Review 1 Derivatives/Antiderivatives

Instructions. In the table 0n page 2, start in row 1 and complete the sentence on the ‘Differentiation’ side. Then move to the “Integration’ side and complete the corresponding sentence (if possible). Follow this procedure through row 13 (row 18, too). For rows 15,16,17,19, start on the ‘Integration’ side. If a cell is shaded, you need not fill it in.

Source of Differentiation Facts. Where can you find the relevant differentiation facts? Well, the inside back cover of your text has them. (It will defeat the purpose of this lab (developing fluency regarding the relationship between derivatives and antiderivatives), however, if you look at the Table of Indefinite Integrals!)

d W.R.T. The symbol indicates that the derivative is taken ‘with respect to’ dx the variable x, which means that the derivative shown is the instantaneous rate of change relative to change in x.

A Family of Functions. Note that ∫ f (x )dx = F(x) + c expresses the idea that the ‘general antiderivative’ (also sometimes called the indefinite integral) of the function f is a family of functions, each a vertical shift (up or down, depending on whether c is positive or negative) of some particular antiderivative, the function F.

You Can Always Check. One helpful aspect to the problem of integrating a function is that one can always check to see if one has an antiderivative ‘just by differentiating’ the proposed antiderivative; of course, this check does depend on determining the derivative correctly! Starting with row 17, complete the “Integration’ side first, and then check your answer by differentiating and show that process on the ‘Differentiation’ side.

Alternate Notations. d d dy [ f (x)] = f (x) = f '(x) , and if y = f(x), [ f (x)] = . dx D x dx dx Differentiation ‘Fact’ Corresponding General Integral Fact (also called ‘indefinite integral) a−1 1. For any constant, a, For any constant, a, ∫ ax dx = d a = dx x (This is called the Power Rule.) (In #15 below, write a more ‘usable’ form of this fact.) 2. For any positive constant, a, For any positive constant, a, d x ( ) = ∫ dx a (In item #16 below, write a more ‘usable’ form of this fact.) 3. As a specific case of #1 above: d 1 = dx 4. As a specific case of #2 above: d x ( ) = dx e d 5. For any constant, a, a = dx d 6. ln(x) = dx 7. For any positive constant, a, d Log (x) = dx a (Not in textbook table, but note ln(x) that Log = .) ax ln(a) d 8. sin(x) = cos(x) and dx ε 1 d cos(x) = sin(x) where and dx ε 2 ε 1 ε 2 are either 1 or negative 1. Determine, using only the graphs of the sine and cosine, which value goes with which function. These graphs appear below this differentiation/integration table. (See last page.) d sin(x) = dx d cos(x) = dx 9. For constants a and b, provided that f and g are differentiable d functions, [af (x) + b(g(x)] = dx

10. As a specific case of #9 above, d for constant a, [af (x)] = dx

11. Provided that f and g are The corresponding antiderivative differentiable functions, idea here is not so obvious – it’s d [ f (x)⋅(g(x)] = called Integration by Parts dx

12. Provided that f and g are differentiable functions, and that d f (x) g(x) is not zero, [ ] = dx g(x) 13. As a specific case of #12 d above, tan(x) = dx

14. For suitably defined and The differentiation rule on the differentiable functions f and g, left is known as the ‘Chain Rule,’ d d d [ f (g(x))] = [ f (g(x))]⋅ [g(x)] , which applies to a composite dx dg(x) dx (chain) function. The which, in simpler notation, and corresponding integration letting y=f(u) and u=g(x), procedure (It’s more a procedure dy = dx than a rule.) is called the method of ‘basic substitution,’ the most critical and widely applicable technique of integration and the subject of section 7.1. More to come! (No written response needed in this column of row #14.) 15. (See instructions given in row 1.)

16. (See instructions given in row 2.) 17. Consider the following integrals: 1 , 1 , ∫ x + 17 dx ∫ 17x + 38 dx 1 1 ∫ 2 dx , and ∫ 2 dx . One of x x +1 these can be determined directly from one of the formulas above, two of these can be done by guessing an antiderivative, differentiating the guess, and then revising the guess if the differentiation check fails; the fourth integral requires special methods, and is addressed later in this review. Do the three “easy” ones!

d 2 18. ln( + ) = dx x 1 x 19. ∫ 2 dx = x +1 Portion of the graph of the (periodic) sine function

What is the rate of change in the sine function for x=0?

Portion of the graph of the (periodic) cosine function

What is the rate of change in the cosine function for x=0?