Differential Calculus

A spherical balloon is inflated with \(r = 3\sqrt[3]{t}\) for \(0 \lt t \le 10\). Find the rate of change of volume in cubic cm at t = 8.

Find the second derivative of \(y = x\ln x\).

The height (in feet) at any time t (in seconds) of a projectile thrown vertically is \(h(t) = -16t^2 + 256t\). What is the projectile’s average velocity for the first 5 seconds of travel?

Evaluate \(\displaystyle\lim_{x \to 3}\dfrac{\sqrt{x^2 - 9}}{2x - 6}\).

A ball is thrown from the top of a 1,200-foot building. The position function expressing the height h of the ball above the ground at any time t is \(h(t) = -16t^2 - 10t + 1{,}200\). Find the average velocity for the first 6 seconds of travel.

Determine the point of inflection of the graph of \(y = x^3 + 6x^2\).

What is the coefficient of the \((x - 1)^3\) term in the Taylor series expansion of \(f(x) = \ln x\) expanded about x = 1?

A balloon is released at eye level and rises at the rate of 5 ft/s. An observer 50 ft away watches the balloon rise. How fast is the angle of elevation increasing 6 seconds after the moment of release?

A rock is dropped down a well that is 256 feet deep. When will it hit the bottom of the well?

If the side of a cube is measured with an error of at most 3 percent, estimate the error in the volume of the cube.

Find the slope of the curve \(x = 3t\), \(y = 9t^2 - 3t\) when t = 1.

Evaluate \(\displaystyle\lim_{x \to 2}\dfrac{x^2 - 4}{x - 2}\).

A runner runs a circular track and a set of data is recorded (time, distance): 68 s, 400 m; 114 s, 600 m; 168 s, 800 m; 209 s, 1,000 m; 256 s, 1,200 m; 322 s, 1,400 m. What is the average velocity from 68 sec to 168 sec?

Water is flowing into a conical vessel 10 ft high and 2 ft in radius at the rate of 50 cu. ft per minute. If the depth of the water is 6 ft, how fast is the radius increasing?

A steel girder 8 m long is moved on rollers along a passageway 4 m wide and into a corridor at right angles with the passageway. Neglecting the width of the girder, how wide must the corridor be?

A girl is flying a kite which is at a height of 120 ft. The wind is carrying the kite horizontally away from the girl at a speed of 10 ft/sec. How fast must the string be let out when the string is 150 feet long?

Find the minimum distance from the point (4, 2) to the parabola \(y^2 = 8x\).

A stone, projected vertically upward with initial velocity 112 ft/sec, moves according to \(s = 112t - 16t^2\), where s is the distance from the starting point. Compute the greatest height reached.

What is the allowable error in measuring the edge of a cube that is intended to hold 8 cu. m, if the error of the computed volume is not to exceed 0.03 cu. m?

Find the value of x for which \(y = 2x^3 - 9x^2 + 12x - 2\) has a maximum value.