=□\int \frac{42 x^{2}}{x^{4}-29 x^{2}+100} d x=\square∫x4−29x2+10042x2​dx=□

Problem 25920

Find the limit: lim⁡x→4π/35+5sin⁡xcos⁡x\lim _{x \rightarrow 4 \pi / 3} \frac{5+5 \sin x}{\cos x}limx→4π/3​cosx5+5sinx​.

Problem 25921

Evaluate the integral: ∫2z3−3z2−17z+23z2−z−12dz\int \frac{2 z^{3}-3 z^{2}-17 z+23}{z^{2}-z-12} d z∫z2−z−122z3−3z2−17z+23​dz.

Problem 25922

Find the derivative of y with respect to θ for y = ln(8 cos^6(θ)).

Problem 25923

Find the length xxx that maximizes the area of a rectangular playground using 120 meters of fencing for three sides.

Problem 25924

Find the absolute extreme values of f(x)=2x2+2cos⁡−1xf(x)=2 x^{2}+2 \cos ^{-1} xf(x)=2x2+2cos−1x on the interval [−1,1][-1,1][−1,1].

Problem 25925

Find the absolute extreme values of f(x)=5(4x)xf(x)=5(4x)^{x}f(x)=5(4x)x on the interval [0.05,1][0.05,1][0.05,1].

Problem 25926

Determine if f(x)=7x4−6x2−3f(x)=7 x^{4}-6 x^{2}-3f(x)=7x4−6x2−3 has a real zero between 1 and 2 using the Intermediate Value Theorem.

Problem 25927

Check if f(x)=7x4−6x2−3f(x)=7 x^{4}-6 x^{2}-3f(x)=7x4−6x2−3 has a real zero between 1 and 2 using the Intermediate Value Theorem.

Problem 25928

Rolle's Theorem doesn't apply to f(x)=∣x∣f(x)=|x|f(x)=∣x∣ on [−a,a][-a, a][−a,a] for any a>0a>0a>0. Explain why this is the case.

Problem 25929

Find the absolute extreme values of f(x)=5csc⁡xf(x)=5 \csc xf(x)=5cscx on the interval [π4,3π4]\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right][4π​,43π​].

Problem 25930

Find the absolute extreme values of f(x)=x3e−xf(x)=x^{3} e^{-x}f(x)=x3e−x on the interval [−1,4][-1,4][−1,4].

Problem 25931

Find points c\mathrm{c}c where the Mean Value Theorem applies for f(x)=x3f(x)=x^{3}f(x)=x3 on [−13,13][-13,13][−13,13].

Problem 25932

Find points ccc where the Mean Value Theorem applies for f(x)=x3f(x)=x^{3}f(x)=x3 on [−4,4][-4,4][−4,4].

Problem 25933

Check if Rolle's theorem applies to f(x)=x(x−3)2f(x)=x(x-3)^{2}f(x)=x(x−3)2 on [0,3][0,3][0,3] and find the guaranteed point(s).

Problem 25934

Find the critical points of the function f(x)=(x−2)5f(x)=(x-2)^{5}f(x)=(x−2)5.

Problem 25935

Find ccc such that f(b)−f(a)b−a=f′(c)\frac{f(b)-f(a)}{b-a}=f^{\prime}(c)b−af(b)−f(a)​=f′(c) for f(x)=tan⁡−1xf(x)=\tan^{-1} xf(x)=tan−1x on [−3,3][-\sqrt{3}, \sqrt{3}][−3​,3​]. Round to the nearest thousandth.

Problem 25936

Determine the absolute max and min of F(x)=x3F(x)=\sqrt[3]{x}F(x)=3x​ for −27≤x≤27-27 \leq x \leq 27−27≤x≤27.

Problem 25937

Evaluate the integral ∫(x4−5x3−5) dx=C\int (x^{4}-5 x^{3}-5) \, dx = \mathrm{C}∫(x4−5x3−5)dx=C.

Problem 25938

Find ccc for the Mean Value Theorem with f(x)=tan⁡−1xf(x) = \tan^{-1}xf(x)=tan−1x on [−3,3][-\sqrt{3}, \sqrt{3}][−3​,3​]: f′(c)=f(3)−f(−3)23f'(c) = \frac{f(\sqrt{3}) - f(-\sqrt{3})}{2\sqrt{3}}f′(c)=23​f(3​)−f(−3​)​.

Problem 25939

Find values of ccc for f(x)=x2+4x+1f(x)=x^{2}+4x+1f(x)=x2+4x+1 in [2,3][2,3][2,3] satisfying f(3)−f(2)3−2=f′(c)\frac{f(3)-f(2)}{3-2}=f^{\prime}(c)3−2f(3)−f(2)​=f′(c).

Problem 25940

Calculate the total revenue from the investment stream R(t)=400e0.08tR(t)=400 e^{0.08 t}R(t)=400e0.08t between 4 and 6 years. Round to the nearest cent.

Find the displacement and distance traveled of a particle with velocity $v(t)=3 t^{2}-42 t+135$ over $[0,10]$ .

Find the critical points of the function $f(x)=\frac{e^{x}+e^{-x}}{6}$ .

Find the critical points of the function $f(x)=\frac{1}{x}-\ln x$ .

Approximate $\int_{2}^{6} f(x) dx$ using a Riemann sum with 2 rectangles and left endpoints. Values: 4, -2, -3.

Evaluate the integral: $\int \frac{48 x^{2}}{x^{4}-26 x^{2}+25} dx$ and find its partial fraction decomposition.

Find the absolute extreme values of $f(x)=24 x^{\frac{1}{2}}-x$ on the interval $[0,576]$ .

Find the min/max, domain, range, and intervals of increase/decrease for $f(x)=4x^2+16x-3$ and $g(x)=-x^2+5x+9$ .

Find the absolute extreme values of $f(x)=2 \sin ^{2} x$ on the interval $[0, \pi]$ .

Estimate total water drained in 3 minutes using average of left- and right-endpoint approximations from $v(t)$ values: $44, 40, 36, 32, 28, 24, 20$ .

Estimate the total extinct marine families from $t=514$ to $t=524$ using $M_{10}$ with $r(t)=\frac{3130}{t+262}$ .

Find the 4th derivative of $f(x) = \cos(\sin(x))$ at $x=0$ using Taylor series: $f^{(4)}(0)$ .

Estimate cardiac output $R$ using $A=4 \mathrm{mg}$ and $c(t)$ values from $t=0$ to $t=10$ . Calculate $R$ in $\mathrm{L} / \mathrm{min}$ .

Evaluate the integral $\int \frac{22 x^{2}}{x^{4}-61 x^{2}+900} d x$ and find its partial fraction decomposition.

Find where the function $h(x)=\frac{5+5 \sin x}{\cos x}$ is continuous and analyze the limits: $\lim _{x \rightarrow \pi / 2^{-}} h(x)$ and $\lim _{x \rightarrow 4 \pi / 3} h(x)$ .

Find the limit: $\lim _{x \rightarrow \pi / 2^{-}} \frac{5+5 \sin x}{\cos x}$ .

Calculate the first four approximations of the largest positive root of $x^{3}+x=17$ using Newton's Method, starting with $x_{0}=3$ . Round to six decimal places.

Evaluate the integral: $\int \frac{54 x^{2}}{x^{4}-45 x^{2}+324} d x$ . Find the result: $\int \frac{54 x^{2}}{x^{4}-45 x^{2}+324} d x=\square$ .

Evaluate the integral: $\int \frac{42 x^{2}}{x^{4}-29 x^{2}+100} d x=\square$

Find the limit: $\lim _{x \rightarrow 4 \pi / 3} \frac{5+5 \sin x}{\cos x}$ .

Evaluate the integral: $\int \frac{2 z^{3}-3 z^{2}-17 z+23}{z^{2}-z-12} d z$ .

Find the length $x$ that maximizes the area of a rectangular playground using 120 meters of fencing for three sides.

Find the absolute extreme values of $f(x)=2 x^{2}+2 \cos ^{-1} x$ on the interval $[-1,1]$ .

Find the absolute extreme values of $f(x)=5(4x)^{x}$ on the interval $[0.05,1]$ .

Determine if $f(x)=7 x^{4}-6 x^{2}-3$ has a real zero between 1 and 2 using the Intermediate Value Theorem.

Check if $f(x)=7 x^{4}-6 x^{2}-3$ has a real zero between 1 and 2 using the Intermediate Value Theorem.

Rolle's Theorem doesn't apply to $f(x)=|x|$ on $[-a, a]$ for any $a>0$ . Explain why this is the case.

Find the absolute extreme values of $f(x)=5 \csc x$ on the interval $\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right]$ .

Find the absolute extreme values of $f(x)=x^{3} e^{-x}$ on the interval $[-1,4]$ .

Find points $\mathrm{c}$ where the Mean Value Theorem applies for $f(x)=x^{3}$ on $[-13,13]$ .

Find points $c$ where the Mean Value Theorem applies for $f(x)=x^{3}$ on $[-4,4]$ .

Check if Rolle's theorem applies to $f(x)=x(x-3)^{2}$ on $[0,3]$ and find the guaranteed point(s).

Find the critical points of the function $f(x)=(x-2)^{5}$ .

Find $c$ such that $\frac{f(b)-f(a)}{b-a}=f^{\prime}(c)$ for $f(x)=\tan^{-1} x$ on $[-\sqrt{3}, \sqrt{3}]$ . Round to the nearest thousandth.

Determine the absolute max and min of $F(x)=\sqrt[3]{x}$ for $-27 \leq x \leq 27$ .

Evaluate the integral $\int (x^{4}-5 x^{3}-5) \, dx = \mathrm{C}$ .

Find $c$ for the Mean Value Theorem with $f(x) = \tan^{-1}x$ on $[-\sqrt{3}, \sqrt{3}]$ : $f'(c) = \frac{f(\sqrt{3}) - f(-\sqrt{3})}{2\sqrt{3}}$ .

Find values of $c$ for $f(x)=x^{2}+4x+1$ in $[2,3]$ satisfying $\frac{f(3)-f(2)}{3-2}=f^{\prime}(c)$ .

Calculate the total revenue from the investment stream $R(t)=400 e^{0.08 t}$ between 4 and 6 years. Round to the nearest cent.

Find the absolute extreme values of $h(x)=\frac{1}{3} x+5$ on the interval $-3 \leq x \leq 4$ .

Interpret $P(20)=80$ and $\left.\frac{d P}{d x}\right|_{x=20}=2$ for advertising profit in thousands.

Find the absolute max and min of $f(x)=5 x^{2 / 3}$ on the interval $[-27, 1]$ .

Find the derivative of $G(x)=(3x^{2}+4x-5)^{6}$ .

Find $c$ such that $\frac{f(7)-f(5)}{7-5}=f'(c)$ for $f(x)=\ln(x-4)$ on $[5,7]$ . Round to the nearest thousandth.

Find the maximum and minimum values of the function $f(x)=x^{2}+x-4$ on the interval $[-4,4]$ .

Find the area between the curves $S(t)=3.5t+9$ and $C(t)=5.2t$ for $t$ years, and interpret the savings in \$. Round answers accordingly.

Bestimme den Tag $x$ , an dem die Funktion $f(x)=(2 x+1) \cdot e^{-0,02 x}$ ihr Maximum erreicht.

Find the limit: $\lim _{x \rightarrow 0^{+}}(9 x+1)^{\cot (x)}$ . Use l'Hospital's Rule if needed.

Find the light intensity $I$ (in thousands of foot-candles) that maximizes the photosynthesis rate $P=\frac{90 I}{I^{2}+I+1}$ .

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{f(4+3 x)+f(4+6 x)}{x}$ given $f(4)=0$ and $f^{\prime}(4)=8$ .

Find the area between the curves $S(t)=3.8t+9$ and $C(t)=5.4t$ for $t$ in years. Round to 2 decimal and nearest integer.

Find the function $f$ if $f'(x) = 1 - 6x$ and $f(0) = 8$ .

Find the general antiderivative of $f(x)=\frac{9-x^{3}+2 x^{5}}{x^{6}}$ .

Find $f(4)$ given $f(1)=12$ , $f^{\prime}$ is continuous, and $\int_{1}^{4} f^{\prime}(x) dx=17$ .

Find $\int_{5}^{9} f(x) d x$ given that $\int_{3}^{9} f(x) d x=8$ and $\int_{3}^{5} f(x) d x=5$ .

Evaluate the integral $\int_{0}^{2} \frac{1}{\sqrt[3]{x-1}} d x$ .

Find the dimensions (in cm) of a box with a square base and volume $13,500 \mathrm{~cm}^{3}$ that minimizes material use.

Find the function $f$ such that $f^{\prime \prime}(x)=10+6x+36x^{2}$ with conditions $f(0)=2$ and $f(1)=14$ .

Find $F^{\prime}(x)$ if $F(x)=\int_{\sin x}^{\cos x}(t+1)^{2024} dt$ .

Bestimme die Maße und maximale Fläche eines rechteckigen Tors in der Seitenwand $f(x)=\frac{1}{25} x^{4}-\frac{2}{3} x^{2}+\frac{9}{5}$ für $[-1,84 ; 1,84]$ .

Find the function $f$ given that $f''(x)=-2+36x-12x^2$ , $f(0)=2$ , and $f'(0)=16$ .

Evaluate the integral from -1 to 2: $\int_{-1}^{2}(x-|x|) d x$ .

Find the general antiderivative of $f(x)=(x+7)(2x-3)$ and check by differentiating. Use $C$ for the constant.

Find the general antiderivative of $f(x)=3 \sqrt{x}+9 \cos (x)$ and check by differentiating. Use $C$ .

Find a function $f$ where $f'(x)=3x^3$ and the line $81x+y=0$ is tangent to $f$ . What is $f(x)=?$

Find the function $f$ such that $f''(x)=x^{-2}$ for $x>0$ , with conditions $f(1)=0$ and $f(7)=0$ .

Determine if the sequence $a_{n}=\sqrt{n+1}-\sqrt{n}$ converges or diverges. If it converges, find the limit.

Find the position of a particle, $s(t)$ , given $a(t)=t^{2}-7t+5$ , $s(0)=0$ , and $s(1)=20$ .

Find $f$ given that $f(x)=f^{\prime \prime}(x)=\frac{8}{9} x^{8 / 9}$ . Use constants $C$ and $D$ .

Check if the series $\sum_{n=1}^{\infty} \frac{\ln n}{n^{4}}$ converges or diverges.