Exam 3: Applications of Differentiation

Find the differential of the function at the indicated number. $f(x)=e^{7 x}+\ln (x+8)$ ; $x=0$

Find the accumulated amount after 7 years on an investment of $2,000 earning an interest rate of 5% per year compounded continuously. Round to the nearest cent.

Calculate $y^{\prime}$ . $\cos (x y)=x^{6}-y$

In calm waters, the oil spilling from the ruptured hull of a grounded tanker spreads in all directions. Assuming that the polluted area is circular, determine how fast the area is increasing when the radius of the circle is 20 ft and is increasing at the rate of $\frac{1}{6}$ ft/sec. Round to the nearest tenth if necessary.

Find the derivative of the function. $f(x)=0.001 x^{3}-0.02 x^{2}+0.5 x-0.3 e^{x}+40$

Find the second derivative of the function. h(t) = (t⁶ + 3) sin t

If $f(x)=\frac{x}{\ln x}$ , find $f^{\prime}\left(e^{4}\right)$ .

Find the derivative of the function. $f(x)$ = sinh ^-1 6x

Find the derivative of the function. $f(x)=5 x-9 \sqrt{x}+e^{x+2}$

Calculate $y^{\prime}$ . $y=\frac{e^{x}}{x^{3}}$

Two chemicals react to form another chemical. Suppose that the amount of chemical formed in time t (in hours) is given by $x(t)=\frac{11\left[1-\left(\frac{2}{3}\right)^{3 t}\right]}{1-\frac{1}{4}\left(\frac{2}{3}\right)^{3 t}}$ where $x(t)$ is measured in pounds. a. Find the rate at which the chemical is formed when $t=4$ Round to two decimal places. b. How many pounds of the chemical are formed eventually?

The displacement of a particle on a vibrating string is given by the equation $s(t)=2+\frac{1}{13} \sin (18 \pi t)$ where s is measured in centimeter and t in seconds. Find the velocity of the particle after t seconds.

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. $y \sin 3 x=x \cos 3 y,\left(\frac{\pi}{3}, \frac{\pi}{6}\right)$

Use the definition of the derivative to find the derivative of the function. $f(x)=7 x^{4}-4 x+6$

Find the derivative of the function. $f(x)=5 x^{7}-7 e^{5}$

Two carts, A and B, are connected by a rope 39 ft long that passes over a pulley (see the figure below). The point Q is on the floor 12 ft directly beneath and between the carts. Cart A is being pulled away from Q at a speed of 2 ft/s. How fast is cart B moving toward Q at the instant when cart A is 5 ft from Q?

The position function of a body moving along a coordinate line is s(t) = 3 sin t + 2 cos t where t is measured in seconds and s(t) in feet. Find the position, velocity, speed, and acceleration of the body when t = $\frac{\pi}{2}$ .

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 the result to the nearest thousandth.

The function $f(x)=8 x \sqrt{x+1}$ satisfies the hypotheses of the Mean Value Theorem on the interval $[0,3]$ . Find all values of c that satisfy the conclusion of the theorem.

Find the derivative of the function. $f(x)=e^{x} \csc ^{2} x$