Calculus

Problem 13001

Calculez la dérivée $h^{\prime}(\pi / 3)$ de la fonction $h(t)=e^{-4 t} \cos (3 t)$ et arrondissez à trois décimales.

See Solution

Problem 13002

Find the derivative of $y$ with respect to $x$ for $y=\frac{\ln x}{1+2 \ln x}$ .

See Solution

Problem 13003

Find the derivative of $f(x)=e^{2 x}$ and $y=x e^{-2 x}$ .

See Solution

Problem 13004

Find the derivative of $y$ with respect to $x$ for $y=\frac{\ln x}{3+5 \ln x}$ .

See Solution

Problem 13005

Find (a) the rate of change of $y$ with respect to $x$ , (b) the relative rate of change of $y$ , and at $x=6$ , find (c) the rate of change of $y$ , (d) the relative rate of change of $y$ , and ( $\theta$ ) the percentage rate of change of $y$ for $y=x^{2}+x-2$ .

See Solution

Problem 13006

Find the derivative of $y$ with respect to $x$ using logarithmic differentiation for $y=x^{5 \sin x}$ .

See Solution

Problem 13007

Find the horizontal asymptote of $f(t)=\frac{0.8 t+1000}{5 t+4}$ for $t \geq 15$ and interpret its meaning.

See Solution

Problem 13008

Find the derivative of $y$ with respect to $x$ using logarithmic differentiation for $y = x^{8} \sin x$ .

See Solution

Problem 13009

Find critical points $A$ and $B$ for $f(x)=-2x^3+33x^2-144x+5$ . Determine if $f'(x)$ is INC or DEC in $(-\infty, A)$ , $[A, B]$ , and $(B, \infty)$ .

See Solution

Problem 13010

Given that $f(x)$ is positive and concave up on interval $I$ , determine the following:
a.) $f^{\prime \prime}(x)>\square$ on $I$ . b.) $g^{\prime \prime}(x)=2\left(A^{2}+B f^{\prime \prime}(x)\right)$ , where $A=\square$ and $B=\square$ . c.) $g^{\prime \prime}(x)>\square$ on $I$ . d.) $g(x)$ is $\square$ on $I$ .

See Solution

Problem 13011

Find the derivative of $y=(\sin 2 x)^{x}$ using logarithmic differentiation.

See Solution

Problem 13012

Find the rate of change of $f(t)=\frac{86 t}{99 t-86}$ for facts remembered after 1 hour and 10 hours.

See Solution

Problem 13013

Prove the harmonic series $\sum_{n=0}^{\infty} \frac{1}{n}$ diverges using partial sums $S_{k}$ . Show $S_{k}$ is increasing and $S_{2^{n}} \geq S_{2^{n-1}}+\frac{1}{2} \geq 1+\frac{n}{2}$ for $n \geq 2$ .

See Solution

Problem 13014

Find the rate of change of facts remembered after 1 hour and 10 hours for $f(t)=\frac{94 t}{99 t-94}$ .

See Solution

Problem 13015

Find the derivative of $f(x) = \frac{-4 x}{(x^{2}+1)^{2}}$ .

See Solution

Problem 13016

Find the rate of change of facts remembered given by $f(t)=\frac{93 t}{99 t-93}$ after 1 hour and 10 hours.

See Solution

Problem 13017

Analyze the function $f(x)=x-\cos (x)$ on $[0,2 \pi]$ :
(a) Find the $y$ -intercept.
(b) Identify increasing/decreasing intervals.
(c) Classify critical points (max, min, neither).
(d) Determine concavity intervals.
(e) Find inflection points.
Provide exact values.

See Solution

Problem 13018

Find the average cost per appliance for 140 units: $c(q)=1300+90q-0.3q^2$ . Also, find the marginal cost at $q=140$ .

See Solution

Problem 13019

Find the average cost of 150 appliances: $c(q)=1300+140q-0.3q^2$ . Also, find the marginal cost at $q=150$ .

See Solution

Problem 13020

Find the marginal-revenue function for the demand $p=\frac{5 q+6}{2 q+9}$ , where revenue $=p q$ .

See Solution

Problem 13021

Find the value of $\lambda$ for which $p(x)=\lambda \sigma^{2} e^{-x}$ is a probability density function on $(0,+\infty)$ .

See Solution

Problem 13022

Find the marginal-revenue function for the demand equation $p=\frac{5 q+6}{2 q+9}$ , where revenue $=p q$ .

See Solution

Problem 13023

Analyze the function $f(x)=\frac{x^{3}}{x^{2}-12}$ :
(a) Find the domain.
(b) Identify asymptotes.
(c) Determine symmetry.
(d) Find increasing/decreasing intervals.
(e) Classify critical points.
(f) Identify concavity.
(g) Locate inflection points.

See Solution

Problem 13024

Find the rate of change of $f(t)=\frac{89 t}{99 t-89}$ after 1 hour and 10 hours. Rate after 1 hour is $\square$ facts/hour.

See Solution

Problem 13025

Find the tangent line equation for $f(x)=\frac{x^{4}}{(x^{2}-3)^{5}}$ at $x=2$ . The equation is $y=\square$ .

See Solution

Problem 13026

Given $f(x)=\frac{x^{3}}{x^{2}-12}$ , find:
(a) Domain of $f$ .
(b) Asymptotes of $f$ .
(c) Symmetry of $f$ .
(d) Intervals of increase/decrease.
(e) Classify critical points.
(f) Concavity intervals.
(g) Inflection points.

See Solution

Problem 13027

Evaluate the integral from $\pi / 3$ to $2 \pi / 3$ of $\csc ^{2}\left(\frac{1}{2} t\right)$ .

See Solution

Problem 13028

Find the marginal cost function for the total-cost function $c=\frac{7 q^{2}}{\sqrt{q^{2}+2}}+6000$ .

See Solution

Problem 13029

Evaluate the integral from 0 to 1: $\int_{0}^{1} \frac{e^{z}+1}{e^{z}+z} d z$

See Solution

Problem 13030

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ for the curve given by $x^{3}+y^{3}+2xy+8=0$ .

See Solution

Problem 13031

Find the number of relative minimum points for the polynomial $g$ given $g^{\prime}(x)=-x^{2}(x+1)^{2}(x-1)^{2}$ . Choose: (A) None (B) One (C) Two (D) Three

See Solution

Problem 13032

Find $\frac{d y}{d x}$ in terms of $x$ and $y$ if $-x y + y - 4 x^{3} = -3$ .

See Solution

Problem 13033

Find the intervals where the function $f$ is increasing, given its derivative $f^{\prime}(x)=\frac{(x+1)^{2}}{x-4}$ . Choose one: (A) $(-1,4)$ (B) $(-\infty,-1)$ (C) $(-\infty,-1)$ and $(4, \infty)$ (D) $(4, \infty)$ (E) entire domain.

See Solution

Problem 13034

Find $\frac{d y}{d x}$ in terms of $x$ and $y$ if $3 - x y = x$ .

See Solution

Problem 13035

A circle's radius decreases at 2 m/s. When the area is $49 \pi$ , find the rate of change of the area, rounded to 3 decimals.

See Solution

Problem 13036

Find the tangent lines to the curve defined by $-1=-x y-2 y$ when $y=-1$ .

See Solution

Problem 13037

Find $f(x)$ that passes through $(1,1)$ and satisfies $f^{\prime}(x)=6 x^{2}+4$ .

See Solution

Problem 13038

Find $\frac{d y}{d x}$ for the equation $\sin(4x^{3} + y^{2}) = y$ in terms of $x$ and $y$ .

See Solution

Problem 13039

Find the number of relative maxima for the polynomial $f$ given its derivative $f^{\prime}(x)=x^{4}(x-2)(x+3)$ . Choose: (A) None (B) One (C) Two (D) Three.

See Solution

Problem 13040

Find the number of relative minimum points for the polynomial $h$ given its derivative $h^{\prime}(x)=(x+2)(x+1)(x-3)$ . Choose: (A) None (B) One (C) Two (D) Three.

See Solution

Problem 13041

A right triangle has legs of 15 in and 20 in. If the short leg decreases by $4 \mathrm{in/sec}$ and the long leg by $8 \mathrm{in/sec}$ , find the hypotenuse's rate of change.

See Solution

Problem 13042

A plane is 6 km high and moves at 500 km/h. Find how fast the angle $\theta$ changes 12 min after passing the radar. $\frac{d \theta}{d t} \approx$ $\mathrm{rad} /$ hour

See Solution

Problem 13043

A circle's radius decreases at 2 m/s. When area = $49 \pi$ , find the area change rate. Round to three decimal places.

See Solution

Problem 13044

Calculate the area under the piecewise function $f(x)=\left\{\begin{array}{ll}x^{2}+4x+5 & x<2 \\ 4x+9 & x \geq 2\end{array}\right.$ from $x=-5$ to $x=4$ .

See Solution

Problem 13045

Find the rate of change of surface area $A=4 \pi r^{2}$ at $t=2$ min when $r=10$ and $dr/dt=40 \mathrm{~cm/min}$ .

See Solution

Problem 13046

A plane is 6 km high and moves at 500 km/h. Find how fast the angle $\theta$ changes 12 min after it passes the radar. $\frac{d \theta}{d t} \approx I$ $\mathrm{rad} /$ hour

See Solution

Problem 13047

A balloon rises vertically, $6 \mathrm{~km}$ away an observer sees it at angle $\frac{\pi}{5}$ , changing at $0.2 \mathrm{rad/min}$ . Find the rising speed.
$\frac{d y}{d t} \approx$ $\mathrm{km} / \mathrm{min}$

See Solution

Problem 13048

Find the derivatives: $f^{\prime}(x)$ for $f(x)=x^{8}$ , $g^{\prime}(x)$ for $g(x)=-6 x^{2}$ , and $h^{\prime}(x)$ for $h(x)=\frac{1}{x^{3}}$ .

See Solution

Problem 13049

Find a function with no inflection point where $f^{\prime \prime}(x)=0$ .

See Solution

Problem 13050

How long will it take for 80% of 60 grams of Cobalt 60 to decay if its half-life is 5.27 years? Round to two decimal places.

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Problem 13051

Given Carbon 14 with an initial amount of $15 \mathrm{~kg}$ and a half-life of 5,700 years: a. How long until $80\%$ decays?

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Problem 13052

Find the derivative of $y$ with respect to $x$ using logarithmic differentiation, where $y = x^{\ln x}$ .

See Solution

Problem 13053

Find the intervals where the function $g$ is increasing, given $g^{\prime}(x)=\frac{x^{2}}{(x-2)^{3}}$ .

See Solution

Problem 13054

Find the intervals where the function $f(x)=x^{6}-3 x^{5}$ is decreasing. Choose from the options given.

See Solution

Problem 13055

Find the tangent line equation to $y=2 \sin x$ at $\left(\frac{\pi}{6}, 1\right)$ in the form $y=m x+b$ .

See Solution

Problem 13056

Find the tangent line equation to $y=2 \sin x$ at $\left(\frac{\pi}{6}, 1\right)$ in the form $y=m x+b$ .

See Solution

Problem 13057

Find intervals where the function $g$ is decreasing, given $g^{\prime}(x)=\frac{x}{x+3}$ . Choose from: (A) $x>0$ (B) $x<-3$ and $-3<x<0$ (C) $-3<x<0$ (D) $x<-3$ (E) entire domain.

See Solution

Problem 13058

Find the second derivative of the function $y=4 \tan (\theta)$ .

See Solution

Problem 13059

Find the second derivative $y^{\prime \prime}$ for the function $y=-2 \cot (x)$ .

See Solution

Problem 13060

Find the derivative of $y$ with respect to $x$ for $y=\cos^{-1}(6x^5)$ .

See Solution

Problem 13061

Find the number of relative minimum points for the polynomial with derivative $g^{\prime}(x)=x^{5}(x+1)(x-1)$ . (A) None (B) One (C) Two (D) Three

See Solution

Problem 13062

Find the value of $x$ where the function $g(x)=-(x+1)(x-1)^{2}$ has a relative minimum. Options: $-\frac{1}{3}$ , 1, -1, $\frac{1}{3}$ .

See Solution

Problem 13063

Find the average rate of change of $q(x)$ from $-r$ to $r$ as $b$ , then show the instantaneous rate at $x=r$ is $2ar+b$ .

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Problem 13064

Un pendule simple de $0,240 \mathrm{~m}$ est relâché à un angle de $3,5^{\circ}$ . (a) Temps pour atteindre la vitesse max ? (b) Temps si l'angle est $1,75^{\circ}$ ?

See Solution

Problem 13065

Find the number of relative minimum points for the polynomial $g$ with derivative $g^{\prime}(x)=-x^{2}(x+1)^{2}(x-1)^{2}$ . (A) None (B) One (C) Two (D) Three

See Solution

Problem 13066

A player runs to first base at $31 \mathrm{ft/s}$ . Find the rate of change of distance to second base when halfway.

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Problem 13067

Prove the average rate of change of $q(x)=a x^{2}+b x+c$ over $r \leq x \leq s$ is $a(r+s)+b$ . Then show it's $b$ for $-r \leq x \leq r$ , and find the instantaneous rate at $x=r$ is $2ar+b$ .

See Solution

Problem 13068

Estimate the rocket's velocity in km/h if $\theta(10)=0.205$ and $\theta(10.5)=0.225$ at 7 km distance. Round to 1 decimal.

See Solution

Problem 13069

Find the point elasticity of demand for the equation $q=400-40p+p^{2}$ at $p=17$ and the change in demand for a 1% price increase.

See Solution

Problem 13070

How much work is required to reduce a $1200 \mathrm{~kg}$ vehicle's speed from $80 \mathrm{~km/h}$ to $50 \mathrm{~km/h}$ ?

See Solution

Problem 13071

A plane at $6 \mathrm{~km}$ altitude moves at $500 \mathrm{~km/h}$ . Find how fast the angle $\theta$ changes after $12 \mathrm{~min}$ . $\frac{d \theta}{d t} \approx$ $\mathrm{rad} /$ hour

See Solution

Problem 13072

Find the point elasticity of demand for $q=500-35p+p^{2}$ at $p=17$ . What is the change in demand if $p$ increases by 1.5%?

See Solution

Problem 13073

Find the power series representation for $f(x)=\ln(8-x)$ .

See Solution

Problem 13074

Find the derivative of $y$ where $y=\sec^{-1}(2x^{2}+1)$ for $x>0$ .

See Solution

Problem 13075

Find $\int_{2}^{15} f(x) d x$ given $\int_{2}^{11} f(x) d x=8$ and $\int_{11}^{15} f(x) d x=7$ .

See Solution

Problem 13076

Find $\frac{d P}{d t}$ for $b=1.4$ , $P=7 \mathrm{kPa}$ , $V=110 \mathrm{~cm}^{3}$ , $\frac{d V}{d t}=50 \mathrm{~cm}^{3}/\mathrm{min}$ . $\frac{d P}{d t} =$ $\mathrm{kPa}/\mathrm{min}$

See Solution

Problem 13077

Calculate the Riemann sum $S_{3}$ for $f(x)=x^{2}-8x-48$ over $[0,9]$ with $c_{1}=1.1, c_{2}=5.1, c_{3}=8.1$ . Find $s_{3}=\square$ .

See Solution

Problem 13078

Evaluate the double integral: $\int_{2}^{3} \int_{-1}^{4}(12 x-18 y) d x d y$ .

See Solution

Problem 13079

Find the area represented by the integral $\int_{0}^{2}|x-1| d x$ .

See Solution

Problem 13080

A balloon's radius decreases at $15 \mathrm{~cm/min}$ . Find the volume change rate when $V=972 \pi \mathrm{cm}^{3}$ .

See Solution

Problem 13081

Find the power series for $f(z)=\frac{z^{3}}{(3-z)^{2}}$ centered at the origin.

See Solution

Problem 13082

Is $\frac{x^{n+1}}{n+1}$ an antiderivative of $x^{n}$ for real $n$ ? Choose: A) all $n$ , B) negative $n$ , C) all $n$ except -1, D) false, E) all $n$ except 0.

See Solution

Problem 13083

Find the power series for the function $f(z)=\frac{1}{2+25 z^{2}}$ .

See Solution

Problem 13084

Find the derivative of $y$ where $y=\ln(\tan^{-1}(2x^5))$ .

See Solution

Problem 13085

Verify demand elasticity and total revenue changes for $0<q<100$ and $100<q<200$ using $p=600-3q$ and $\eta=\frac{p}{q} \cdot \frac{d q}{d p}$ . Find $\frac{d q}{d p}$ .

See Solution

Problem 13086

Differentiate the function $f(x)=(x+4) e^{-8 x+3}$ and provide $f^{\prime}(x)$ in factored form.

See Solution

Problem 13087

Find the value of the integral $\int_{-5}^{-2} h(u) d u$ given that $\int_{-5}^{5} h(u) d u=11$ and $\int_{-2}^{5} h(u) d u=9.4$ .

See Solution

Problem 13088

Find the interval of convergence for the series of $f(x)=\tan^{-1}\left(\frac{x}{2}\right)$ from the series of $1/(1+x^2)$ .

See Solution

Problem 13089

Find derivatives for $g(x)=\frac{f(x)}{1+f(x)}$ and $h(x)=x f(x) \cos (x)$ at $x=2$ given $f(2)=-5$ and $f^{\prime}(2)=-1$ .

See Solution

Problem 13090

Given functions $f$ , $g$ , and their derivatives, compute:
(a) $(f g)^{\prime}(1)$ (b) $(f / g)^{\prime}(3)$ (c) $(g h)^{\prime}(2)$ , where $h(x)=\sin(x)$ (d) $(f / k)^{\prime}(4)$ , where $k(x)=\sqrt{x}$

See Solution

Problem 13091

Calculate the integral $\int_{-52}^{-34}[4 f(x)+5 g(x)-h(x)] d x$ using given values: $\int_{-52}^{-34} f(x) d x=10$ , $\int_{-52}^{-34} g(x) d x=22$ , $\int_{-52}^{-34} h(x) d x=35$ .

See Solution

Problem 13092

Find the function $L(x)$ from $L^{\prime}(x) = 3800x^{-1/2}$ and evaluate $L(100)$ for labor-hours for 100 units.

See Solution

Problem 13093

Evaluate the integral $I=\int_{0}^{3} 2 e^{-x^{2}} d x$ using the Taylor series for $e^{-x^{2}}$ .

See Solution

Problem 13094

Compute the derivatives: (a) $(f g)^{\prime}(1)$ (b) $(f / g)^{\prime}(3)$ (c) $(g h)^{\prime}(2)$ , with $h(x)=\sin (x)$ (d) $(f / k)^{\prime}(4)$ , with $k(x)=\sqrt{x}$

See Solution

Problem 13095

Given $f(4)=9$ and $f'(4)=2$ , find $(f h)'(4)$ where $h(x)=\sqrt{x}$ . Round your answer to three decimal places.

See Solution

Problem 13096

Find the derivative of $y = 7 x^{5} \sin^{-1}(7 x^{5}) + \sqrt{1 - 49 x^{10}}$ with respect to $x$ .

See Solution

Problem 13097

Calculate the Riemann sum $S_{3}$ for $f(x)=x^{2}-6x-40$ on $[0,6]$ with 3 equal subintervals and $c_{1}=0.7$ , $c_{2}=2.7$ , $c_{3}=5.1$ .

See Solution

Problem 13098

Find the derivative of $f(t)=3^{t^{3}}$ .

See Solution

Problem 13099

Find the derivative $f^{\prime}(x)$ of the function $f(x)=(4x+3)^{-1}$ and calculate $f^{\prime}(4)$ .

See Solution

Problem 13100

Find $\int_{1}^{6} f(x) d x$ given $\int_{1}^{7} f(x) d x=8$ and $\int_{6}^{7} f(x) d x=3.2$ .

See Solution

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Calculus

Problem 13001

Calculez la dérivée h′(π/3)h^{\prime}(\pi / 3)h′(π/3) de la fonction h(t)=e−4tcos⁡(3t)h(t)=e^{-4 t} \cos (3 t)h(t)=e−4tcos(3t) et arrondissez à trois décimales.

Problem 13002

Find the derivative of yyy with respect to xxx for y=ln⁡x1+2ln⁡xy=\frac{\ln x}{1+2 \ln x}y=1+2lnxlnx​.

Problem 13003

Find the derivative of f(x)=e2xf(x)=e^{2 x}f(x)=e2x and y=xe−2xy=x e^{-2 x}y=xe−2x.

Problem 13004

Find the derivative of yyy with respect to xxx for y=ln⁡x3+5ln⁡xy=\frac{\ln x}{3+5 \ln x}y=3+5lnxlnx​.

Problem 13005

Find (a) the rate of change of yyy with respect to xxx, (b) the relative rate of change of yyy, and at x=6x=6x=6, find (c) the rate of change of yyy, (d) the relative rate of change of yyy, and (θ\thetaθ) the percentage rate of change of yyy for y=x2+x−2y=x^{2}+x-2y=x2+x−2.

Problem 13006

Find the derivative of yyy with respect to xxx using logarithmic differentiation for y=x5sin⁡xy=x^{5 \sin x}y=x5sinx.

Problem 13007

Find the horizontal asymptote of f(t)=0.8t+10005t+4f(t)=\frac{0.8 t+1000}{5 t+4}f(t)=5t+40.8t+1000​ for t≥15t \geq 15t≥15 and interpret its meaning.

Problem 13008

Find the derivative of yyy with respect to xxx using logarithmic differentiation for y=x8sin⁡xy = x^{8} \sin xy=x8sinx.

Problem 13009

Find critical points AAA and BBB for f(x)=−2x3+33x2−144x+5f(x)=-2x^3+33x^2-144x+5f(x)=−2x3+33x2−144x+5. Determine if f′(x)f'(x)f′(x) is INC or DEC in (−∞,A)(-\infty, A)(−∞,A), [A,B][A, B][A,B], and (B,∞)(B, \infty)(B,∞).

Problem 13010

Problem 13011

Find the derivative of y=(sin⁡2x)xy=(\sin 2 x)^{x}y=(sin2x)x using logarithmic differentiation.

Problem 13012

Find the rate of change of f(t)=86t99t−86f(t)=\frac{86 t}{99 t-86}f(t)=99t−8686t​ for facts remembered after 1 hour and 10 hours.

Problem 13013

Problem 13014

Find the rate of change of facts remembered after 1 hour and 10 hours for f(t)=94t99t−94f(t)=\frac{94 t}{99 t-94}f(t)=99t−9494t​.

Problem 13015

Find the derivative of f(x)=−4x(x2+1)2f(x) = \frac{-4 x}{(x^{2}+1)^{2}}f(x)=(x2+1)2−4x​.

Problem 13016

Find the rate of change of facts remembered given by f(t)=93t99t−93f(t)=\frac{93 t}{99 t-93}f(t)=99t−9393t​ after 1 hour and 10 hours.

Problem 13017

Problem 13018

Find the average cost per appliance for 140 units: c(q)=1300+90q−0.3q2c(q)=1300+90q-0.3q^2c(q)=1300+90q−0.3q2. Also, find the marginal cost at q=140q=140q=140.

Problem 13019

Find the average cost of 150 appliances: c(q)=1300+140q−0.3q2c(q)=1300+140q-0.3q^2c(q)=1300+140q−0.3q2. Also, find the marginal cost at q=150q=150q=150.

Problem 13020

Find the marginal-revenue function for the demand p=5q+62q+9p=\frac{5 q+6}{2 q+9}p=2q+95q+6​, where revenue =pq=p q=pq.

Problem 13021

Find the value of λ\lambdaλ for which p(x)=λσ2e−xp(x)=\lambda \sigma^{2} e^{-x}p(x)=λσ2e−x is a probability density function on (0,+∞)(0,+\infty)(0,+∞).

Problem 13022

Find the marginal-revenue function for the demand equation p=5q+62q+9p=\frac{5 q+6}{2 q+9}p=2q+95q+6​, where revenue =pq=p q=pq.

Problem 13023

Analyze the function f(x)=x3x2−12f(x)=\frac{x^{3}}{x^{2}-12}f(x)=x2−12x3​: (a) Find the domain. (b) Identify asymptotes. (c) Determine symmetry. (d) Find increasing/decreasing intervals. (e) Classify critical points. (f) Identify concavity. (g) Locate inflection points.

Problem 13024

Find the rate of change of f(t)=89t99t−89f(t)=\frac{89 t}{99 t-89}f(t)=99t−8989t​ after 1 hour and 10 hours. Rate after 1 hour is □\square□ facts/hour.

Problem 13025

Find the tangent line equation for f(x)=x4(x2−3)5f(x)=\frac{x^{4}}{(x^{2}-3)^{5}}f(x)=(x2−3)5x4​ at x=2x=2x=2. The equation is y=□y=\squarey=□.

Problem 13026

Given f(x)=x3x2−12f(x)=\frac{x^{3}}{x^{2}-12}f(x)=x2−12x3​, find:(a) Domain of fff. (b) Asymptotes of fff. (c) Symmetry of fff. (d) Intervals of increase/decrease. (e) Classify critical points. (f) Concavity intervals. (g) Inflection points.

Problem 13027

Evaluate the integral from π/3\pi / 3π/3 to 2π/32 \pi / 32π/3 of csc⁡2(12t)\csc ^{2}\left(\frac{1}{2} t\right)csc2(21​t).

Problem 13028

Find the marginal cost function for the total-cost function c=7q2q2+2+6000c=\frac{7 q^{2}}{\sqrt{q^{2}+2}}+6000c=q2+2​7q2​+6000.

Problem 13029

Evaluate the integral from 0 to 1: ∫01ez+1ez+zdz\int_{0}^{1} \frac{e^{z}+1}{e^{z}+z} d z∫01​ez+zez+1​dz

Problem 13030

Find dy dx\frac{\mathrm{d} y}{\mathrm{~d} x} dxdy​ for the curve given by x3+y3+2xy+8=0x^{3}+y^{3}+2xy+8=0x3+y3+2xy+8=0.

Problem 13031

Find the number of relative minimum points for the polynomial ggg given g′(x)=−x2(x+1)2(x−1)2g^{\prime}(x)=-x^{2}(x+1)^{2}(x-1)^{2}g′(x)=−x2(x+1)2(x−1)2. Choose: (A) None (B) One (C) Two (D) Three

Problem 13032

Find dydx\frac{d y}{d x}dxdy​ in terms of xxx and yyy if −xy+y−4x3=−3-x y + y - 4 x^{3} = -3−xy+y−4x3=−3.

Problem 13033

Problem 13034

Find dydx\frac{d y}{d x}dxdy​ in terms of xxx and yyy if 3−xy=x3 - x y = x3−xy=x.

Problem 13035

A circle's radius decreases at 2 m/s. When the area is 49π49 \pi49π, find the rate of change of the area, rounded to 3 decimals.

Problem 13036

Find the tangent lines to the curve defined by −1=−xy−2y-1=-x y-2 y−1=−xy−2y when y=−1y=-1y=−1.

Problem 13037

Find f(x)f(x)f(x) that passes through (1,1)(1,1)(1,1) and satisfies f′(x)=6x2+4f^{\prime}(x)=6 x^{2}+4f′(x)=6x2+4.

Problem 13038

Find dydx\frac{d y}{d x}dxdy​ for the equation sin⁡(4x3+y2)=y\sin(4x^{3} + y^{2}) = ysin(4x3+y2)=y in terms of xxx and yyy.

Problem 13039

Find the number of relative maxima for the polynomial fff given its derivative f′(x)=x4(x−2)(x+3)f^{\prime}(x)=x^{4}(x-2)(x+3)f′(x)=x4(x−2)(x+3). Choose: (A) None (B) One (C) Two (D) Three.

Problem 13040

Find the number of relative minimum points for the polynomial hhh given its derivative h′(x)=(x+2)(x+1)(x−3)h^{\prime}(x)=(x+2)(x+1)(x-3)h′(x)=(x+2)(x+1)(x−3). Choose: (A) None (B) One (C) Two (D) Three.

Problem 13041

A right triangle has legs of 15 in and 20 in. If the short leg decreases by 4in/sec4 \mathrm{in/sec}4in/sec and the long leg by 8in/sec8 \mathrm{in/sec}8in/sec, find the hypotenuse's rate of change.

Problem 13042

Calculez la dérivée $h^{\prime}(\pi / 3)$ de la fonction $h(t)=e^{-4 t} \cos (3 t)$ et arrondissez à trois décimales.

Find the derivative of $y$ with respect to $x$ for $y=\frac{\ln x}{1+2 \ln x}$ .

Find the derivative of $f(x)=e^{2 x}$ and $y=x e^{-2 x}$ .

Find the derivative of $y$ with respect to $x$ for $y=\frac{\ln x}{3+5 \ln x}$ .

Find (a) the rate of change of $y$ with respect to $x$ , (b) the relative rate of change of $y$ , and at $x=6$ , find (c) the rate of change of $y$ , (d) the relative rate of change of $y$ , and ( $\theta$ ) the percentage rate of change of $y$ for $y=x^{2}+x-2$ .

Find the derivative of $y$ with respect to $x$ using logarithmic differentiation for $y=x^{5 \sin x}$ .

Find the horizontal asymptote of $f(t)=\frac{0.8 t+1000}{5 t+4}$ for $t \geq 15$ and interpret its meaning.

Find the derivative of $y$ with respect to $x$ using logarithmic differentiation for $y = x^{8} \sin x$ .

Find critical points $A$ and $B$ for $f(x)=-2x^3+33x^2-144x+5$ . Determine if $f'(x)$ is INC or DEC in $(-\infty, A)$ , $[A, B]$ , and $(B, \infty)$ .

Find the derivative of $y=(\sin 2 x)^{x}$ using logarithmic differentiation.

Find the rate of change of $f(t)=\frac{86 t}{99 t-86}$ for facts remembered after 1 hour and 10 hours.

Find the rate of change of facts remembered after 1 hour and 10 hours for $f(t)=\frac{94 t}{99 t-94}$ .

Find the derivative of $f(x) = \frac{-4 x}{(x^{2}+1)^{2}}$ .

Find the rate of change of facts remembered given by $f(t)=\frac{93 t}{99 t-93}$ after 1 hour and 10 hours.

Find the average cost per appliance for 140 units: $c(q)=1300+90q-0.3q^2$ . Also, find the marginal cost at $q=140$ .

Find the average cost of 150 appliances: $c(q)=1300+140q-0.3q^2$ . Also, find the marginal cost at $q=150$ .

Find the marginal-revenue function for the demand $p=\frac{5 q+6}{2 q+9}$ , where revenue $=p q$ .

Find the value of $\lambda$ for which $p(x)=\lambda \sigma^{2} e^{-x}$ is a probability density function on $(0,+\infty)$ .

Find the marginal-revenue function for the demand equation $p=\frac{5 q+6}{2 q+9}$ , where revenue $=p q$ .

Analyze the function $f(x)=\frac{x^{3}}{x^{2}-12}$ :
(a) Find the domain.
(b) Identify asymptotes.
(c) Determine symmetry.
(d) Find increasing/decreasing intervals.
(e) Classify critical points.
(f) Identify concavity.
(g) Locate inflection points.

Find the rate of change of $f(t)=\frac{89 t}{99 t-89}$ after 1 hour and 10 hours. Rate after 1 hour is $\square$ facts/hour.

Find the tangent line equation for $f(x)=\frac{x^{4}}{(x^{2}-3)^{5}}$ at $x=2$ . The equation is $y=\square$ .

Given $f(x)=\frac{x^{3}}{x^{2}-12}$ , find:
(a) Domain of $f$ .
(b) Asymptotes of $f$ .
(c) Symmetry of $f$ .
(d) Intervals of increase/decrease.
(e) Classify critical points.
(f) Concavity intervals.
(g) Inflection points.

Evaluate the integral from $\pi / 3$ to $2 \pi / 3$ of $\csc ^{2}\left(\frac{1}{2} t\right)$ .

Find the marginal cost function for the total-cost function $c=\frac{7 q^{2}}{\sqrt{q^{2}+2}}+6000$ .

Evaluate the integral from 0 to 1: $\int_{0}^{1} \frac{e^{z}+1}{e^{z}+z} d z$

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ for the curve given by $x^{3}+y^{3}+2xy+8=0$ .

Find the number of relative minimum points for the polynomial $g$ given $g^{\prime}(x)=-x^{2}(x+1)^{2}(x-1)^{2}$ . Choose: (A) None (B) One (C) Two (D) Three

Find $\frac{d y}{d x}$ in terms of $x$ and $y$ if $-x y + y - 4 x^{3} = -3$ .

Find $\frac{d y}{d x}$ in terms of $x$ and $y$ if $3 - x y = x$ .

A circle's radius decreases at 2 m/s. When the area is $49 \pi$ , find the rate of change of the area, rounded to 3 decimals.

Find the tangent lines to the curve defined by $-1=-x y-2 y$ when $y=-1$ .

Find $f(x)$ that passes through $(1,1)$ and satisfies $f^{\prime}(x)=6 x^{2}+4$ .

Find $\frac{d y}{d x}$ for the equation $\sin(4x^{3} + y^{2}) = y$ in terms of $x$ and $y$ .

Find the number of relative maxima for the polynomial $f$ given its derivative $f^{\prime}(x)=x^{4}(x-2)(x+3)$ . Choose: (A) None (B) One (C) Two (D) Three.

Find the number of relative minimum points for the polynomial $h$ given its derivative $h^{\prime}(x)=(x+2)(x+1)(x-3)$ . Choose: (A) None (B) One (C) Two (D) Three.

A right triangle has legs of 15 in and 20 in. If the short leg decreases by $4 \mathrm{in/sec}$ and the long leg by $8 \mathrm{in/sec}$ , find the hypotenuse's rate of change.

A plane is 6 km high and moves at 500 km/h. Find how fast the angle $\theta$ changes 12 min after passing the radar. $\frac{d \theta}{d t} \approx$ $\mathrm{rad} /$ hour

A circle's radius decreases at 2 m/s. When area = $49 \pi$ , find the area change rate. Round to three decimal places.

Calculate the area under the piecewise function $f(x)=\left\{\begin{array}{ll}x^{2}+4x+5 & x<2 \\ 4x+9 & x \geq 2\end{array}\right.$ from $x=-5$ to $x=4$ .

Find the rate of change of surface area $A=4 \pi r^{2}$ at $t=2$ min when $r=10$ and $dr/dt=40 \mathrm{~cm/min}$ .

A plane is 6 km high and moves at 500 km/h. Find how fast the angle $\theta$ changes 12 min after it passes the radar. $\frac{d \theta}{d t} \approx I$ $\mathrm{rad} /$ hour

A balloon rises vertically, $6 \mathrm{~km}$ away an observer sees it at angle $\frac{\pi}{5}$ , changing at $0.2 \mathrm{rad/min}$ . Find the rising speed.
$\frac{d y}{d t} \approx$ $\mathrm{km} / \mathrm{min}$

Find the derivatives: $f^{\prime}(x)$ for $f(x)=x^{8}$ , $g^{\prime}(x)$ for $g(x)=-6 x^{2}$ , and $h^{\prime}(x)$ for $h(x)=\frac{1}{x^{3}}$ .

Find a function with no inflection point where $f^{\prime \prime}(x)=0$ .

Given Carbon 14 with an initial amount of $15 \mathrm{~kg}$ and a half-life of 5,700 years: a. How long until $80\%$ decays?

Find the derivative of $y$ with respect to $x$ using logarithmic differentiation, where $y = x^{\ln x}$ .

Find the intervals where the function $g$ is increasing, given $g^{\prime}(x)=\frac{x^{2}}{(x-2)^{3}}$ .

Find the intervals where the function $f(x)=x^{6}-3 x^{5}$ is decreasing. Choose from the options given.

Find the tangent line equation to $y=2 \sin x$ at $\left(\frac{\pi}{6}, 1\right)$ in the form $y=m x+b$ .

Find the tangent line equation to $y=2 \sin x$ at $\left(\frac{\pi}{6}, 1\right)$ in the form $y=m x+b$ .

Find intervals where the function $g$ is decreasing, given $g^{\prime}(x)=\frac{x}{x+3}$ . Choose from: (A) $x>0$ (B) $x<-3$ and $-3<x<0$ (C) $-3<x<0$ (D) $x<-3$ (E) entire domain.

Find the second derivative of the function $y=4 \tan (\theta)$ .

Find the second derivative $y^{\prime \prime}$ for the function $y=-2 \cot (x)$ .

Find the derivative of $y$ with respect to $x$ for $y=\cos^{-1}(6x^5)$ .

Find the number of relative minimum points for the polynomial with derivative $g^{\prime}(x)=x^{5}(x+1)(x-1)$ . (A) None (B) One (C) Two (D) Three

Find the value of $x$ where the function $g(x)=-(x+1)(x-1)^{2}$ has a relative minimum. Options: $-\frac{1}{3}$ , 1, -1, $\frac{1}{3}$ .

Find the average rate of change of $q(x)$ from $-r$ to $r$ as $b$ , then show the instantaneous rate at $x=r$ is $2ar+b$ .

Un pendule simple de $0,240 \mathrm{~m}$ est relâché à un angle de $3,5^{\circ}$ . (a) Temps pour atteindre la vitesse max ? (b) Temps si l'angle est $1,75^{\circ}$ ?

Find the number of relative minimum points for the polynomial $g$ with derivative $g^{\prime}(x)=-x^{2}(x+1)^{2}(x-1)^{2}$ . (A) None (B) One (C) Two (D) Three

A player runs to first base at $31 \mathrm{ft/s}$ . Find the rate of change of distance to second base when halfway.

Prove the average rate of change of $q(x)=a x^{2}+b x+c$ over $r \leq x \leq s$ is $a(r+s)+b$ . Then show it's $b$ for $-r \leq x \leq r$ , and find the instantaneous rate at $x=r$ is $2ar+b$ .

Estimate the rocket's velocity in km/h if $\theta(10)=0.205$ and $\theta(10.5)=0.225$ at 7 km distance. Round to 1 decimal.

Find the point elasticity of demand for the equation $q=400-40p+p^{2}$ at $p=17$ and the change in demand for a 1% price increase.

How much work is required to reduce a $1200 \mathrm{~kg}$ vehicle's speed from $80 \mathrm{~km/h}$ to $50 \mathrm{~km/h}$ ?

A plane at $6 \mathrm{~km}$ altitude moves at $500 \mathrm{~km/h}$ . Find how fast the angle $\theta$ changes after $12 \mathrm{~min}$ . $\frac{d \theta}{d t} \approx$ $\mathrm{rad} /$ hour

Find the point elasticity of demand for $q=500-35p+p^{2}$ at $p=17$ . What is the change in demand if $p$ increases by 1.5%?

Find the power series representation for $f(x)=\ln(8-x)$ .

Find the derivative of $y$ where $y=\sec^{-1}(2x^{2}+1)$ for $x>0$ .