Differential Calculus

Find the average rate of change of the area of a square with respect to its side x as x changes from 4 to 7.

If the distance x from the point of departure at time t is defined by the equation \(x = -16t^2 + 5000t + 5000\), what is the initial velocity?

Find the derivative of the function \(f(x) = (2x - 3)^2\).

Find the value of x for which \(y = 4 + 3x - x^3\) will have a maximum value.

A rectangular trough is 8 feet long, 2 feet across the top and 4 feet deep. If water flows in at a rate of 2 ft³/min, how fast is the surface rising when the water is 1 ft deep?

Find the point(s) on the graph of \(y = x^2\) at which the tangent line is parallel to the line \(y = 6x - 1\).

Find the average rate of change of the area of a square with respect to its side x as x changes from 4 to 7.

Find the point of inflection of the curve \(y = x^3 + 3x^2 - 1\).

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