Exam 3: Topics in Deifferentiation

A kite is 150 feet high and is moving horizontally away from a boy at the rate of 25 feet per second. How fast is the string being paid out when the kite is 390 feet from him?

If , find .

A shark, looking for dinner, is swimming parallel to a straight beach and 80 feet offshore. The shark is swimming at the constant speed of 17 feet per second. At time t = 0, the shark is directly opposite a lifeguard station. How fast is the shark moving away from the lifeguard station when the distance between them is 170 feet?

Use implicit differentiation to find if xy = ln(x tan y).

A metal cone contracts as it cools. Assume the height of the cone is 20cm and the radius at the base of the cone is 5cm. If the height of the cone is decreasing at 4.0 *10^-5cm per second, at what rate is the volume of the cone decreasing when its height is 18.8cm? (Assume the radius is held constant.)

Find dy/dx by differentiating implicitly. xy = y⁵

Answer true or false. If y + ln(xy) = 9, .

Evaluate. .

Find the formula for dy if y = cot x.

If x² + y² = 33, find dy/dx.

The magnetic force F acting on a particle is given by , where r is the distance from the magnetic source and k is a constant. r is measured to be 2cm with a possible error of ±8%. Estimate the percentage error in F and r.

Answer true or false. If y = $\pi$ ^{cos x - sin x} , .

Find if .

Gravitational force is inversely proportional to the distance between two objects squared. If F = at a distance d = 2m, how fast is the force diminishing if the objects are moving away from each other at 0.1 m/s?

Evaluate.

The power in watts for a circuit is given by P = I ²R. How fast is the power changing if the resistance, R, of the circuit is 1,400 $\Omega$ , the current, I, is 1.5A, and the current is decreasing with respect to time at a rate of 0.025 A/s. (Assume R is a constant.)

Answer true or false. A circular spill is spreading so that when its radius r is 1 m, dr = 0.005 m. The corresponding change in the area covered by the spill, A, is, to the nearest hundredth of a square meter, 0.008 m².

Evaluate.

Find the formula for $\Delta$ y if y = 3x⁴ - 9.

A spherical balloon is inflated so that its volume is increasing at the rate of 30 cubic feet per minute. How fast is the surface area of the balloon increasing when the radius is 5 feet? [Use and S = 4 $\pi$ 0r².]