Math 142 (Summer 2018) Business Calculus 6.1 Notes

Transcription

1 Math 142 (Summer 2018) Business Calculus 6.1 Notes Antiderivatives Why? So far in the course we have studied derivatives. Differentiation is the process of going from a function f to its derivative f. In this section, we reverse the process. That is, we start with f and try to determine the function f. It turns out that we can only determine f up to a constant. Later in the chapter we will see many applications of this process known as antidifferentiation. Antiderivatives Suppose we have a function f, say f(x) = x 2. Our goal will be to find another function F such that the derivative F of F is f. That is, we want to find a function F such that F (x) = f(x) = x 2. We know from the power rule that differentiating x to a power reduces the power by one. Thus we might begin by guessing that F (x) should be x 3. Since we know how to take derivatives, we can check if this is correct. We find that d dx (x3 ) = 3x 2 = 3f(x). We are off by a factor of 3. To correct this, we will try dividing by this number. Set F (x) = 1 3 x3. Now we find that as we desired. F (x) = d ( ) 1 dx 3 x3 = 1 3 3x2 = x 2 = f(x), We ve done it! We ve solved our problem. However, we may stop to ask: Is F (x) = x 3 /3 the only function that satisfies F (x) = f(x)? After remembering the sum/difference rule and the fact that the derivative of a constant is zero, we realize that the functions x 3 /3 + 1 and x 3 /3 + 2 also have derivative x 2. In fact, we can add any constant to x 3 /3. Thus d dx ( 1 3 x3 + C ) = x 2, for any constant C. This leads to a follow up question: Is every function whose derivative is x 2 of the form x 3 /3 + C for some constant C? The answer to this is yes! We will see that this is always true in an upcoming theorem. First we need some terminology. 1

2 Definition (Antiderivative) If F (x) = f(x), then F is called an antiderivative of f. If we can find one antiderivative, then every other antiderivative can be obtained by adding constants to the original antiderivative. This is a consequence of the following theorem. Theorem (Only constants differentiate to zero) If F (x) = 0 on (a, b), then F (x) = C for some constant C on (a, b). We obtain the following theorem, which is called Theorem 1 in the textbook, as a consequence. Theorem (Antiderivatives only differ by a constant) If F (x) = G (x) on (a, b), then F (x) = G(x) + C for some constant C on (a, b). The main use of this theorem is to prove the theorem at the bottom of the page. Examples 1. Find all antiderivatives of x 2. The function x 3 /3 + C is called the general form of the antiderivative of x 2. Definition (The indefinite integral) The collection of all antiderivatives of a function f is called the indefinite integral and is denoted by f(x) dx. This is read as the indefinite integral of f(x) with respect to x. Here f(x) is called the integrand. The integral symbol is an elongated S, and it stands for sum. We can use the second theorem above to obtain the following theorem. Theorem (Finding an indefinite integral requires finding one antiderivative) If F is an antiderivative of f, then f(x) dx = F (x) + C, where C is an arbitrary constant and is called the constant of integration. 2

3 Rules of integration Each rule for derivatives leads to a rule for indefinite integrals. We list several integration rules in the theorem below. Theorem (Some integration rules) Suppose f and g are functions and n and k are constants. kf(x) dx = k f(x) dx (f(x) ± g(x)) dx = f(x) dx ± g(x) dx (Constant multiple rule) (Sum/difference rule) x n dx = 1 n + 1 xn+1 + C, n 1 (Power rule) e x dx = e x + C 1 dx = ln x + C x (Indefinite integral of exponential function) (Indefinite integral of 1/x) The proof in each case is very easy. Just use the corressponding rule for derivatives. We ll prove the power rule here, and leave the rest as exercises. Proof. We just need to show that the derivative of the right hand side is the integrand: ( ) d 1 dx n + 1 xn+1 + C = 1 n + 1 (n + 1)xn + 0 = x n. This proof highlights something important. To check that a function is an antiderivative, we need only take the derivative of it and compare it with the integrand. 2. Find x 5 dx. 3. Find 1 x dx. 3

4 4. Find 3t 9 dt. 5. Find x π dx. 6. Find (u 2 + u 3 ) du. 7. Find (2x x2 ) dx. 4

5 8. Find (3e x 4x) dx. 9. Find 2 x dx. 10. Find the revenue function R(x) if marginal revenue is 2x 10. [Hint: To find C, recall that R(0) = 0 since zero sales results in zero revenue.] 5

6 11. Find the cost function C(x) if the marginal cost is 10e x + 40x 100 and fixed costs are Exercises 1. Do exercises from the textbook. 2. (Textbook exercise ) If a ball is thrown upward with an initial velocity of 30 ft/sec, then from physics it can be shown that the velocity (neglecting air resistance) is given by v(t) = 32t Find s(t) if the ball is thrown from 15 feet above ground level and s(t) measures the height of the ball in feet. 3. (Textbook exercise ) A painting by one of the masters is purchased by a museum for $1, 000, 000 and increases in value at a rate given by V (t) = 100e t, where t is measured in years from the time of purchase. What will the painting be worth in 10 years? 4. Get more practice with these word problems by doing exercises from the text. 5. Explain in complete sentences the difference between an antiderivative of f and the indefinite integral f(x) dx. 6. Find the antiderivatives of e 2x. [Hint: Start by finding the derivative of e 2x.] 7. Find the antiderivatives of a x, where a > 1 is a constant. [Hint: Start by finding the derivative of a x.] 8. Prove the rest of the integration rules using the corresponding rule for derivatives. 6