Differential Calculus

The cost C of a product is a function of the quantity x of the product: \(C(x)=x^2-4000x+50\) Find the quantity for which the cost is minimum.

Water is pouring into a swimming pool. After t hours, there are \(t+\sqrt{t}\) gallons in the pool. At what rate is the water pouring into the pool when t = 9 hours?

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

The tangent line to the function h(x) at (6, −1) intercepts the y-axis at y = 4. Find \(h'(6)\).

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.

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 \(y = x^3 + 3x^2 - 1\).

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?

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 \(y = x^3 + 6x^2\).

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?