Calculus

Problem 11801

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

See Solution

Problem 11802

Calculate the integral: $\int\left(\frac{5}{\sqrt{x}}+5 \sqrt{x}\right) dx$ .

See Solution

Problem 11803

Das Profil einer Skischanze wird durch $f(x)=\frac{1}{120} x^{2}-x+60$ (für $0 \leq x \leq 30$ ) beschrieben. Berechne die mittlere Steigung.

See Solution

Problem 11804

Find the first and second derivatives of $f(x)=\ln(x^{2}+1)$ , its domain, critical numbers, concavity intervals, and inflection points.

See Solution

Problem 11805

Find the integral: $\int\left(4 x^{\frac{1}{3}}+2 x^{-\frac{2}{3}}+11\right) d x$ .

See Solution

Problem 11806

Find the indefinite integral: $\int \frac{3 x^{3}+4 x^{2}}{x} d x$ .

See Solution

Problem 11807

Given $f(x)=\frac{1}{6-x}$ , find critical points, intervals of increase/decrease, asymptotes, and concavity for graphing.

See Solution

Problem 11808

Given $f(x)=\frac{1}{6-x}$ , find its derivatives, critical numbers, domain, asymptotes, and graph $f(x)$ .

See Solution

Problem 11809

Ein ICE hat das Weg-Zeit-Gesetz $s(t)=0,7 t^{2}$ . Berechne: a) Weg in 20 s und mittlere Geschwindigkeit, b) mittlere Geschwindigkeit in [19; 20], c) in [19,99; 20], d) Zeit für 300 km/h.

See Solution

Problem 11810

Calculate the volume of the solid formed by rotating the area between $y = Vx + 1$ , $y = 0$ , $x=0$ , $x=3$ around the x-axis.

See Solution

Problem 11811

Berechnen Sie das Volumen des Rotationskörpers, der durch $y=\sqrt{x+1}$ , $y=0$ , $x=0$ , und $x=3$ um die x-Achse entsteht.

See Solution

Problem 11812

Find where the function $f(x)=\frac{x^{2}-35}{x-6}, x \neq 6$ is increasing/decreasing and identify local extrema.

See Solution

Problem 11813

How many years for Strontium-90 from Chernobyl to reduce by $98.7\%$ ? Use its half-life of $30$ years. Round to the nearest year.

See Solution

Problem 11814

Given that $f^{\prime}(x)$ is continuous and $f(x)$ has a local extreme at $x=9$ , find $f^{\prime}(2)$ , $f^{\prime}(9)$ , $f^{\prime}(15)$ . Is $f(x)$ increasing or decreasing before and after $x=9$ , and is it a max or min?

See Solution

Problem 11815

Find the first and second derivatives of $f(x)=x^{3}+3x^{2}-105x+24$ . Determine intervals where $f$ is increasing, decreasing, concave up, and concave down.

See Solution

Problem 11816

Analyze the function with derivative $f'(x)=(x+3)e^{-2x}$ to find where $f$ is increasing or decreasing and locate local max/min points.

See Solution

Problem 11817

Find the derivative of the function $f(t) = 12 e^{-t}$ .

See Solution

Problem 11818

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\left(\sin ^{-1}(x+3)\right)^{5}$ .

See Solution

Problem 11819

Find the local minimum and maximum of $f(x)=2 x^{3}-33 x^{2}+60 x+2$ . Calculate $f^{\prime}(x)$ and find $x$ and $y$ values.

See Solution

Problem 11820

Find $\left(f^{-1}\right)^{\prime}(1)$ for the one-to-one function $f(x)=2 x+\cos (x)$ using the formula $\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}(x)\right)}$ .

See Solution

Problem 11821

Find the first time $t > 0$ when the velocity of the particle, given by $x(t)=12 e^{-t} \sin t$ , is zero.

See Solution

Problem 11822

Find the acceleration of a particle at $t=3$ given its velocity $v(t)=7-(1.01)^{-t^{2}}$ .

See Solution

Problem 11823

Find the local minimum and maximum of $f(x)=2 x^{3}-39 x^{2}+216 x+4$ . Calculate $f^{\prime}(x)$ and the values at those points.

See Solution

Problem 11824

Find the lengths of two curves from the list below. Only submit two.
(a) $y=\frac{x^{3}}{6}+\frac{1}{2 x}$ for $1 \leq x \leq 2$ . (b) $y=\frac{x^{2}}{2}-\frac{\ln x}{4}$ for $2 \leq x \leq 4$ . (c) $y=\ln(1-x^{2})$ for $0 \leq x \leq \frac{1}{2}$ . (d) $x=\frac{4}{3} t^{\frac{3}{2}}, y=\frac{1}{2}(t-1)^{2}$ for $0 \leq t \leq 4$ . (e) $x=r(t-\sin t), y=r(1-\cos t)$ for $0 \leq t \leq 2 \pi$ . ( $r$ is a constant.)

See Solution

Problem 11825

Find when the speed of the particle, given by $s(t)=2 t^{3}-24 t^{2}+90 t+7$ , is increasing for $t \geq 0$ .

See Solution

Problem 11826

Find when the speed of a particle, with velocity $v(t)=-t^{3}+2 t^{2}+2^{-t}$ for $t \geq 0$ , is increasing.

See Solution

Problem 11827

Verify Theorem 1 for $f=2z^2-x^2-y^2$ over the box $0 \leq x \leq 1, 0 \leq y \leq 2, 0 \leq z \leq 4$ and for $f=x^2-y^2$ over the cylinder $x^2+y^2 \leq 4$ , $0 \leq z \leq 1$ .

See Solution

Problem 11828

Show that for harmonic functions $f$ and $g$ in domain $D$ , $\iint_{S} g \frac{\partial g}{\partial n} d A = \iiint_{T} |\operatorname{grad} g|^{2} d V$ .

See Solution

Problem 11829

Find the moment of inertia for a rectangle ( $0 \leq x \leq L$ , $0 \leq y \leq 2L$ ) rotated about the $z$ -axis. Calculate $M$ , $dI$ , integrate for $I$ , and express $I$ as $I = C M L^2$ . What is $C$ (to two decimal places)?

See Solution

Problem 11830

Sand leaks from a bag modeled by $S(t)=K\left(1-\frac{t^{2}}{\mu}\right)^{3}$ .
1. (a) Find initial sand amount. (b) Determine leak rate at time $t$ . (c) For $\mu=6$ , analyze leak rate at $t=2$ (speeding up or slowing down).

See Solution

Problem 11831

Calculate the limit as $n$ approaches infinity: $\operatorname{Lim}_{n \rightarrow \infty} \frac{7 n^{3}+8 n^{5}-n^{2}+1}{2-n^{2}-3 n^{3}+4 n^{4}-5 n^{5}}$

See Solution

Problem 11832

Find where the function $f(x)=e^{-x^{2} / 32}$ is concave up or down and identify inflection points.

See Solution

Problem 11833

A circular pool expands at $16 \pi \mathrm{in}^{2} / \mathrm{sec}$ . Find the radius expansion rate when the radius is 4 inches.

See Solution

Problem 11834

Gegeben ist die Funktion $f(x)=x^{2}-3 x$ . Skizzieren Sie den Graphen für $-1 \leq x \leq 4$ und bestimmen Sie Steigung und Winkel bei $x=2$ .

See Solution

Problem 11835

A balloon expands at $60 \pi \mathrm{in}^{3}/\mathrm{sec}$ . Find the surface area growth rate when the radius is 4 in.

See Solution

Problem 11836

Evaluate the triple integral $\iiint_{E} f(x, y, z) d V$ where $f(x, y, z)=e^{\sqrt{x^{2}+y^{2}}}$ over $E=\{(x, y, z) \mid 1 \leq x^{2}+y^{2} \leq 9, y \leq 0, x \leq y \sqrt{3}, 4 \leq z \leq 5\}$ .

See Solution

Problem 11837

Rewrite the function $f(x, y, z)=x+y$ and region $E$ in spherical coordinates, then convert and evaluate the integral $\iiint_{E} f dV$ .

See Solution

Problem 11838

Evaluate the integral $\int_{0}^{\frac{\pi}{5}} \frac{3}{\cos ^{2} 7 x} d x$ .

See Solution

Problem 11839

Convert the integral to spherical coordinates:
$\int_{-9}^{9} \int_{0}^{\sqrt{81-x^{2}}} \int_{-\sqrt{81-x^{2}}}^{\sqrt{81-x^{2}}-y^{2}} (x^{2}+y^{2}+z^{2})^{2} dz \, dy \, dx$
using $x = \rho\sin\varphi\cos\theta$ , $y = \rho\sin\varphi\sin\theta$ , $z = \rho\cos\varphi$ , and $dV = \rho^2\sin\varphi d\rho d\varphi d\theta$ .

See Solution

Problem 11840

Is the integral range from 0 to 5 correct for the potential $v=2 \pi \sigma \int_{0}^{5} \frac{R}{R} d R$ with $\sigma=1.25$ and $r=0.025 \mathrm{~m}$ ?

See Solution

Problem 11841

1. A moving object's kinetic energy is $E=\frac{1}{2} m v^{2}$ . Find $P=\frac{dE}{dt}$ using $\frac{dm}{dt}$ , units of $P$ ? Show $P=\frac{1}{2} \rho A v^{3}$ , then calculate $P$ for a turbine with blades $3 \mathrm{~m}$ and $v=10 \mathrm{~m/s}$ .

See Solution

Problem 11842

Calculate the integral for the electrostatic potential on a disc:
$v=2 \pi \sigma \int_{0}^{5} \frac{R}{\sqrt{R^{2}+0.025^{2}}} d R$
with $\sigma=1.25$ .

See Solution

Problem 11843

Estimate $2.004^{5}$ using the linearization of $f(x)=x^{5}$ at $x=2$ . What is the slope and linearization?

See Solution

Problem 11844

Estimate $2004^{5}$ using the linearization of $f(x)=x^{5}$ at 2. Find the slope and tangent line. Then estimate.

See Solution

Problem 11845

Evaluate the integral $\iint_{R}(x^{2}+16 y^{2})^{2} d A$ over the region $R$ bounded by the ellipse $x^{2}+16 y^{2}=1$ using the transformation $x=u, 4y=v$ .

See Solution

Problem 11846

Find the derivative of $f(x)=x(2x-7)^{3}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 11847

Find the derivative of $f(t)=\frac{e^{t^5+3}+3}{e^{3t}}$ .

See Solution

Problem 11848

Find the derivative of the function $f(t)=\frac{e^{5+3}+3}{e^{5}}$ .

See Solution

Problem 11849

Find the price $p$ that maximizes revenue using the elasticity of demand $E=\frac{p}{360-5 p}$ . Round to the nearest dollar.

See Solution

Problem 11850

A lighthouse is $500 \mathrm{~m}$ from shore, rotating at 4 rev/min. Find the beam's speed at point $Q$ ( $200 \mathrm{~m}$ left of $P$ ). How does this speed change between $P$ and $Q$ ?

See Solution

Problem 11851

A 48 cm wire is cut into two pieces to form squares.
(a) Find the area function $A(x)$ for side length $x$ . (b) What side length $x$ minimizes the area? (c) What is the minimum area?

See Solution

Problem 11852

Find the limit as $x$ approaches -1 for $\frac{x^{2}-1}{2x^{2}+3x+1}$ .

See Solution

Problem 11853

Find the derivative of the function $f(t)=\frac{4t}{\ln(4t)}$ . What is $f'(t)$ ?

See Solution

Problem 11854

Find the limit as $x$ approaches -1 for $\frac{x^{2}-1}{2 x^{2}+3 x+1}$ . Enter the result or "inf"/"-inf".

See Solution

Problem 11855

Find the derivative of the function $f(t)=\frac{t^{7}-t^{2}+4 t}{\ln (4 t)}$ . What is $f^{\prime}(t)$ ?

See Solution

Problem 11856

Show that the equation $5-e^{\sin(3x)}=2x^2$ has a solution using the Intermediate Value Theorem. Define $f(x)$ and find $[a, b]$ where $f(a)<0$ and $f(b)>0$ .

See Solution

Problem 11857

Find the limit: $\lim _{x \rightarrow 4} \frac{7-x}{(x-4)^{2}}$ .

See Solution

Problem 11858

Find the limit as $x$ approaches 4 for the expression $\frac{7-x}{(x-4)^{2}}$ .

See Solution

Problem 11859

Find where the function $f(x)=2 x^{4}+48 x^{3}+432 x^{2}-12 x-7$ is concave up or down and identify inflection points.

See Solution

Problem 11860

Compute the derivatives: a) $f(x)=\sin(2x+3)$ , b) $g(x)=\frac{e^{3x}}{x^2+1}$ , c) $h(x)=(2x+3)^{2x+3}$ .

See Solution

Problem 11861

Find where the function $f(x)=2 x^{4}+48 x^{3}+432 x^{2}-12 x-7$ is concave up or down, and identify inflection points.

See Solution

Problem 11862

Estimate $\ln(6.007)$ using the linearization of $f(x)=\ln(x)$ at $x=6$ . Find the slope and tangent line equation.

See Solution

Problem 11863

Find where the function $f(x)=2 x^{4}+48 x^{3}+432 x^{2}-12 x-7$ is concave up or down and identify inflection points.

See Solution

Problem 11864

Find where the function $f(x)=2 x^{4}+24 x^{3}+108 x^{2}-24 x-5$ is concave up or down and identify inflection points.

See Solution

Problem 11865

Find local maxima and minima, and determine intervals of increase and decrease for $y=x^{3}-12x+2$ .

See Solution

Problem 11866

Estimate $\ln(6.007)$ using the linearization of $f(x)=\ln(x)$ at $x=6$ .
a) Find the slope at $x=6$ . b) Determine the tangent line equation: $L(x)=$ . c) Estimate $\ln(6.007)$ using $L\left(6+\frac{7}{1000}\right)=$ .
The variables found in your answer wire $[x]$ .

See Solution

Problem 11867

Cost Function: Given $C(x)=5 \cdot x^{2}+5 \cdot x+125$ , find $\hat{C}_{n}$ , $C^{\prime}(x)$ , and estimate cost for 6th item.

See Solution

Problem 11868

Differentiate implicitly to find $\frac{d y}{d x}$ for the equation $x^{4}+y^{4}=16 x y$ .

See Solution

Problem 11869

Given the function $f(x)=e^{9 x}+e^{-x}$ , find where $f$ is increasing and decreasing, and determine its local minimum value.

See Solution

Problem 11870

Find $a$ such that $\int_{a}^{a^{2}} (a x^{2}-7 a^{2} x-\frac{7 a^{3}}{3}) \, dx = 0$ , with $a$ not an integer.

See Solution

Problem 11871

Given the demand function $q=D(p)=\frac{600}{(p+8)^{5}}$ , find: a) elasticity, b) elasticity at $p=6$ (elastic, inelastic, unit), c) $p$ for max total revenue.

See Solution

Problem 11872

Déterminez la dérivée $C^{\prime}(t)$ de la fonction de coût $C(t)$ pour la publicité télévisée pendant 1980-2010.

See Solution

Problem 11873

Coût d'une pub télé de 30s de 1980 à 2010 : $C(t)=\begin{cases}50t+325 & 0 \leq t<15 \\ 150t-1175 & 15 \leq t \leq 30\end{cases}$ . a) $C(t)$ est-elle continue? b) Trouver $C'(t)$ .

See Solution

Problem 11874

Déterminez la longévité à long terme $s(t)$ donnée par $s(t)=\frac{1105}{14+13 \times 2^{-0.04 t}}$ pour $t \geq 0$ .

See Solution

Problem 11875

Fonction de coût $C(q)=m q+b$ . Trouvez $\lim_{q \to \infty} \bar{C}(q)=\lim_{q \to \infty} \frac{C(q)}{q}$ . Est-ce que la règle de l'Hôpital s'applique ? Justifiez. Interprétez le résultat.

See Solution

Problem 11876

Evaluate $f(1)$ and $f(2)$ for $f(x)=-x^{3}+2x^{2}-2x+7$ . Does the Intermediate Value Theorem confirm a zero between 1 and 2? YES $\bigcirc$ NO

See Solution

Problem 11877

Find $f(2)$ and $f(3)$ for $f(x)=x^{3}-5x^{2}+8x-8$ . Does the IVT ensure a zero between $x=2$ and $x=3$ ? $\text { YES } \bigcirc \text { NO }$

See Solution

Problem 11878

Evaluate $f(1)$ and $f(2)$ for $f(x)=-x^{3}+2x^{2}-2x+7$ . Does the IVT confirm a zero between 1 and 2?

See Solution

Problem 11879

Calculate the area under the curve $y=\frac{5}{\sqrt{x}}$ from $x=1$ to $x=25$ .

See Solution

Problem 11880

Find the area between $y=e^{x}$ , $y=0$ , $x=1$ , and $x=6$ . Use a graphing tool to check your answer.

See Solution

Problem 11881

Calculate the area under the curve $y=\frac{5}{\sqrt{x}}$ for all values of $x$ on the x-axis.

See Solution

Problem 11882

Find the critical point of the function $f(x)=e^{x}(x-3)$ . Enter the value of $x$ .

See Solution

Problem 11883

Find the absolute maximum and minimum of $f(x)=x-\frac{64 x}{x+4}$ on $[0,13]$ . What are the values of $x$ ?

See Solution

Problem 11884

Find the change in revenue $R$ when $x$ increases by 4 from $x=13$ , given $\frac{d R}{d x}=83-4x$ .

See Solution

Problem 11885

Given $f^{\prime \prime}(x)=2 x-1$ and $x=1$ is critical, determine if it's a local/absolute min/max using the Second Derivative Test.

See Solution

Problem 11886

Find the net change and average rate of change between points $(-1,1)$ and $(5,4)$ .
(a) Net change: $4 - 1$
(b) Average rate: $\frac{4 - 1}{5 - (-1)}$

See Solution

Problem 11887

Find the intervals where the function $f(x)=\frac{x^{4}}{4}-3 x^{3}-1$ is concave down: $(0,6)$ , $(-\infty, 0)$ , $(-\infty, 6)$ , $(6, \infty)$ , or never.

See Solution

Problem 11888

If $f(x)$ has a local minimum at $x=c$ with $f^{\prime}(x)<0$ for $x<c$ and $f^{\prime}(x)>0$ for $x>c$ , then $x=c$ is: - an absolute minimum - an absolute maximum - a local maximum - a local minimum

See Solution

Problem 11889

Is it TRUE or FALSE that $f^{\prime \prime}(p)=0$ implies $p$ is an inflection point of $f$ ?

See Solution

Problem 11890

Find the inflection point(s) of $f(x)=x^{3/5}(x-3)$ : $x=\frac{-3}{4}$ , $x=0$ , $x=3$ , $x=\frac{9}{8}$ , or none?

See Solution

Problem 11891

Find the critical numbers of the function $g(x)=x^{1/9}-x^{-8/9}$ at $x=-8$ .

See Solution

Problem 11892

Find the absolute max and min of $f(x)=x-\frac{64 x}{x+4}$ on $[0,13]$ . What are the $x$ values for each?

See Solution

Problem 11893

Given a twice differentiable function $g(x)$ , which statement about $f(x)=g(x)-x^{3}$ is TRUE?

See Solution

Problem 11894

Given $g(0)=2$ , $g'(0)=-1$ , $g''(0)=-3$ , determine the truth about $f(0) = \sin(0) + g(0)$ .

See Solution

Problem 11895

Find the inflection point(s) of $f(x)=x^{3/5}(x-3)$ . Choose from: $x=\frac{-3}{4}$ , $x=0$ , $x=3$ , $x=\frac{9}{8}$ , or none.

See Solution

Problem 11896

Find the local maximum and minimum of the function $f(x)=x^{3}-48x$ . What are the $x$ values for each?

See Solution

Problem 11897

Find the absolute max and min of $f(x)=x-\frac{64 x}{x+4}$ on $[0,13]$ . What are the $x$ values?

See Solution

Problem 11898

Show why $0$ is a critical number for $h(t)=t^{3/4}-6t^{1/4}$ . Should you test $0$ in the function or its derivative? Explain the exponent's sign role.

See Solution

Problem 11899

Find the tangent line equation to $y=3 x^{2}-2 x+5$ at $x=1$ . Choose from the options given.

See Solution

Problem 11900

Find the derivative of $f(x) = x^{5} e^{-7x}$ and show that the derivative is undefined at $x=0$ .

See Solution

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Calculus

Problem 11801

Evaluate the integral: ∫(2x+2x)dx\int\left(\frac{2}{\sqrt{x}}+2 \sqrt{x}\right) d x∫(x​2​+2x​)dx

Problem 11802

Calculate the integral: ∫(5x+5x)dx\int\left(\frac{5}{\sqrt{x}}+5 \sqrt{x}\right) dx∫(x​5​+5x​)dx.

Problem 11803

Das Profil einer Skischanze wird durch f(x)=1120x2−x+60f(x)=\frac{1}{120} x^{2}-x+60f(x)=1201​x2−x+60 (für 0≤x≤300 \leq x \leq 300≤x≤30) beschrieben. Berechne die mittlere Steigung.

Problem 11804

Find the first and second derivatives of f(x)=ln⁡(x2+1)f(x)=\ln(x^{2}+1)f(x)=ln(x2+1), its domain, critical numbers, concavity intervals, and inflection points.

Problem 11805

Find the integral: ∫(4x13+2x−23+11)dx\int\left(4 x^{\frac{1}{3}}+2 x^{-\frac{2}{3}}+11\right) d x∫(4x31​+2x−32​+11)dx.

Problem 11806

Find the indefinite integral: ∫3x3+4x2xdx\int \frac{3 x^{3}+4 x^{2}}{x} d x∫x3x3+4x2​dx.

Problem 11807

Given f(x)=16−xf(x)=\frac{1}{6-x}f(x)=6−x1​, find critical points, intervals of increase/decrease, asymptotes, and concavity for graphing.

Problem 11808

Given f(x)=16−xf(x)=\frac{1}{6-x}f(x)=6−x1​, find its derivatives, critical numbers, domain, asymptotes, and graph f(x)f(x)f(x).

Problem 11809

Ein ICE hat das Weg-Zeit-Gesetz s(t)=0,7t2s(t)=0,7 t^{2}s(t)=0,7t2. Berechne: a) Weg in 20 s und mittlere Geschwindigkeit, b) mittlere Geschwindigkeit in [19; 20], c) in [19,99; 20], d) Zeit für 300 km/h.

Problem 11810

Calculate the volume of the solid formed by rotating the area between y=Vx+1y = Vx + 1y=Vx+1, y=0y = 0y=0, x=0x=0x=0, x=3x=3x=3 around the x-axis.

Problem 11811

Berechnen Sie das Volumen des Rotationskörpers, der durch y=x+1y=\sqrt{x+1}y=x+1​, y=0y=0y=0, x=0x=0x=0, und x=3x=3x=3 um die x-Achse entsteht.

Problem 11812

Find where the function f(x)=x2−35x−6,x≠6f(x)=\frac{x^{2}-35}{x-6}, x \neq 6f(x)=x−6x2−35​,x=6 is increasing/decreasing and identify local extrema.

Problem 11813

How many years for Strontium-90 from Chernobyl to reduce by 98.7%98.7\%98.7%? Use its half-life of 303030 years. Round to the nearest year.

Problem 11814

Given that f′(x)f^{\prime}(x)f′(x) is continuous and f(x)f(x)f(x) has a local extreme at x=9x=9x=9, find f′(2)f^{\prime}(2)f′(2), f′(9)f^{\prime}(9)f′(9), f′(15)f^{\prime}(15)f′(15). Is f(x)f(x)f(x) increasing or decreasing before and after x=9x=9x=9, and is it a max or min?

Problem 11815

Find the first and second derivatives of f(x)=x3+3x2−105x+24f(x)=x^{3}+3x^{2}-105x+24f(x)=x3+3x2−105x+24. Determine intervals where fff is increasing, decreasing, concave up, and concave down.

Problem 11816

Analyze the function with derivative f′(x)=(x+3)e−2xf'(x)=(x+3)e^{-2x}f′(x)=(x+3)e−2x to find where fff is increasing or decreasing and locate local max/min points.

Problem 11817

Find the derivative of the function f(t)=12e−tf(t) = 12 e^{-t}f(t)=12e−t.

Problem 11818

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=(sin⁡−1(x+3))5f(x)=\left(\sin ^{-1}(x+3)\right)^{5}f(x)=(sin−1(x+3))5.

Problem 11819

Find the local minimum and maximum of f(x)=2x3−33x2+60x+2f(x)=2 x^{3}-33 x^{2}+60 x+2f(x)=2x3−33x2+60x+2. Calculate f′(x)f^{\prime}(x)f′(x) and find xxx and yyy values.

Problem 11820

Find (f−1)′(1)\left(f^{-1}\right)^{\prime}(1)(f−1)′(1) for the one-to-one function f(x)=2x+cos⁡(x)f(x)=2 x+\cos (x)f(x)=2x+cos(x) using the formula (f−1)′(x)=1f′(f−1(x))\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}(x)\right)}(f−1)′(x)=f′(f−1(x))1​.

Problem 11821

Find the first time t>0t > 0t>0 when the velocity of the particle, given by x(t)=12e−tsin⁡tx(t)=12 e^{-t} \sin tx(t)=12e−tsint, is zero.

Problem 11822

Find the acceleration of a particle at t=3t=3t=3 given its velocity v(t)=7−(1.01)−t2v(t)=7-(1.01)^{-t^{2}}v(t)=7−(1.01)−t2.

Problem 11823

Find the local minimum and maximum of f(x)=2x3−39x2+216x+4f(x)=2 x^{3}-39 x^{2}+216 x+4f(x)=2x3−39x2+216x+4. Calculate f′(x)f^{\prime}(x)f′(x) and the values at those points.

Problem 11824

Problem 11825

Find when the speed of the particle, given by s(t)=2t3−24t2+90t+7s(t)=2 t^{3}-24 t^{2}+90 t+7s(t)=2t3−24t2+90t+7, is increasing for t≥0t \geq 0t≥0.

Problem 11826

Find when the speed of a particle, with velocity v(t)=−t3+2t2+2−tv(t)=-t^{3}+2 t^{2}+2^{-t}v(t)=−t3+2t2+2−t for t≥0t \geq 0t≥0, is increasing.

Problem 11827

Problem 11828

Show that for harmonic functions fff and ggg in domain DDD, ∬Sg∂g∂ndA=∭T∣grad⁡g∣2dV\iint_{S} g \frac{\partial g}{\partial n} d A = \iiint_{T} |\operatorname{grad} g|^{2} d V∬S​g∂n∂g​dA=∭T​∣gradg∣2dV.

Problem 11829

Find the moment of inertia for a rectangle (0≤x≤L0 \leq x \leq L0≤x≤L, 0≤y≤2L0 \leq y \leq 2L0≤y≤2L) rotated about the zzz-axis. Calculate MMM, dIdIdI, integrate for III, and express III as I=CML2I = C M L^2I=CML2. What is CCC (to two decimal places)?

Problem 11830

Sand leaks from a bag modeled by S(t)=K(1−t2μ)3S(t)=K\left(1-\frac{t^{2}}{\mu}\right)^{3}S(t)=K(1−μt2​)3. 1. (a) Find initial sand amount. (b) Determine leak rate at time ttt. (c) For μ=6\mu=6μ=6, analyze leak rate at t=2t=2t=2 (speeding up or slowing down).

Problem 11831

Calculate the limit as nnn approaches infinity: Lim⁡n→∞7n3+8n5−n2+12−n2−3n3+4n4−5n5\operatorname{Lim}_{n \rightarrow \infty} \frac{7 n^{3}+8 n^{5}-n^{2}+1}{2-n^{2}-3 n^{3}+4 n^{4}-5 n^{5}}Limn→∞​2−n2−3n3+4n4−5n57n3+8n5−n2+1​

Problem 11832

Find where the function f(x)=e−x2/32f(x)=e^{-x^{2} / 32}f(x)=e−x2/32 is concave up or down and identify inflection points.

Problem 11833

A circular pool expands at 16πin2/sec16 \pi \mathrm{in}^{2} / \mathrm{sec}16πin2/sec. Find the radius expansion rate when the radius is 4 inches.

Problem 11834

Gegeben ist die Funktion f(x)=x2−3xf(x)=x^{2}-3 xf(x)=x2−3x. Skizzieren Sie den Graphen für −1≤x≤4-1 \leq x \leq 4−1≤x≤4 und bestimmen Sie Steigung und Winkel bei x=2x=2x=2.

Problem 11835

A balloon expands at 60πin3/sec60 \pi \mathrm{in}^{3}/\mathrm{sec}60πin3/sec. Find the surface area growth rate when the radius is 4 in.

Problem 11836

Problem 11837

Rewrite the function f(x,y,z)=x+yf(x, y, z)=x+yf(x,y,z)=x+y and region EEE in spherical coordinates, then convert and evaluate the integral ∭EfdV\iiint_{E} f dV∭E​fdV.

Problem 11838

Evaluate the integral ∫0π53cos⁡27xdx\int_{0}^{\frac{\pi}{5}} \frac{3}{\cos ^{2} 7 x} d x∫05π​​cos27x3​dx.

Problem 11839

Problem 11840

Is the integral range from 0 to 5 correct for the potential v=2πσ∫05RRdRv=2 \pi \sigma \int_{0}^{5} \frac{R}{R} d Rv=2πσ∫05​RR​dR with σ=1.25\sigma=1.25σ=1.25 and r=0.025 mr=0.025 \mathrm{~m}r=0.025 m?

Problem 11841

Problem 11842

Calculate the integral for the electrostatic potential on a disc: v=2πσ∫05RR2+0.0252dR v=2 \pi \sigma \int_{0}^{5} \frac{R}{\sqrt{R^{2}+0.025^{2}}} d R v=2πσ∫05​R2+0.0252​R​dR with σ=1.25\sigma=1.25σ=1.25.

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

Calculate the integral: $\int\left(\frac{5}{\sqrt{x}}+5 \sqrt{x}\right) dx$ .

Das Profil einer Skischanze wird durch $f(x)=\frac{1}{120} x^{2}-x+60$ (für $0 \leq x \leq 30$ ) beschrieben. Berechne die mittlere Steigung.

Find the first and second derivatives of $f(x)=\ln(x^{2}+1)$ , its domain, critical numbers, concavity intervals, and inflection points.

Find the integral: $\int\left(4 x^{\frac{1}{3}}+2 x^{-\frac{2}{3}}+11\right) d x$ .

Find the indefinite integral: $\int \frac{3 x^{3}+4 x^{2}}{x} d x$ .

Given $f(x)=\frac{1}{6-x}$ , find critical points, intervals of increase/decrease, asymptotes, and concavity for graphing.

Given $f(x)=\frac{1}{6-x}$ , find its derivatives, critical numbers, domain, asymptotes, and graph $f(x)$ .

Ein ICE hat das Weg-Zeit-Gesetz $s(t)=0,7 t^{2}$ . Berechne: a) Weg in 20 s und mittlere Geschwindigkeit, b) mittlere Geschwindigkeit in [19; 20], c) in [19,99; 20], d) Zeit für 300 km/h.

Calculate the volume of the solid formed by rotating the area between $y = Vx + 1$ , $y = 0$ , $x=0$ , $x=3$ around the x-axis.

Berechnen Sie das Volumen des Rotationskörpers, der durch $y=\sqrt{x+1}$ , $y=0$ , $x=0$ , und $x=3$ um die x-Achse entsteht.

Find where the function $f(x)=\frac{x^{2}-35}{x-6}, x \neq 6$ is increasing/decreasing and identify local extrema.

How many years for Strontium-90 from Chernobyl to reduce by $98.7\%$ ? Use its half-life of $30$ years. Round to the nearest year.

Given that $f^{\prime}(x)$ is continuous and $f(x)$ has a local extreme at $x=9$ , find $f^{\prime}(2)$ , $f^{\prime}(9)$ , $f^{\prime}(15)$ . Is $f(x)$ increasing or decreasing before and after $x=9$ , and is it a max or min?

Find the first and second derivatives of $f(x)=x^{3}+3x^{2}-105x+24$ . Determine intervals where $f$ is increasing, decreasing, concave up, and concave down.

Analyze the function with derivative $f'(x)=(x+3)e^{-2x}$ to find where $f$ is increasing or decreasing and locate local max/min points.

Find the derivative of the function $f(t) = 12 e^{-t}$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\left(\sin ^{-1}(x+3)\right)^{5}$ .

Find the local minimum and maximum of $f(x)=2 x^{3}-33 x^{2}+60 x+2$ . Calculate $f^{\prime}(x)$ and find $x$ and $y$ values.

Find $\left(f^{-1}\right)^{\prime}(1)$ for the one-to-one function $f(x)=2 x+\cos (x)$ using the formula $\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}(x)\right)}$ .

Find the first time $t > 0$ when the velocity of the particle, given by $x(t)=12 e^{-t} \sin t$ , is zero.

Find the acceleration of a particle at $t=3$ given its velocity $v(t)=7-(1.01)^{-t^{2}}$ .

Find the local minimum and maximum of $f(x)=2 x^{3}-39 x^{2}+216 x+4$ . Calculate $f^{\prime}(x)$ and the values at those points.

Find when the speed of the particle, given by $s(t)=2 t^{3}-24 t^{2}+90 t+7$ , is increasing for $t \geq 0$ .

Find when the speed of a particle, with velocity $v(t)=-t^{3}+2 t^{2}+2^{-t}$ for $t \geq 0$ , is increasing.

Show that for harmonic functions $f$ and $g$ in domain $D$ , $\iint_{S} g \frac{\partial g}{\partial n} d A = \iiint_{T} |\operatorname{grad} g|^{2} d V$ .

Find the moment of inertia for a rectangle ( $0 \leq x \leq L$ , $0 \leq y \leq 2L$ ) rotated about the $z$ -axis. Calculate $M$ , $dI$ , integrate for $I$ , and express $I$ as $I = C M L^2$ . What is $C$ (to two decimal places)?

Sand leaks from a bag modeled by $S(t)=K\left(1-\frac{t^{2}}{\mu}\right)^{3}$ .
1. (a) Find initial sand amount. (b) Determine leak rate at time $t$ . (c) For $\mu=6$ , analyze leak rate at $t=2$ (speeding up or slowing down).

Calculate the limit as $n$ approaches infinity: $\operatorname{Lim}_{n \rightarrow \infty} \frac{7 n^{3}+8 n^{5}-n^{2}+1}{2-n^{2}-3 n^{3}+4 n^{4}-5 n^{5}}$

Find where the function $f(x)=e^{-x^{2} / 32}$ is concave up or down and identify inflection points.

A circular pool expands at $16 \pi \mathrm{in}^{2} / \mathrm{sec}$ . Find the radius expansion rate when the radius is 4 inches.

Gegeben ist die Funktion $f(x)=x^{2}-3 x$ . Skizzieren Sie den Graphen für $-1 \leq x \leq 4$ und bestimmen Sie Steigung und Winkel bei $x=2$ .

A balloon expands at $60 \pi \mathrm{in}^{3}/\mathrm{sec}$ . Find the surface area growth rate when the radius is 4 in.

Rewrite the function $f(x, y, z)=x+y$ and region $E$ in spherical coordinates, then convert and evaluate the integral $\iiint_{E} f dV$ .

Evaluate the integral $\int_{0}^{\frac{\pi}{5}} \frac{3}{\cos ^{2} 7 x} d x$ .

Is the integral range from 0 to 5 correct for the potential $v=2 \pi \sigma \int_{0}^{5} \frac{R}{R} d R$ with $\sigma=1.25$ and $r=0.025 \mathrm{~m}$ ?

Calculate the integral for the electrostatic potential on a disc:
$v=2 \pi \sigma \int_{0}^{5} \frac{R}{\sqrt{R^{2}+0.025^{2}}} d R$
with $\sigma=1.25$ .

Estimate $2.004^{5}$ using the linearization of $f(x)=x^{5}$ at $x=2$ . What is the slope and linearization?

Estimate $2004^{5}$ using the linearization of $f(x)=x^{5}$ at 2. Find the slope and tangent line. Then estimate.

Evaluate the integral $\iint_{R}(x^{2}+16 y^{2})^{2} d A$ over the region $R$ bounded by the ellipse $x^{2}+16 y^{2}=1$ using the transformation $x=u, 4y=v$ .

Find the derivative of $f(x)=x(2x-7)^{3}$ . What is $f^{\prime}(x)$ ?

Find the derivative of $f(t)=\frac{e^{t^5+3}+3}{e^{3t}}$ .

Find the derivative of the function $f(t)=\frac{e^{5+3}+3}{e^{5}}$ .

Find the price $p$ that maximizes revenue using the elasticity of demand $E=\frac{p}{360-5 p}$ . Round to the nearest dollar.

A lighthouse is $500 \mathrm{~m}$ from shore, rotating at 4 rev/min. Find the beam's speed at point $Q$ ( $200 \mathrm{~m}$ left of $P$ ). How does this speed change between $P$ and $Q$ ?

A 48 cm wire is cut into two pieces to form squares.
(a) Find the area function $A(x)$ for side length $x$ . (b) What side length $x$ minimizes the area? (c) What is the minimum area?

Find the limit as $x$ approaches -1 for $\frac{x^{2}-1}{2x^{2}+3x+1}$ .

Find the derivative of the function $f(t)=\frac{4t}{\ln(4t)}$ . What is $f'(t)$ ?

Find the limit as $x$ approaches -1 for $\frac{x^{2}-1}{2 x^{2}+3 x+1}$ . Enter the result or "inf"/"-inf".

Find the derivative of the function $f(t)=\frac{t^{7}-t^{2}+4 t}{\ln (4 t)}$ . What is $f^{\prime}(t)$ ?

Show that the equation $5-e^{\sin(3x)}=2x^2$ has a solution using the Intermediate Value Theorem. Define $f(x)$ and find $[a, b]$ where $f(a)<0$ and $f(b)>0$ .

Find the limit: $\lim _{x \rightarrow 4} \frac{7-x}{(x-4)^{2}}$ .

Find the limit as $x$ approaches 4 for the expression $\frac{7-x}{(x-4)^{2}}$ .

Find where the function $f(x)=2 x^{4}+48 x^{3}+432 x^{2}-12 x-7$ is concave up or down and identify inflection points.

Compute the derivatives: a) $f(x)=\sin(2x+3)$ , b) $g(x)=\frac{e^{3x}}{x^2+1}$ , c) $h(x)=(2x+3)^{2x+3}$ .

Find where the function $f(x)=2 x^{4}+48 x^{3}+432 x^{2}-12 x-7$ is concave up or down, and identify inflection points.

Estimate $\ln(6.007)$ using the linearization of $f(x)=\ln(x)$ at $x=6$ . Find the slope and tangent line equation.

Find where the function $f(x)=2 x^{4}+48 x^{3}+432 x^{2}-12 x-7$ is concave up or down and identify inflection points.

Find where the function $f(x)=2 x^{4}+24 x^{3}+108 x^{2}-24 x-5$ is concave up or down and identify inflection points.

Find local maxima and minima, and determine intervals of increase and decrease for $y=x^{3}-12x+2$ .

Cost Function: Given $C(x)=5 \cdot x^{2}+5 \cdot x+125$ , find $\hat{C}_{n}$ , $C^{\prime}(x)$ , and estimate cost for 6th item.

Differentiate implicitly to find $\frac{d y}{d x}$ for the equation $x^{4}+y^{4}=16 x y$ .

Given the function $f(x)=e^{9 x}+e^{-x}$ , find where $f$ is increasing and decreasing, and determine its local minimum value.

Find $a$ such that $\int_{a}^{a^{2}} (a x^{2}-7 a^{2} x-\frac{7 a^{3}}{3}) \, dx = 0$ , with $a$ not an integer.

Given the demand function $q=D(p)=\frac{600}{(p+8)^{5}}$ , find: a) elasticity, b) elasticity at $p=6$ (elastic, inelastic, unit), c) $p$ for max total revenue.

Déterminez la dérivée $C^{\prime}(t)$ de la fonction de coût $C(t)$ pour la publicité télévisée pendant 1980-2010.

Coût d'une pub télé de 30s de 1980 à 2010 : $C(t)=\begin{cases}50t+325 & 0 \leq t<15 \\ 150t-1175 & 15 \leq t \leq 30\end{cases}$ . a) $C(t)$ est-elle continue? b) Trouver $C'(t)$ .

Déterminez la longévité à long terme $s(t)$ donnée par $s(t)=\frac{1105}{14+13 \times 2^{-0.04 t}}$ pour $t \geq 0$ .