1. MAT 213 Brief Calculus Section 4.1 Approximating Change

2. Recall that when we “zoomed in” on a differentiable graph, it became almost linear, no matter how much curve there was in the original graph Therefore a tangent line at x = a can be a good approximation for a function near a Let’s take a look at the function and its tangent line at x = 1

4. Let’s zoom in

6. Let’s zoom in again

8. Around x = 1 both graphs look almost identical Let’s find the tangent line at x = 0 and use it to approximate f(.5), f(.9), f(1.1), f(1.5), and f(2) We will then compare these to their actual function values

9. Around x = 1 both graphs look almost identical Let’s find the tangent line at x = 0 and use it to approximate f(.5), f(.9), f(1.1), f(1.5), and f(2) xf(x)f(x)f’(x) 0.50.250 0.90.810.8 1.11.211.2 1.52.252 243

10. The Tangent Line Approximation Suppose f is differentiable at a. Then, for values of x near a, the tangent line approximation to f(x) is f(x) ≈ f(a) + f’(a)(x - a) The expression f(a) + f’(a)(x - a) is called the Local Linearization of f near x=a. (We are thinking of a as fixed, so that both f(a) and f’(a) are constant) The error, E(x) in the approximation is defined by: E(x) = f(x) - f(a) ≈ f’(a)(x - a) actual approximation

11. Now let’s use the same two graphs to talk about change

13. ΔxΔx ΔyΔy

14. ΔxΔx ΔyΔy Now is the slope of our tangent line

15. ΔxΔx ΔyΔy f’ is ALSO the slope of our tangent line

16. Δx = h f(x+h) – f(x) Notice that f(x+h) – f(x) is close to Δy

17. Δx = h f(x+h) – f(x) Notice that f(x+h) – f(x) is close to Δy ΔyΔy

18. Δx = h f(x+h) – f(x) So Δy ≈ f(x+h) – f(x) ΔyΔy

19. Δx = h f(x+h) – f(x) ΔyΔy

20. –Using our results we have –Which can be rewritten to as –Which approximates the change in the function values by multiplying the derivative by a small change in inputs, h –Alternatively we can write –Which says the output at x + h is approximately the output at f plus the approximate change in f

21. Marginal Analysis Often a companies decision to continue to produce goods is based on how much additional revenue they gain versus the additional cost The Marginal Cost is the change in total cost of adding one more unit Therefore it can be approximated by the instantaneous rate of change Marginal Cost = MC = C’(q) Marginal Revenue = MR = R’(q) Marginal Profit = MP = P’(q)

22. Example What is the marginal cost of q if fixed costs are $3000 and the variable cost is $225 per item? What is the marginal revenue if you charge $375 per item? What is your marginal profit?

23. EXAMPLES a. Find the tangent line approximation for each of the following b. Does the approximation give you an upper or lower-estimate? Pg 239, #22