Math Review and Lessons in Calculus

1. Math Review and Lessons in Calculus

2. Agenda • Rules of Exponents • Functions • Inverses • Limits • Calculus

3. Rules of Exponents • x0 = 1 • xa * xb = x(a+b) • x2 * x3 = x5 • xa / xb = x(a-b) • x5 / x2 = x3 • x-a = 1 / xa • x-5 = 1 / x5

4. Rules of Exponents Cont.

5. Rules of Exponents Cont. • xaya= (xy)a • x2y2= (xy)2 • 52 * 42= (5 * 4)2

6. Exponent Example • Assume that x1½ x2 ¼ = 1. • Solve x2 as a function of x1.

7. Exponent Example Cont.

8. Functions • A function describes how a set of unique variables is mapped/transformed into another set of unique variables. • Examples: • Linear: y = f(x) = ax + b • Quadratic: y = f(x) = ax2 + bx + c • Cubic: y = f(x) = ax3 + bx2 + cx + d • Exponential: y = f(x) = aex • Logarithmic: y = f(x) = aln(x)

9. Functions Cont. • A function is comprised of independent and dependent variables. • The independent variable is usually transformed by the function, whereas the dependent variable is the “output” of the function. • In the examples above y is a variable that is usually known as the dependant variable, while x is considered the independent variable.

10. Functions Cont. • Functions can map more than one independent variable into the dependent variable. • Example: • y = f(x1, x2) = a*x1 + b*x2 + c • It is possible to represent a function visually by using a graph.

11. Some Rules for Functions • Functions can be summed and then evaluated. • Functions can be subtracted from each other and then evaluated. • Functions can be multiplied by each other and then evaluated. • Functions can be divided by each other and then evaluated assuming the denominator is not equal to zero.

12. Composition of Functions • When one function is evaluated inside another function, it is said that you are performing a composition of functions. • Mathematically, we represent a composition of functions in two ways: • f(g(x)) • (fg)(x) • Note this is NOT multiplication of the two functions.

13. Example of Composition • Suppose f(x) = 2x + 1 and g(x) = 3x+3, then: • f(g(x))=2(3x+3) + 1 = 6x + 6 + 1 = 6x+7 • g(f(x))=3(2x+1) + 3 = 6x + 3 + 3 = 6x+6 • Note that f(g(x)) does not necessarily equal g(f(x)).

14. Inverse Functions • An inverse function is a function that can map the dependent variable into the independent variable. • In essence, it is a function that can reverse the independent and dependent variables. • Example: • y = ax + b has as its inverse function x = y/a – b/a.

15. Example of Finding the Inverse of a Linear Equation • y = 5x + 10 • Subtract 10 from both sides • y – 10 = 5x • Divide both sides by 5 • (y/5) – 2 = x • x = (y/5) – 2 • This is the inverse of the first equation above.

16. Quadratic Formula • The quadratic formula is an equation that allows you to solve for all the x values that would make the following equation true: • ax2 + bx + c = 0

17. Using the Quadratic Formula to Find the Inverse of a Quadratic Equation • Suppose you had the following equation: • y = qx2 + rx + s • If we transform the above equation into the following, we can use the quadratic equation to find the inverse: • qx2 + rx + s-y = 0

18. Using the Quadratic Formula to Find the Inverse of a Quadratic Equation Cont. • Define a = q, b = r, and c = s-y • Substituting these relationships into the above equation gives the following: • ax2 + bx + c = 0

19. Inverse of a Quadratic Equation Example • Suppose y = 2x2 + 4x + 8 • By rearrangement: 2x2 + 4x + 8 – y = 0

20. Finding the Intersection of Two Equations • At the point where two equations meet, they will have the same values for the dependent and independent variables for each equation. • Example: • y = 5x + 7 and y = 3x + 11 intersect at y = 17 and x = 2.

21. Finding the Intercepts of a Curve or Line • The vertical intercept is where the curve crosses or touches the y-axis. • To find the vertical intercept, you set x = 0, and solve for y. • The horizontal intercept is where the curve crosses or touches the x -axis. • To find the horizontal intercept, you set y = 0, and solve for x.

22. Slope of a Line • From algebra, it is known that the slope (m) of a line is defined as the rise over the run, i.e., the change in the y value divided by the change in the x value.

23. Slope of the Line Cont. • The slope of a line is constant. • The slope of a line gives you the average rate of change between two points.

24. Needed Terminology • A secant line is a line that passes through two points on a curve. • A tangent line is a line that touches a curve at just one point. • In essence, it gives you the slope of the curve at the one point. • A tangent line can tell you the instantaneous rate of change at a point.

25. Limits • A number L is said to be the limit of a function f(x) at point t, if as you get closer to t, f(x) gets closer to L.

26. Example of a Limit

27. Finding the Tangent Line • One way to find the tangent line of a curve at a given point is to examine secant lines that have corresponding points that get closer to each other. • This is known as examining the limits.

28. Graphical Representation of the Secant and the Tangent Lines Y Function: y = f(x) = x2 Secant Line f(x+x) Tangent Line f(x) X x x+x

29. Defining the Derivative • According to Varian, “the derivative is the limit of the rate of change of y with respect to x as the change in x goes to zero. • Suppose y =f(x), then the derivative is defined as the following:

30. Equivalency of Derivative Notation • There are many ways that are used to represent the derivative. • Suppose that y = f(x), then the derivative can be represented in the following ways:

31. Example of Using Limits for a linear Equation

32. Example of Using Limits for a Quadratic

33. Example of Using Limits for a Cubic

34. Differentiation Rules • Constant Rule • Power Rule • Constant Times a Function Rule • Sum and Difference Rule • Product and Quotient Rule • The Chain Rule • Generalized Power Rule • Exponential Rule

35. Constant Rule • The constant rule states that the derivative of a constant function is zero.

36. Power Rule • Suppose y=f(x)=xn, then the derivative of f(x) is the following: • f’(x)=nxn-1

37. Constant Times a function Rule • Suppose y=f(x)=ag(x), then the derivative of f(x) is the following: • f’(x)=a*g’(x)

38. Sum and Difference Rule • Suppose y=K(x)=f(x)  g(x), then the derivative of K(x) is: • K’(x) = f’(x)  g’(x)

39. The Product Rule • Suppose y=K(x)=f(x) * g(x), then the derivative of K(x) is: • K’(x) = f’(x) * g(x) + f (x) * g’(x)

40. Quotient Rule • Suppose y=K(x)=f(x) / g(x), then the derivative of K(x) is: • K’(x) = [f’(x) * g(x) - f (x) * g’(x)]/ (g(x))2

41. Chain Rule • Suppose y=K(x)=f(g(x)), then the derivative of K(x) is: • K’(x) = f’(g(x))*g’(x)

42. Generalized Power Rule • Suppose y=K(x)=[f(x)]n, then the derivative of K(x) is: • K’(x) = n[f(x)]n-1f’(x)

43. Exponential Rule • Suppose y = K(x) = ef(x), then the derivative of K(x) is: • K’(x)=f’(x)ef(x) • K(x) = ex ; K’(x) = ex • K(x) = e2x ; K’(x) = 2e2x

44. The Second Derivative • It is useful in calculus to look at the derivative of a function that has already been differentiating. • This is known as taking the second derivative. • The second derivative is usually represented by f’’(x).

45. Second Derivative Cont. • Suppose that y = f(x). • The second derivative can also be represented as the following:

46. Partial Derivative • Suppose that y = f(x1,x2). • The partial derivative of y is defined as the following:

47. Example of Taking a Partial Derivative Using Limits • Suppose that y = f(x1,x2) = 5x12+4x1x2+3x22 . • The partial derivative of y w.r.t. x1 is defined as the following:

48. Notes on Partial Derivatives • The partial derivative has the same rules as the derivative. • The key to working with partial derivatives is to keep in mind that you are holding all other variables constant except the one that you are changing.