Math Review and Lessons in Calculus

Agenda Rules of Exponents Functions Inverses Limits Calculus

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

Rules of Exponents Cont.

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

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

Exponent Example Cont.

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)

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.

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.

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.

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)).

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.

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.

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

Suppose you had the following equation: 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

Define a = q, b = r, and c = s-y 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

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

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.

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.

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.

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.

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.

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.

Example of a Limit

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.

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

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:

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:

Example of Using Limits for a linear Equation

Example of Using Limits for a Quadratic

Example of Using Limits for a Cubic

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

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

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

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)

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)

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)

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

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

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

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

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).

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

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

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:

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.