Calculus (414 problems)

Suppose than an ostrich 5ft tall is walking at a speed of 4ft/s directly toward a street light 10 ft high. How fast is the tip of the ostrich’s shadow moving along the ground?

A spherical tank of radius 10 ft is being filled with water at the rate of 200 gal/min. How fast is the water level rising when the maximum depth of water in the tanl is 5ft?

A circle is inscribed in a square as shown in the figure:

The circumference of the circle is increasing at a constant rate of 6 inches per second. As the circle, the square expands to maintain the condition of tangency

(a) Find the rate at which perimeter of the square is increasing

(b) At the instant when the area of the circle is in², find the rate of increase in the area enclosed between the circle and the square.

A rocket is launched vertically upward from a point 9 miles west of an observer on the ground. What is the speed of the rocket when the angle of elevation (from the horizontal) of the observer’s line of sight to the rocket is 50 degrees and is increasing at 1 degree per second?

Water is being pumped into an inverted conical tank. The tank has height 13 meters and the diameter at the top is 7 meters. If the water level is rising at a rate of 25 centimeters per minute when the height of the water is 4.5 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute

When air expands adiabatically (without gaining or losing heat), its pressure P and volume V are related by the equation, where C is a constant. Suppose that at a certain instant the volume is 560 cubic centimeters and the pressure is 97 kPa and is decreasing at a rate of 11 kPa/minute. At what rate in cubic centimeters per minute is the volume increasing at this instant?

A 10-ft ladder is leaning against a wall. The bottom of the ladder begins to slide away from the wall at a speed of 3 ft/s. Find the rate at which the top of the ladder is moving when it is 6 ft from the ground.

Write equations of the two lines through (3, 4) that are tangent to the parabola.

Prove that (Assume that)

Find the area of the given region:

(a) Inside r = 1 and outside

(b) Common interior ofand

Ifand , what is the function ?

Given parametric equations

a) Findand the slope of the tangent at t = 1.

b) Find .

Find the length of the astroid

for.

Find the one-sided limit.

Find the limit. Please explain your answer.

Evaluate the following limits

(a)

(b)

(c)

(d)

(e)

(f)

Show that .

Use the fact that the power series, centered at 0, for is , find the power series for the following function with center at c.

(a)

(b)

Use the series of to approximate the definite integral with an error less than 0.0005