MA110 Exam-AID Session

1. MA110 Exam-AID Session By: Riley Furoy & Amanda Nichol

2. Agenda • Functions • Symmetry • Increasing/Decreasing • Composite • Inverse • Exponent Laws • Logarithmic Functions • Laws of Logarithms • Change of Base Formula • The Natural Logarithm • Limits • Limit Laws • Horizontal & Vertical Asymptotes • The Squeeze Theorem • Continuity • Intermediate Value Theorem • Tangent & Secant Lines • Derivatives • Differentiation Rules • Differentiation & Continuity

3. FUNCTIONS

4. Functions A function f: D E x1 x2 x3 . . . f(x1) f(x2) f(x3) . . . {x1,x2,…} is the domain {f(x1),f(x2),… } is the range

5. Symmetry Even Function Odd Function

6. Increasing/Decreasing Increasing Function Decreasing Function

7. Composite Functions A function f(g(x)): x1 x2 x3 . . . g(x1) g(x2) g(x3) . . . f(g(x1)) f(g(x2)) f(g(x3)) . . . {x1,x2,…} is the domain of g {g(x1), g(x2),… } is the range of g; a subset of this is the domain of f ∘ g {f(g(x1)), f(g(x2)),… } is the range of f ∘ g

8. Example e.g. Find the domain of the function f∘ ggiven: and

9. EXPONENT LAWS

10. Exponent Laws 1) 3) 2) 4)

11. LOGARITHMIC FUNCTIONS

12. Logarithmic Functions logax = y ⇔ ay = x Cancellation Equations: for all x Є R for all x > 0 Laws of Logarithms: 1) 2) 3)

13. Logarithmic Functions (ctd.) In the special case where a = e, we have the natural logarithm: Cancellation Equations: for all x Є R for all x > 0 Change of Base Formula:

14. Example e.g. Solve

15. Example e.g. Solve

16. Inverse Functions y=x A One-to-One Function… …and its Inverse Function The inverse is essentially a reflection along the line y=x.

17. Inverse Functions (ctd.) Cancellation Equations: f -1(f(x)) = x for all x in A f(f -1(x)) = x for all x in B Note:

18. Inverse Functions (ctd.) Steps to Finding the Inverse: Step 1 – Write y = f(x) Step 2 – Interchange x and y Step 3 – Solve this equation for x in terms of y The resulting equation is y = f -1(x)

19. Example e.g. Find the inverse of

20. LIMITS

21. The Limit of a Function Let f(x) be defined for all x in an open interval containing the number a (except possibly at a itself). Then we write If f(x) can be made arbitrarily close to L whenever x is sufficiently close (but not equal to)a. In order for the limit to exist at a:

22. Limit Laws Addition Law Subtraction Law Constant Law Multiplication Law Division Law(Holds only if the bottom limit is not zero)

23. Limit Laws (ctd.) Power Law Root Law

24. Asymptotes The line x = a is a vertical asymptote of y = f(x) if at least one of the following conditions is true: The line y = a is a horizontal asymptote of y = f(x) if at least one of the following conditions is true:

25. Example e.g. Determine if has any vertical or horizontal asymptotes.

26. The Squeeze Theorem If f(x) ≤ g(x) ≤ h(x) when x is near a (except possibly at a) and then:

27. Example e.g. Find given for all x ∈ (0, 2).

28. CONTINUITY

29. Continuity A function f is continuous at a if This implies that 3 conditions must be met: f(a) is defined exists A function is continuous on an interval if it is continuous at every point on the interval.

30. Continuity (ctd.) Any polynomial function is continuous everywhere; that is, it is continuous on . Similarly, any rational function is continuous wherever it is defined; that it, it is continuous on its domain. If f and g are continuous at a, and c is a constant, then the following functions are also continuous at a: 1) f + g 2) f – g 3) c ∙ f 4) f ∙ g 5) If g(a)≠ 0

31. Example e.g. Let . Determine the value(s) of x at which f is undefined and use your answer to determine the domain of f.

32. Example if x < 2 if x ≥ 2 e.g. Given Determine the value of the constant c for which f is continuous at x = 2.

33. Intermediate Value Theorem Suppose that f is continuous on the closed interval (a,b) and let N be any number between f(a) and f(b), where f(a) ≠ f(b). Then there exists a number c in (a,b)such that f(c) = N.

34. Example e.g. Use the Intermediate Value Theorem to show that the equation (where x > 0) has at least one solution on the interval (1,e).

35. TANGENT & SECANT LINES

36. Tangent & Secant Lines Taking points P(a, f(a)) and Q(x, f(x)) for any function, the slope of the line between these two points (the secant) is If we take (and make Q closer and closer to P), the slopeof the secant line converges to the slope of the tangent line at P.

37. Tangent & Secant Lines (ctd.) The tangent measures the instantaneous rate of change of the function, and its slope is

38. THE DERIVATIVE

39. The Derivative The derivative of a function f at a number a, denoted f’(a), is if this limit exists.

40. Example e.g. Use the limit definition to find the slope of the tangent line l to the curve at the point .

41. Differentiation Rules The method for finding the derivative outlined above, known as “first principles”, can be tedious for complicated functions. Therefore there are some shortcuts that we use: Power Rule: Constant Rule:

42. Differentiation Rules (ctd.) Product Rule: Quotient Rule: Chain Rule: Exponential Differentiation: Logarithmic Differentiation:

43. Example We can use the derivative laws to solve the previous example. To do so, find the derivative of f and substitute in the x-coordinate of the point P. e.g. Find the slope of the tangent line l to the curve at the point .

44. Differentiation & Continuity If a function is differentiable at a, then it is continuous at a. Note that the converse of this theorem is NOT true. For example, Which of the above functions are differentiable at a?

45. Differentiation & Continuity Answer: NONE The first graph has a “cusp”, and so the slope of the tangent is different on either side of a. The second graph is not continuous, and therefore cannot be differentiable. The third graph has a vertical tangent, so the derivative does not exist at a.

46. Study Tips • Practice, Practice, Practice • Go over past tests and all examples given by us and teacher

47. ADDITIONAL EXAMPLES