Exam 3: Applications of Differentiation

Find the derivative of the function. $f(x)$ = 3 sin x + 9x - 9

A turkey is removed from the oven when its temperature reaches $175^{\circ} \mathrm{F}$ and is placed on a table in a room where the temperature is $70^{\circ} \mathrm{F}$ . After 10 minutes the temperature of the turkey is $160^{\circ} \mathrm{F}$ and after 20 minutes it is $150^{\circ} \mathrm{F}$ . Use a linear approximation to predict the temperature of the turkey after $30$ minutes.

Find the derivative function. $f(x)=\frac{4}{x^{5}}-\frac{8}{x^{4}}-\frac{3}{x^{3}}+700$

Differentiate the function. $G(u)=\ln \sqrt{\frac{10 u+100}{10 u-100}}$

The position function of a particle is given by $s=t^{3}-10.5 t^{2}-2 t, \mathrm{t} \geq 0$ When does the particle reach a velocity of 1 m/s?

A body moves along a coordinate line in such a way that its position function at any time t is given by $S(t)=t \sqrt{1-t^{2}}$ where $s(t)$ is measured in feet and t in seconds. Find the velocity and acceleration of the body when $t=\frac{1}{2}$ .

If f is the focal length of a convex lens and an object is placed at a distance v from the lens, then its image will be at a distance u from the lens, where f ,v ,and u are related by the lens equation $\frac{1}{f}=\frac{1}{v}+\frac{8}{u}$ . Find the rate of change of v with respect to u.

Find $y^{\prime}$ by implicit differentiation. $10 \cos x \sin y=14$

A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 32 ft/s. At what rate is his distance from second base decreasing when he is halfway to first base? Round the result to the nearest hundredth.

The volume of a cube is increasing at a rate of $10 \mathrm{~cm}^{3} / \min$ . How fast is the surface area increasing when the length of an edge is $90 \mathrm{~cm}$ .

Suppose the daily total cost (in dollars) of manufacturing x televisions is $C(x)=0.0004 x^{3}-0.08 x^{2}+160 x+7000$ What is the marginal cost when x = 300? What is the actual cost incurred in manufacturing the 301st television?

Find the rate of change of y with respect of x at the indicated value of x. t = csc x - 18 cos x; $x=\frac{\pi}{6}$

Find the points on the curve $y=2 x^{3}+3 x^{2}-12 x+1$ where the tangent is horizontal.

If two resistors with resistances $R_{1}$ and $R_{2}$ are connected in parallel, as in the figure, then the total resistance $R$ measured in ohms ( $\Omega$ ), is given by $\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}$ . If $R_{1}$ and $R_{2}$ are increasing at rates of $0.1 ~\Omega / s$ and $0.2 ~\Omega / s$ respectively, how fast is $R$ changing when $R_{1}=75$ and $R_{2}=100$ ? Round your answer to the nearest thousandth.

Find the second derivative of the function. $f(x)=x\left(2 x^{2}-1\right)^{6}$

Differentiate. $g(x)=x^{9} \cos x$

Find the equation of the tangent to the curve at the given point. $y=\sqrt{4+4 \sin x}, \quad(0,4)$

The top of a ladder slides down a vertical wall at a rate of $0.1$ m/s . At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 m/s . How long is the ladder?

A plane flying horizontally at an altitude of 1 mi and a speed of $520$ mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station.

Find $d^{2} y / d x^{2}$ in terms of x and y. $x^{6}-y^{6}=1$