Differentiation

1. Mathematics for Business Decisions, part II Differentiation Math 115b Ekstrom Math 115b

2. Differentiation, Part I Rate • What comes to mind when you think of “rate” Ekstrom Math 115b

3. Differentiation, Part I Properties of Graphs • Describe the graph: • Where is the function… • increasing? • decreasing? • decreasing the fastest? Ekstrom Math 115b

4. Differentiation, Part I Properties, cont. • Describe f(x). Where is f: • positive? • negative? • zero? • increasing? • decreasing? Ekstrom Math 115b

5. Differentiation, Part I Rate of Change • Rate of change of a linear function is called “slope” • Denoted as m in y = mx + b • How is it defined? • What if the function is not linear? Ekstrom Math 115b

6. Differentiation, Part I Rate of Change, cont. • Consider the function from earlier: • Can we define a “slope” of this line? Ekstrom Math 115b

7. Differentiation, Part I Example Data • Consider the following set of data points (Tucson temperatures before, during, and after a thunderstorm): Ekstrom Math 115b

8. Differentiation, Part I Example, cont. • Perhaps plotting the data will give us a better description: • What is the rate of change of the temperature at 4:29 (16:29)? Ekstrom Math 115b

9. Differentiation, Part I Finding the “slope” at a point • So what do we want to do? • To evaluate the rate of change (slope) of f (x) at x, we should find the slope between the points before and after the point in question: for some h. Ekstrom Math 115b

10. Differentiation, Part I Slope at a point • As h gets smaller and smaller, the approximation of the slope gets better and better. • The derivative of a function is slope of a tangent line at a point on any curve, and can be calculated by: • It is usually denoted as or Ekstrom Math 115b

11. Differentiation, Part I Algebra Review • What does f (x + h) mean? • Ex. • Soln: • It means you evaluate the function at the quantity, x + h. Do NOT simply add h to f(x)! This will ultimately lead to a slope of 1. Ekstrom Math 115b

12. Differentiation, Part I Algebra Review, cont. • Example: Calculate the derivative of the function f (x) = 5x + 2 using the difference quotient. • Solution: Surprised? Ekstrom Math 115b

13. Differentiation, Part I Example calculations • Calculate the derivatives of the following functions: Ekstrom Math 115b

14. Differentiation, Part I Difference Quotient • The derivative of a function is the slope of the line tangent to any point on the curve, f (x). • It is calculated by finding the limit: • This gives an instantaneous rate of change of the function, f (x). Ekstrom Math 115b

15. Differentiation, Part I Instantaneous Rate of Change • What do we mean by instantaneous? • If h was one unit, and we calculated the difference quotient, then we would be finding the average rate of change between the points before and after the point in question. • We want h to be smaller and smaller (closer and closer to 0) so that the length 2h is approximately 0 so our quotient will stabilize. Ekstrom Math 115b

16. Differentiation, Part I Tangent Line To visualize the tangent line, think of a bird’s eye view of a curvy road at night. The headlights of a car traveling along this road will not follow the curves of the pavement. The path of the headlights represents the tangent line to the curvy road. Ekstrom Math 115b

17. Differentiation, Part I Tangent Line • The equation of the tangent line should be y = mx+ b • Slope of tangent line is equal to the derivative at every point x • m = f ’(x), where m is the slope of the tangent line • Since we know the slope and a point on the line, we can find the equation of the tangent line • If the derivative at the point exists, then the tangent line to the graph of f at the point (a, f (a)) has the equation Ekstrom Math 115b

18. Differentiation, Part I First Example • Find the slope of the line tangent to the graph of at the point (3, f (3)). • Find an equation for the tangent line at that point. Ekstrom Math 115b

19. Differentiation, Part I Second Example • Let f (x) = x3 + 6 • Find the equation of the line tangent to f (x) at the point (-1, f (-1)). • Luckily, you don’t have to do this by hand every time • Differentiating.xls Ekstrom Math 115b

20. Differentiation, Part I Graphing the Derivative • Want to get a sketch of the derivative graph • Interpretation of derivative is slope of tangent line • What does an ordered pair represent on the derivative graph? • How can you obtain the ordered pairs? Ekstrom Math 115b

21. Differentiation, Part I Algebraic Rules • If f (x) = k (constant), then f (x) = 0 • If f (x) is linear, f (x) = mx + b, and f (x) = m • Why? • If f (x) = a  g(x), then f (x) =a g(x) • If f (x) = g(x)  h(x), then f (x) = g(x)  h(x) • Specifically, since P(q) = R(q) - C(q), then P(q) = R(q) – C (q) ANDP(q) = 0 when R(q) = C (q) Ekstrom Math 115b

22. Derivitive Rules Power Rule f’ (xn) = nxn-1 Product Rule f’(u∙v) = u ∙ f’(v) + v ∙ f’(u) Quotient Rule f’ (u/v) = v ∙ f’(u) – u ∙ f’(v) v2