Calculus

Problem 26001

Find the limit: $\lim _{x=\frac{\pi}{4}} \sec (2 x)$ .

See Solution

Problem 26002

(d) When should the store hire an extra employee: 6:00 a.m.-12:00 p.m. or 12:00 p.m.-6:00 p.m.? Discuss average rates of change. (e) Is the assistant manager right that no customers entered or left between 5:00 p.m. and 5:30 p.m. if the average rate of change is zero?

See Solution

Problem 26003

Gegeben ist $f(x)=2 x^{2}+x-3$ . Bestimmen Sie $f^{\prime}(x)$ und berechnen Sie die lokale Änderungsrate für $x = -2, -0,25, 0, 0,5, 3$ .

See Solution

Problem 26004

Find the average rate of change of $f(x)=4 x^{2}-6 x+1$ from $x_{1}=1$ to $x_{2}=5$ .

See Solution

Problem 26005

Untersuche die Funktion $f_{t}(x)=x^{4}-t x^{2}+1$ auf $y$ -Achsenabschnitt, Extrema, Wendepunkte und finde $t$ für A $(2;2)$ .

See Solution

Problem 26006

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=x^{2}-3x+5$ .

See Solution

Problem 26007

A radioactive material decays as $A(t)=500 e^{-0.0245 t}$ . Find the decay rate, graph, amount after 20 years, time for 400g, and half-life.

See Solution

Problem 26008

A radioactive material decays as $A(t)=500 e^{-0.0244 t}$ . Find the decay rate, graph, amount after 30 years, time for 400g, and half-life.

See Solution

Problem 26009

A radioactive material decays as $A(t)=A_0 e^{-0.0244t}$ . Given 800g, find: (a) decay rate, (c) amount after 10 years, (d) time for 600g, (e) half-life.

See Solution

Problem 26010

Untersuchen Sie die Funktion $f(x)=|x|$ auf Stetigkeit und Differenzierbarkeit.

See Solution

Problem 26011

A radioactive material decays as $A(t)=800 e^{-0.0244 t}$ . Find decay rate, graph, amount after 30 years, time for 600g, and half-life.

See Solution

Problem 26012

A swimming pool's chlorine ( $\mathrm{HOCl}$ ) decays from 2.8 ppm to 2.5 ppm in 24 hours. What is the level after 72 hours and when will it reach 1.0 ppm?

See Solution

Problem 26013

A stone is thrown with an initial velocity of 36 m/s from 42 m height. Find $t$ for max height and velocity when it hits the ground.

See Solution

Problem 26014

A mass on a spring is displaced by $A$ , released at $t=0$ . Its position is $x(t)=\frac{4}{3} A e^{-\gamma t}-\frac{1}{3} A e^{-4 \gamma t}$ . Sketch the displacement and velocity over time, and find when velocity is most negative. Calculate $e^{-\gamma t}$ at $t=\frac{1}{\gamma}$ to 2 s.f.

See Solution

Problem 26015

Bestimmen Sie a, sodass $f(x)=-\frac{1}{12} x^{3}+a x^{2}+4 i x$ in $x=2$ ein Extremum hat. Welche Art ist es?

See Solution

Problem 26016

A mass on a spring is displaced by $A$ and released. Its displacement is $x(t)=\frac{4}{3} A e^{-\gamma t}-\frac{1}{3} A e^{-4\gamma t}$ . Sketch $x(t)$ and find when velocity is most negative.

See Solution

Problem 26017

Compound $A$ decays from $0.30 \mathrm{M}$ to $0.25 \mathrm{M}$ in 30 min. Find remaining amount after 2 hrs and time for $0.10 \mathrm{M}$ .

See Solution

Problem 26018

Insect population $\mathrm{P}(t)=700 e^{0.06 t}$ : (a) Find $\mathrm{P}(0)$ , (b) growth rate, (c) $\mathrm{P}(10)$ , (d) when $\mathrm{P}=1050$ , (e) when $\mathrm{P}$ doubles.

See Solution

Problem 26019

Find the area between the curves $y=x^{2}-x-2$ and $y=3$ for $3 \leq x \leq 5$ .

See Solution

Problem 26020

Find the derivative of the function $f(x)=\left(2 x^{2}+5\right)^{7}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 26021

Find the rectangle dimensions for a track with a 2000 m perimeter to maximize the rectangle's area.

See Solution

Problem 26022

Find the derivative of the function $f(x)=\frac{5-x}{x^{3}+2}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 26023

Find the derivative $f^{\prime}(\sqrt{2 \pi})$ for the function $f(x)=\sin \left(x^{2}+\pi\right)$ .

See Solution

Problem 26024

Find the tangent line for $f(x)=\sqrt[3]{256 x^{3}+256}$ at $x=1$ in $m x+b$ form. Approximate $f(1.1)$ and find the error.

See Solution

Problem 26025

Find $\frac{d y}{d x}$ at the point $(2,1)$ for the equation $x^{2}+x y-3 y=3$ .

See Solution

Problem 26026

Cost function: $C(x)=40000+300 x+x^{2}$ . Find: a) Cost at $x=2000$ , b) Average cost at $x=2000$ , c) Marginal cost at $x=2000$ , d) Production level minimizing average cost, e) Minimal average cost.

See Solution

Problem 26027

Find the limit: $\lim _{x \rightarrow 3} \frac{\ln x - \ln 3}{x - 3}$ .

See Solution

Problem 26028

Find the average rate of change of $h(x)=x^{2}-9x$ from 8 to 9 and the secant line through $(8, h(8))$ and $(9, h(9))$ .

See Solution

Problem 26029

For the function $F(x)=-x^{4}+10 x^{2}+96$ , find if it's even/odd, a second max value, and area from $x=-4$ to $x=0$ .

See Solution

Problem 26030

Gegeben ist $f(x)=x^{3}+0,5 x^{2}$ . a) Finde $f^{\prime}(x)$ . b) Berechne die Änderungsrate von $f$ mit $f'$ .

See Solution

Problem 26031

Find all numbers $x$ in the interval $(-1,2)$ where $f'(x)=3x^2+8x-1$ equals the average rate of change of $f(x)=x^3+4x^2-x+8$ .

See Solution

Problem 26032

Find the total distance in meters traveled by a particle with velocity $v(t)=3 e^{-t / 2} \sin (2 t)$ from $t=0$ to $t=2$ .

See Solution

Problem 26033

Find all numbers $x$ in the interval $(-1,3)$ where $f'(x)=3x^2+6x-1$ equals the average rate of change of $f(x)=x^3+3x^2-x+6$ .

See Solution

Problem 26034

Evaluate the integral: $\int \frac{\sqrt{64-x^{2}}}{x} d x$

See Solution

Problem 26035

Find intervals where the function $f(x)=x^{3}+3 x^{2}-24 x$ is decreasing and concave down.

See Solution

Problem 26036

Find intervals where $f(x)=3 x^{3}+27 x^{2}+72 x$ is decreasing and concave down.

See Solution

Problem 26037

Find the average rate of change for $f(x)=x^{2}+5x-2$ over the interval $-3 \leq x \leq -1$ .

See Solution

Problem 26038

Find the average rate of change of $f(x)=x^{2}-3x-5$ from $x=-2$ to $x=3$ .

See Solution

Problem 26039

Find the limit: $\lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0}$ for $f(x) = x^2 + 2x$ .

See Solution

Problem 26040

Find the derivative $\frac{d y}{d x}$ for the equation $y=2 x^{5}+2 x^{4}-x^{3}-4 x$ .

See Solution

Problem 26041

Given values of the function $f$ and its derivative, find $g'(1)$ , $g'(-3)$ , and the tangent line to $f^{-1}$ at $x=1$ .

See Solution

Problem 26042

Find the derivative of the function $y=\frac{1}{2 x^{3}}$ with respect to $x$ and simplify without negative exponents.

See Solution

Problem 26043

Find $g^{\prime}(1)$ if $g(x)=f^{-1}(x)$ and $f(1)=8, f(2)=1, f^{\prime}(1)=3, f^{\prime}(2)=-5$ .

See Solution

Problem 26044

Find $g^{\prime}(1)$ if $g(x)=f^{-1}(x)$ and $f(1)=8, f(2)=1, f^{\prime}(1)=3, f^{\prime}(2)=-5$ .

See Solution

Problem 26045

Find the absolute max and min of $f(x)=\frac{\sqrt{x}}{48+x^{2}}$ for $0 \leq x \leq 6$ .

See Solution

Problem 26046

Find the absolute maximum and minimum of $f(x)=\frac{\sqrt{x}}{108+x^{2}}$ for $0 \leq x \leq 8$ .

See Solution

Problem 26047

Find $g^{\prime}(3)$ where $f(x)=\sqrt{3 x-5}$ and $g$ is the inverse of $f$ .

See Solution

Problem 26048

Find the derivatives of these functions: 1. $f(x)=\sec^{-1}(2x)$ , 2. $f(x)=\operatorname{arccot}(x^3)$ , 3. $f(x)=\sin^{-1}(3x^2)$ .

See Solution

Problem 26049

Find the derivative of the function: $y = \operatorname{arccot}(x^{3})$ .

See Solution

Problem 26050

Find the derivative of $f(x) = \sin^{-1}(3x^2)$ .

See Solution

Problem 26051

Find the tangent line equations for these curves at specified points: 1. $y=\tan^{-1}(x)$ at $x=-1$ ; 2. $y=\arcsin(2x)$ at $x=\frac{1}{4}$ .

See Solution

Problem 26052

Find the tangent line equation for $y=\arcsin (x)$ at $x=\frac{\sqrt{2}}{2}$ .

See Solution

Problem 26053

A firm produces goods A and B with demand functions $p = 600 - 18x - 3y$ and $q = 300 - 3x - 4y$ .
(a) Find the profit function.
(b) Maximize profit and find optimal $x$ and $y$ . Show it's a maximum.
(c) Plot profit function for $0 \leq x \leq 2x^$ and $0 \leq y \leq 2y^$ .
(d) Determine fixed costs to eliminate profits and where cost equals profit at optimal $x$ and $y$ .
(e) If production is limited to 9 units total, find max profit and optimal $x$ and $y$ .
(f) Explain the Lagrange multiplier's value.
(g) Verify results from (b) and (e).
(h) Use $\mathbb{R}$ to see profit change with a one-unit production increase and compare to Lagrange multiplier from (f).

See Solution

Problem 26054

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ of $y=\frac{x}{2+x^{2}}$ at $x=1$ .

See Solution

Problem 26055

Find the derivative $\frac{d y}{d x}$ for the equation $y=-5 x^{5}+3 x^{4}-5 x^{2}+6$ .

See Solution

Problem 26056

Calculate the sum: $\sum_{n=1}^{\infty} 24\left(-\frac{3}{5}\right)^{n-1}$ .

See Solution

Problem 26057

Find the maximum of $f(x)=-x^{3}+2x^{2}+2$ for $-2 \leq x \leq 3$ . Options: a) -7, b) 3.185, c) 18, d) 2.

See Solution

Problem 26058

Find $\frac{d y}{d x}$ when $x=2$ for the equation $x^{2} y+y^{2}+4=0$ . Choices: (A) -2 (B) -1 (C) 0 (D) 2 (E) nonexistent.

See Solution

Problem 26059

Find the acceleration of the object at $t=4$ s for the position function $s(t)=8t^{3}-4t^{2}$ . Options: a) $352 \mathrm{~m/s}^{2}$ b) $184 \mathrm{~m/s}^{2}$ c) $248 \mathrm{~m/s}^{2}$ d) $56 \mathrm{~m/s}^{2}$ .

See Solution

Problem 26060

Find the derivative of $f(x)=\cos(e^{2x})$ . What is $f^{\prime}(x)$ ? Choices: (A) $\sin e^{2 x}$ , (B) $2 \sin e^{2 x}$ , (C) $-\sin e^{2 x}$ , (D) $-2 \sin e^{2 x}$ , (E) $-2 e^{2 x} \sin e^{2 x}$ .

See Solution

Problem 26061

Find the derivative of the function $f(x)=\frac{5 \sqrt[4]{x}}{4}$ , expressed in radical form.

See Solution

Problem 26062

Find the derivative $\frac{d y}{d x}$ for the function $y=(-4 x^{2}-1)(-3+2 x^{2}-4 x^{-3})$ .

See Solution

Problem 26063

Find the derivative of the function $f(x)=\frac{1}{3 \sqrt{x^{3}}}$ and simplify without negative exponents.

See Solution

Problem 26064

Find the derivative of the function $f(x)=3 x^{-3} \sin x$ , denoted as $f^{\prime}(x)$ .

See Solution

Problem 26065

Find the derivative $f^{\prime}(x)$ of the function $f(x)=(-3+10 x^{2}-10 x^{-1})(-6 x+2)$ .

See Solution

Problem 26066

Find the derivative of the function $y=(2+2x^{2}-x^{3})(9+2x^{2})$ .

See Solution

Problem 26067

Find the derivative of the function $f(x)=x^{3} \cos x$ , denoted as $f^{\prime}(x)$ .

See Solution

Problem 26068

Calcula $h^{\prime}(-1)$ si $h(x)=x^{2} p(x)$ , con $p(-1)=4$ y $p^{\prime}(-1)=2$ .

See Solution

Problem 26069

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

See Solution

Problem 26070

Find the derivative $\frac{d y}{d x}$ of the function $y=\frac{5 x-2}{4-x^{4}}$ in simplified form.

See Solution

Problem 26071

Find the derivative of the function $y=\frac{x^{4}+2}{4-3 x^{4}}$ in simplified form.

See Solution

Problem 26072

Calculate the derivative of the function: $\frac{d}{d x}(-9 \cos x)$ .

See Solution

Problem 26073

Find the tangent line equation for $y=\tan^{-1} x$ at $x=\sqrt{3}$ .

See Solution

Problem 26074

Differentiate $\sin x - 2$ with respect to $x$ .

See Solution

Problem 26075

Differentiate the function $4 \cos x - 5$ with respect to $x$ .

See Solution

Problem 26076

Find the derivative of the function $y=\frac{x}{4-3 x^{4}}$ and simplify it.

See Solution

Problem 26077

Find the derivative $\frac{d y}{d x}$ of the function $y=\frac{x}{4-3 x^{4}}$ in simplified form.

See Solution

Problem 26078

Find the limit: $\lim _{u \rightarrow \infty} \frac{\sqrt{2 u-5}+1}{1+\sqrt[3]{2 u}}$

See Solution

Problem 26079

Find the tangent line equation for $f(x)=x^{2}+3$ at $x=4$ .

See Solution

Problem 26080

Find the tangent line equation for $f(x)=-2 x^{2}-4 x+3$ at $x=-4$ .

See Solution

Problem 26081

Find the tangent line equation of $f(x)=2 x^{3}+x+5$ at $x=-2$ .

See Solution

Problem 26082

Find the displacement and total distance for the velocity function $v(t)=t^{2}-t-42$ over $1 \leq t \leq 16$ .

See Solution

Problem 26083

How high is a bridge if an object takes 27.04 s to fall to the ground? Use $h = \frac{1}{2}gt^2$ with $g \approx 9.81 \, m/s^2$ .

See Solution

Problem 26084

Find $\frac{d^{2} y}{d x^{2}}$ at $x=1$ for the equation $\frac{d^{2} y}{d x^{2}}=\frac{-40 x-40^{2} x^{3}+6 x^{5}}{(2+x^{2})^{4}}$ .

See Solution

Problem 26085

Evaluate $\frac{d^{2} y}{d x^{2}}=\frac{-40 x-40 x^{3}+6 x^{5}}{(2+x^{2})^{4}}$ at $x=1$ .

See Solution

Problem 26086

Evaluate the second derivative $\frac{d^{2} y}{d x^{2}}$ at $x = 1$ for $\frac{d^{2} y}{d x^{2}} = \frac{-40 x - 40 x^{3} + 6 x^{5}}{(2 + x^{2})^{4}}$ .

See Solution

Problem 26087

Find the inverse $g(x)$ of $f(x)=x^{3}-1$ , graph both, and calculate their derivatives and slopes at specific points.

See Solution

Problem 26088

The Sun's angle decreases at $.25 \mathrm{rad} /$ hour. Find the shadow length increase for a $400 \mathrm{ft}$ . building at $\pi / 6$ .

See Solution

Problem 26089

Find when the speed of a particle, with velocity $v(t)=6 t^{2}-24 t+18$ , is decreasing for $t \geq 0$ .

See Solution

Problem 26090

Find the derivative of the function $f(x)=4 \sin (\sqrt{5 x-1})$ , denoted as $f^{\prime}(x)$ .

See Solution

Problem 26091

Find the derivative of the function $f(x)=-\sin \left(\sqrt{6+5 x^{2}}\right)$ , i.e., calculate $f^{\prime}(x)$ .

See Solution

Problem 26092

Find the inverse $g(x)$ of $f(x)=\sqrt{x+1}$ , graph both, and calculate their derivatives and slopes at given points.

See Solution

Problem 26093

Calculate the integral $\int_{0}^{\frac{\pi}{2}} x \sin^{3} x \, dx$ .

See Solution

Problem 26094

Find the derivative of the function $f(x)=\frac{2 x^{3}+3}{1+x}$ in simplified form.

See Solution

Problem 26095

Find $\left(f^{-1}\right)^{\prime}(2)$ for $f(x)=x^{3}+2x-1$ and $\left(f^{-1}\right)^{\prime}(6)$ for $f(x)=x^{3}-\frac{4}{x}$ .

See Solution

Problem 26096

Find the derivatives of the inverse function $g$ at points: a. $g^{\prime}(-1)$ , b. $g^{\prime}(0)$ , c. $g^{\prime}(1)$ , d. $g^{\prime}(2)$ , e. $g^{\prime}(3)$ . Given $f$ with values and derivatives at specific points.

See Solution

Problem 26097

Auf einem Volksfest wird die Besucherzahl durch $b(t)=20 t^{3}-300 t^{2}+1000 t$ beschrieben.
a) Finde $B(t)$ für die Besucheranzahl. b) Wie viele Besucher sind nach 3 Stunden da? c) Bestimme die durchschnittliche Besucheranzahl zwischen 11 und 14 Uhr. d) Berechne die maximale Besucheranzahl. e) Wann steigt die Besucherzahl am schnellsten? f) Nenne Grenzen des Modells bezüglich Zeit und Besucherzahlen.

See Solution

Problem 26098

4. If $f(2)=3$ and $f'(2)=4$ , find $\left(f^{-1}\right)'(3)$ .
5. For point $(1,2)$ on $f(x)=x^3+2x-1$ , find $\left(f^{-1}\right)'(2)$ .
6. Given $f(x)=x^3-\frac{4}{x}$ for $x>0$ , find $\left(f^{-1}\right)'(6)$ .

See Solution

Problem 26099

Find $g^{\prime}$ for inverse functions $f$ and $g$ given: 7. $f(2)=5$ , $f^{\prime}(2)=\frac{-2}{3}$ ; 8. $f(x)=x^{5}+2x^{3}-1$ , $f(1)=2$ ; 9. $f(x)=e^{x-2}$ .

See Solution

Problem 26100

Given the piecewise function $f(x)$ , find the values of $g(x) = \int_{-5}^{x} f(t) dt$ for: (a) $g(-8)$ , (b) $g(-4)$ , (c) $g(2)$ , (d) $g(6)$ .

See Solution

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Calculus

Problem 26001

Find the limit: lim⁡x=π4sec⁡(2x)\lim _{x=\frac{\pi}{4}} \sec (2 x)limx=4π​​sec(2x).

Problem 26002

(d) When should the store hire an extra employee: 6:00 a.m.-12:00 p.m. or 12:00 p.m.-6:00 p.m.? Discuss average rates of change. (e) Is the assistant manager right that no customers entered or left between 5:00 p.m. and 5:30 p.m. if the average rate of change is zero?

Problem 26003

Gegeben ist f(x)=2x2+x−3f(x)=2 x^{2}+x-3f(x)=2x2+x−3. Bestimmen Sie f′(x)f^{\prime}(x)f′(x) und berechnen Sie die lokale Änderungsrate für x=−2,−0,25,0,0,5,3x = -2, -0,25, 0, 0,5, 3x=−2,−0,25,0,0,5,3.

Problem 26004

Find the average rate of change of f(x)=4x2−6x+1f(x)=4 x^{2}-6 x+1f(x)=4x2−6x+1 from x1=1x_{1}=1x1​=1 to x2=5x_{2}=5x2​=5.

Problem 26005

Untersuche die Funktion ft(x)=x4−tx2+1f_{t}(x)=x^{4}-t x^{2}+1ft​(x)=x4−tx2+1 auf yyy-Achsenabschnitt, Extrema, Wendepunkte und finde ttt für A (2;2)(2;2)(2;2).

Problem 26006

Find the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for the function f(x)=x2−3x+5f(x)=x^{2}-3x+5f(x)=x2−3x+5.

Problem 26007

A radioactive material decays as A(t)=500e−0.0245tA(t)=500 e^{-0.0245 t}A(t)=500e−0.0245t. Find the decay rate, graph, amount after 20 years, time for 400g, and half-life.

Problem 26008

A radioactive material decays as A(t)=500e−0.0244tA(t)=500 e^{-0.0244 t}A(t)=500e−0.0244t. Find the decay rate, graph, amount after 30 years, time for 400g, and half-life.

Problem 26009

A radioactive material decays as A(t)=A0e−0.0244tA(t)=A_0 e^{-0.0244t}A(t)=A0​e−0.0244t. Given 800g, find: (a) decay rate, (c) amount after 10 years, (d) time for 600g, (e) half-life.

Problem 26010

Untersuchen Sie die Funktion f(x)=∣x∣f(x)=|x|f(x)=∣x∣ auf Stetigkeit und Differenzierbarkeit.

Problem 26011

A radioactive material decays as A(t)=800e−0.0244tA(t)=800 e^{-0.0244 t}A(t)=800e−0.0244t. Find decay rate, graph, amount after 30 years, time for 600g, and half-life.

Problem 26012

A swimming pool's chlorine (HOCl\mathrm{HOCl}HOCl) decays from 2.8 ppm to 2.5 ppm in 24 hours. What is the level after 72 hours and when will it reach 1.0 ppm?

Problem 26013

A stone is thrown with an initial velocity of 36 m/s from 42 m height. Find ttt for max height and velocity when it hits the ground.

Problem 26014

Problem 26015

Bestimmen Sie a, sodass f(x)=−112x3+ax2+4ixf(x)=-\frac{1}{12} x^{3}+a x^{2}+4 i xf(x)=−121​x3+ax2+4ix in x=2x=2x=2 ein Extremum hat. Welche Art ist es?

Problem 26016

A mass on a spring is displaced by AAA and released. Its displacement is x(t)=43Ae−γt−13Ae−4γtx(t)=\frac{4}{3} A e^{-\gamma t}-\frac{1}{3} A e^{-4\gamma t}x(t)=34​Ae−γt−31​Ae−4γt. Sketch x(t)x(t)x(t) and find when velocity is most negative.

Problem 26017

Compound AAA decays from 0.30M0.30 \mathrm{M}0.30M to 0.25M0.25 \mathrm{M}0.25M in 30 min. Find remaining amount after 2 hrs and time for 0.10M0.10 \mathrm{M}0.10M.

Problem 26018

Insect population P(t)=700e0.06t\mathrm{P}(t)=700 e^{0.06 t}P(t)=700e0.06t: (a) Find P(0)\mathrm{P}(0)P(0), (b) growth rate, (c) P(10)\mathrm{P}(10)P(10), (d) when P=1050\mathrm{P}=1050P=1050, (e) when P\mathrm{P}P doubles.

Problem 26019

Find the area between the curves y=x2−x−2y=x^{2}-x-2y=x2−x−2 and y=3y=3y=3 for 3≤x≤53 \leq x \leq 53≤x≤5.

Problem 26020

Find the derivative of the function f(x)=(2x2+5)7f(x)=\left(2 x^{2}+5\right)^{7}f(x)=(2x2+5)7. What is f′(x)f^{\prime}(x)f′(x)?

Problem 26021

Find the rectangle dimensions for a track with a 2000 m perimeter to maximize the rectangle's area.

Problem 26022

Find the derivative of the function f(x)=5−xx3+2f(x)=\frac{5-x}{x^{3}+2}f(x)=x3+25−x​. What is f′(x)f^{\prime}(x)f′(x)?

Problem 26023

Find the derivative f′(2π)f^{\prime}(\sqrt{2 \pi})f′(2π​) for the function f(x)=sin⁡(x2+π)f(x)=\sin \left(x^{2}+\pi\right)f(x)=sin(x2+π).

Problem 26024

Find the tangent line for f(x)=256x3+2563f(x)=\sqrt[3]{256 x^{3}+256}f(x)=3256x3+256​ at x=1x=1x=1 in mx+bm x+bmx+b form. Approximate f(1.1)f(1.1)f(1.1) and find the error.

Problem 26025

Find dydx\frac{d y}{d x}dxdy​ at the point (2,1)(2,1)(2,1) for the equation x2+xy−3y=3x^{2}+x y-3 y=3x2+xy−3y=3.

Problem 26026

Cost function: C(x)=40000+300x+x2C(x)=40000+300 x+x^{2}C(x)=40000+300x+x2. Find: a) Cost at x=2000x=2000x=2000, b) Average cost at x=2000x=2000x=2000, c) Marginal cost at x=2000x=2000x=2000, d) Production level minimizing average cost, e) Minimal average cost.

Problem 26027

Find the limit: lim⁡x→3ln⁡x−ln⁡3x−3\lim _{x \rightarrow 3} \frac{\ln x - \ln 3}{x - 3}limx→3​x−3lnx−ln3​.

Problem 26028

Find the average rate of change of h(x)=x2−9xh(x)=x^{2}-9xh(x)=x2−9x from 8 to 9 and the secant line through (8,h(8))(8, h(8))(8,h(8)) and (9,h(9))(9, h(9))(9,h(9)).

Problem 26029

For the function F(x)=−x4+10x2+96F(x)=-x^{4}+10 x^{2}+96F(x)=−x4+10x2+96, find if it's even/odd, a second max value, and area from x=−4x=-4x=−4 to x=0x=0x=0.

Problem 26030

Gegeben ist f(x)=x3+0,5x2f(x)=x^{3}+0,5 x^{2}f(x)=x3+0,5x2. a) Finde f′(x)f^{\prime}(x)f′(x). b) Berechne die Änderungsrate von fff mit f′f'f′.

Problem 26031

Find all numbers xxx in the interval (−1,2)(-1,2)(−1,2) where f′(x)=3x2+8x−1f'(x)=3x^2+8x-1f′(x)=3x2+8x−1 equals the average rate of change of f(x)=x3+4x2−x+8f(x)=x^3+4x^2-x+8f(x)=x3+4x2−x+8.

Problem 26032

Find the total distance in meters traveled by a particle with velocity v(t)=3e−t/2sin⁡(2t)v(t)=3 e^{-t / 2} \sin (2 t)v(t)=3e−t/2sin(2t) from t=0t=0t=0 to t=2t=2t=2.

Problem 26033

Find all numbers xxx in the interval (−1,3)(-1,3)(−1,3) where f′(x)=3x2+6x−1f'(x)=3x^2+6x-1f′(x)=3x2+6x−1 equals the average rate of change of f(x)=x3+3x2−x+6f(x)=x^3+3x^2-x+6f(x)=x3+3x2−x+6.

Problem 26034

Evaluate the integral: ∫64−x2xdx\int \frac{\sqrt{64-x^{2}}}{x} d x∫x64−x2​​dx

Problem 26035

Find intervals where the function f(x)=x3+3x2−24xf(x)=x^{3}+3 x^{2}-24 xf(x)=x3+3x2−24x is decreasing and concave down.

Problem 26036

Find intervals where f(x)=3x3+27x2+72xf(x)=3 x^{3}+27 x^{2}+72 xf(x)=3x3+27x2+72x is decreasing and concave down.

Problem 26037

Find the average rate of change for f(x)=x2+5x−2f(x)=x^{2}+5x-2f(x)=x2+5x−2 over the interval −3≤x≤−1-3 \leq x \leq -1−3≤x≤−1.

Problem 26038

Find the average rate of change of f(x)=x2−3x−5f(x)=x^{2}-3x-5f(x)=x2−3x−5 from x=−2x=-2x=−2 to x=3x=3x=3.

Problem 26039

Find the limit: lim⁡x→x0f(x)−f(x0)x−x0\lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0}limx→x0​​x−x0​f(x)−f(x0​)​ for f(x)=x2+2xf(x) = x^2 + 2xf(x)=x2+2x.

Problem 26040

Find the derivative dydx\frac{d y}{d x}dxdy​ for the equation y=2x5+2x4−x3−4xy=2 x^{5}+2 x^{4}-x^{3}-4 xy=2x5+2x4−x3−4x.

Find the limit: $\lim _{x=\frac{\pi}{4}} \sec (2 x)$ .

Gegeben ist $f(x)=2 x^{2}+x-3$ . Bestimmen Sie $f^{\prime}(x)$ und berechnen Sie die lokale Änderungsrate für $x = -2, -0,25, 0, 0,5, 3$ .

Find the average rate of change of $f(x)=4 x^{2}-6 x+1$ from $x_{1}=1$ to $x_{2}=5$ .

Untersuche die Funktion $f_{t}(x)=x^{4}-t x^{2}+1$ auf $y$ -Achsenabschnitt, Extrema, Wendepunkte und finde $t$ für A $(2;2)$ .

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=x^{2}-3x+5$ .

A radioactive material decays as $A(t)=500 e^{-0.0245 t}$ . Find the decay rate, graph, amount after 20 years, time for 400g, and half-life.

A radioactive material decays as $A(t)=500 e^{-0.0244 t}$ . Find the decay rate, graph, amount after 30 years, time for 400g, and half-life.

A radioactive material decays as $A(t)=A_0 e^{-0.0244t}$ . Given 800g, find: (a) decay rate, (c) amount after 10 years, (d) time for 600g, (e) half-life.

Untersuchen Sie die Funktion $f(x)=|x|$ auf Stetigkeit und Differenzierbarkeit.

A radioactive material decays as $A(t)=800 e^{-0.0244 t}$ . Find decay rate, graph, amount after 30 years, time for 600g, and half-life.

A swimming pool's chlorine ( $\mathrm{HOCl}$ ) decays from 2.8 ppm to 2.5 ppm in 24 hours. What is the level after 72 hours and when will it reach 1.0 ppm?

A stone is thrown with an initial velocity of 36 m/s from 42 m height. Find $t$ for max height and velocity when it hits the ground.

Bestimmen Sie a, sodass $f(x)=-\frac{1}{12} x^{3}+a x^{2}+4 i x$ in $x=2$ ein Extremum hat. Welche Art ist es?

A mass on a spring is displaced by $A$ and released. Its displacement is $x(t)=\frac{4}{3} A e^{-\gamma t}-\frac{1}{3} A e^{-4\gamma t}$ . Sketch $x(t)$ and find when velocity is most negative.

Compound $A$ decays from $0.30 \mathrm{M}$ to $0.25 \mathrm{M}$ in 30 min. Find remaining amount after 2 hrs and time for $0.10 \mathrm{M}$ .

Insect population $\mathrm{P}(t)=700 e^{0.06 t}$ : (a) Find $\mathrm{P}(0)$ , (b) growth rate, (c) $\mathrm{P}(10)$ , (d) when $\mathrm{P}=1050$ , (e) when $\mathrm{P}$ doubles.

Find the area between the curves $y=x^{2}-x-2$ and $y=3$ for $3 \leq x \leq 5$ .

Find the derivative of the function $f(x)=\left(2 x^{2}+5\right)^{7}$ . What is $f^{\prime}(x)$ ?

Find the derivative of the function $f(x)=\frac{5-x}{x^{3}+2}$ . What is $f^{\prime}(x)$ ?

Find the derivative $f^{\prime}(\sqrt{2 \pi})$ for the function $f(x)=\sin \left(x^{2}+\pi\right)$ .

Find the tangent line for $f(x)=\sqrt[3]{256 x^{3}+256}$ at $x=1$ in $m x+b$ form. Approximate $f(1.1)$ and find the error.

Find $\frac{d y}{d x}$ at the point $(2,1)$ for the equation $x^{2}+x y-3 y=3$ .

Cost function: $C(x)=40000+300 x+x^{2}$ . Find: a) Cost at $x=2000$ , b) Average cost at $x=2000$ , c) Marginal cost at $x=2000$ , d) Production level minimizing average cost, e) Minimal average cost.

Find the limit: $\lim _{x \rightarrow 3} \frac{\ln x - \ln 3}{x - 3}$ .

Find the average rate of change of $h(x)=x^{2}-9x$ from 8 to 9 and the secant line through $(8, h(8))$ and $(9, h(9))$ .

For the function $F(x)=-x^{4}+10 x^{2}+96$ , find if it's even/odd, a second max value, and area from $x=-4$ to $x=0$ .

Gegeben ist $f(x)=x^{3}+0,5 x^{2}$ . a) Finde $f^{\prime}(x)$ . b) Berechne die Änderungsrate von $f$ mit $f'$ .

Find all numbers $x$ in the interval $(-1,2)$ where $f'(x)=3x^2+8x-1$ equals the average rate of change of $f(x)=x^3+4x^2-x+8$ .

Find the total distance in meters traveled by a particle with velocity $v(t)=3 e^{-t / 2} \sin (2 t)$ from $t=0$ to $t=2$ .

Find all numbers $x$ in the interval $(-1,3)$ where $f'(x)=3x^2+6x-1$ equals the average rate of change of $f(x)=x^3+3x^2-x+6$ .

Evaluate the integral: $\int \frac{\sqrt{64-x^{2}}}{x} d x$

Find intervals where the function $f(x)=x^{3}+3 x^{2}-24 x$ is decreasing and concave down.

Find intervals where $f(x)=3 x^{3}+27 x^{2}+72 x$ is decreasing and concave down.

Find the average rate of change for $f(x)=x^{2}+5x-2$ over the interval $-3 \leq x \leq -1$ .

Find the average rate of change of $f(x)=x^{2}-3x-5$ from $x=-2$ to $x=3$ .

Find the limit: $\lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0}$ for $f(x) = x^2 + 2x$ .

Find the derivative $\frac{d y}{d x}$ for the equation $y=2 x^{5}+2 x^{4}-x^{3}-4 x$ .

Given values of the function $f$ and its derivative, find $g'(1)$ , $g'(-3)$ , and the tangent line to $f^{-1}$ at $x=1$ .

Find the derivative of the function $y=\frac{1}{2 x^{3}}$ with respect to $x$ and simplify without negative exponents.

Find $g^{\prime}(1)$ if $g(x)=f^{-1}(x)$ and $f(1)=8, f(2)=1, f^{\prime}(1)=3, f^{\prime}(2)=-5$ .

Find $g^{\prime}(1)$ if $g(x)=f^{-1}(x)$ and $f(1)=8, f(2)=1, f^{\prime}(1)=3, f^{\prime}(2)=-5$ .

Find the absolute max and min of $f(x)=\frac{\sqrt{x}}{48+x^{2}}$ for $0 \leq x \leq 6$ .

Find the absolute maximum and minimum of $f(x)=\frac{\sqrt{x}}{108+x^{2}}$ for $0 \leq x \leq 8$ .

Find $g^{\prime}(3)$ where $f(x)=\sqrt{3 x-5}$ and $g$ is the inverse of $f$ .

Find the derivatives of these functions: 1. $f(x)=\sec^{-1}(2x)$ , 2. $f(x)=\operatorname{arccot}(x^3)$ , 3. $f(x)=\sin^{-1}(3x^2)$ .

Find the derivative of the function: $y = \operatorname{arccot}(x^{3})$ .

Find the derivative of $f(x) = \sin^{-1}(3x^2)$ .

Find the tangent line equations for these curves at specified points: 1. $y=\tan^{-1}(x)$ at $x=-1$ ; 2. $y=\arcsin(2x)$ at $x=\frac{1}{4}$ .

Find the tangent line equation for $y=\arcsin (x)$ at $x=\frac{\sqrt{2}}{2}$ .

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ of $y=\frac{x}{2+x^{2}}$ at $x=1$ .

Find the derivative $\frac{d y}{d x}$ for the equation $y=-5 x^{5}+3 x^{4}-5 x^{2}+6$ .

Calculate the sum: $\sum_{n=1}^{\infty} 24\left(-\frac{3}{5}\right)^{n-1}$ .

Find the maximum of $f(x)=-x^{3}+2x^{2}+2$ for $-2 \leq x \leq 3$ . Options: a) -7, b) 3.185, c) 18, d) 2.

Find $\frac{d y}{d x}$ when $x=2$ for the equation $x^{2} y+y^{2}+4=0$ . Choices: (A) -2 (B) -1 (C) 0 (D) 2 (E) nonexistent.

Find the acceleration of the object at $t=4$ s for the position function $s(t)=8t^{3}-4t^{2}$ . Options: a) $352 \mathrm{~m/s}^{2}$ b) $184 \mathrm{~m/s}^{2}$ c) $248 \mathrm{~m/s}^{2}$ d) $56 \mathrm{~m/s}^{2}$ .

Find the derivative of the function $f(x)=\frac{5 \sqrt[4]{x}}{4}$ , expressed in radical form.

Find the derivative $\frac{d y}{d x}$ for the function $y=(-4 x^{2}-1)(-3+2 x^{2}-4 x^{-3})$ .

Find the derivative of the function $f(x)=\frac{1}{3 \sqrt{x^{3}}}$ and simplify without negative exponents.

Find the derivative of the function $f(x)=3 x^{-3} \sin x$ , denoted as $f^{\prime}(x)$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=(-3+10 x^{2}-10 x^{-1})(-6 x+2)$ .

Find the derivative of the function $y=(2+2x^{2}-x^{3})(9+2x^{2})$ .

Find the derivative of the function $f(x)=x^{3} \cos x$ , denoted as $f^{\prime}(x)$ .

Calcula $h^{\prime}(-1)$ si $h(x)=x^{2} p(x)$ , con $p(-1)=4$ y $p^{\prime}(-1)=2$ .

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

Find the derivative $\frac{d y}{d x}$ of the function $y=\frac{5 x-2}{4-x^{4}}$ in simplified form.

Find the derivative of the function $y=\frac{x^{4}+2}{4-3 x^{4}}$ in simplified form.

Calculate the derivative of the function: $\frac{d}{d x}(-9 \cos x)$ .

Find the tangent line equation for $y=\tan^{-1} x$ at $x=\sqrt{3}$ .

Differentiate $\sin x - 2$ with respect to $x$ .