13.1 Antiderivatives and Indefinite Integrals

The Antiderivative The reverse operation of finding a derivative is called the antiderivative. A function F is an antiderivative of a function f if F ’(x) = f (x). 1)Find the antiderivative of f(x) = 5 Find several functions that have the derivative of 5 Answer: 5x; 5x+ 1; 5x -3; 2) Find the antiderivative of f(x) = x 2 Find several functions that have the derivative of x 2 Answer: Theorem 1: If a function has more than one antiderivative, then the antiderivatives differ by at most a constant.

The graphs of antiderivatives are vertical translations of each other. For example: f(x) = 2x Find several functions that are the antiderivatives for f(x) Answer: x 2, x 2 + 1, x 2 + 3, x 2 - 2, x 2 + c (c is any real number)

The symbol  is called an integral sign, and the function f (x) is called the integrand. The symbol dx indicates that anti- differentiation is performed with respect to the variable x. By the previous theorem, if F(x) is any antiderivative of f, then The arbitrary constant C is called the constant of integration. Indefinite Integrals Let f (x) be a function. The family of all functions that are antiderivatives of f (x) is called the indefinite integral and has the symbol

Indefinite Integral Formulas and Properties (power rule) It is important to note that property 4 states that a constant factor can be moved across an integral sign. A variable factor cannot be moved across an integral sign.

Example 1: A) B) C)

Example 1 (continue) D)

Example 1 (continue) E)

Example 2 A)

Example 2 (continue) B)

Example 2 (continue) C)

Example 2 (continue) D)

Example 2 (continue) E)

Example 3 Find the equation of the curve that passes through (2,6) if its slope is given by dy/dx = 3x 2 at any point x. The curve that has the derivative of 3x 2 is Since we know that the curve passes through (2, 6), we can find out C Therefore, the equation is y = x 3 - 2

Example 4 Find the revenue function R(x) when the marginal revenue is R’(x) = x and no revenue results at a 0 production level. What is the revenue at a production of 1000 units? The marginal revenue is the derivative of the function so to find the revenue function, we need to find the antiderivative of that function So R(x) = 400x -.2x 2, we know need to find R(1000) Therefore, the revenue at a production level of 1000 units is $200,000

Example 5 The current monthly circulation of the magazine is 640,000 copies. Due to the competition from a new magazine, the monthly circulation is expected to decrease at a rate of C’(t)= -6000t 1/3 copies per month, t is the # of months. How long will it take the circulation of the magazine to decrease to 460,000 copies per month? We must solve this equation: C(t) = 460,000 with C(0) = 640,000 To find the function C(t), take the antiderivative So, it takes about 16 months