Differential Calculus

If the first derivative of a function is a constant, then the function is

A function f(x) is said to be an even function if its graph is symmetric with respect to

The notation f’(x) was invented by

If a function f(x) is concave downward on the interval (1,10), then f(8) and f(3)

If a tangent to a curve y = f(x) is horizontal at x = a, then f’(a) is

For a function y = f(x), if f”(x) = -f(x), then the function is

The derivative of an increasing function f(x) must be

At the maximum point, the value of the 2nd derivative of a function is

At the inflection point, the value of y” is

The function y = f(x) has a maximum value of x = 2 if f’(2) = 0 and f”(2) is

At the maximum point, the tangent line is

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?

A point on the curve where the second derivative of a function is equal to zero is called

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.