Math Review and Lessons in Calculus

Math Review and Lessons in Calculus

Agenda Rules o Eponents Functions Inverses Limits Calculus

Rules o Eponents 0 Zero Eponent Rule a * b ab Product Rule * 3 5 a / b a-b Quotient Rule 5 / 3 -a / a Negative Eponent Rule -5 / 5 3

Rules o Eponents Cont. Power Rules m / n n m /3 3 a b ab 5 *5 0

Rules o Eponents Cont. a y a y a Product Rule y y 5 * 5 * 5

Important Rule to Remember Regarding Eponents The product rule works when two items are multiplied, not added or subtracted, i.e., y n n y n E.g., y y y y y 6

7 Eponent Eample Assume that ½ ¼. Solve as a unction o. 0

8 Eponent Eample Cont.

Functions A unction describes how a set o unique variables is mapped/transormed into another set o unique variables. Eamples: Linear: y a b Quadratic: y a b c Cubic: y a 3 b c d Eponential: y ae Logarithmic: y aln 9

Functions Cont. A unction is comprised o independent and dependent variables. The independent variable is usually transormed by the unction, whereas the dependent variable is the output o the unction. In the eamples above y is a variable that is usually known as the dependant variable, while is considered the independent variable. 0

Functions Cont. Functions can map more than one independent variable into the dependent variable. Eample: y, a* b* c It is possible to represent a unction visually by using a graph.

Composition o Functions When one unction is evaluated inside another unction, it is said that you are perorming a composition o unctions. Mathematically, we represent a composition o unctions in two ways: g g Note this is NOT multiplication o the two unctions.

Eample o Composition Suppose and g 33, then: g33 6 6 67 g3 3 6 3 3 66 Note that g does not necessarily equal g. 3

Inverse Functions An inverse unction is a unction that can map the dependent variable into the independent variable. In essence, it is a unction that can reverse the independent and dependent variables. Eample: y a b has as its inverse unction y/a b/a.

Eample o Finding the Inverse o a Linear Equation y 5 0 Subtract 0 rom both sides y 0 5 Divide both sides by 5 y/5 y/5 This is the inverse o the irst equation above. 5

Quadratic Formula The quadratic ormula is an equation that allows you to solve or all the values that would make the ollowing equation true: a b c 0 b ± b ac a 6

Using the Quadratic Formula to Find the Inverse o a Quadratic Equation Suppose you had the ollowing equation: y q r s I we transorm the above equation into the ollowing, we can use the quadratic equation to ind the inverse: q r s-y 0 7

Using the Quadratic Formula to Find the Inverse o a Quadratic Equation Cont. Deine a q, b r, and c s-y Substituting these relationships into the above equation gives the ollowing: a b c 0 r ± r q s y q 8

Inverse o a Quadratic Equation Eample Suppose y 8 By rearrangement: 8 y 0 ± **8 * y ± 6 8*8 y ± 6 6 8* y ± 8 8* y 9

Finding the Intersection o Two Equations At the point where two equations meet, they will have the same values or the dependent and independent variables or each equation. Eample: y 5 7 and y 3 intersect at y 7 and. 0

Finding the Intercepts o a Curve or Line The vertical intercept is where the curve crosses or touches the y-ais. To ind the vertical intercept, you set 0, and solve or y. The horizontal intercept is where the curve crosses or touches the -ais. To ind the horizontal intercept, you set y 0, and solve or.

Slope o a Line From algebra, it is known that the slope m o a line is deined as the rise over the run, i.e., the change in the y value divided by the change in the value. m y y y

Slope o the Line Cont. The slope o a line is constant. The slope o a line gives you the average rate o change between two points. 3

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 o the curve at the one point. A tangent line can tell you the instantaneous rate o change at a point.

Limits A number L is said to be the limit o a unction at point t, i as you get closer to t, gets closer to L. lim L t 5

6 Eample o a Limit 0 0 0. 0.8 0.. 0. 0.6 0.. 0.6 0. 0.6.6 0.8 0. 0.8.8 0 0 lim lim y y y

Finding the Tangent Line One way to ind the tangent line o a curve at a given point is to eamine secant lines that have corresponding points that get closer to each other. This is known as eamining the limits. 7

Graphical Representation o the Secant and the Tangent Lines Y Function: y Secant Line Tangent Line X 8

Deining the Derivative According to Varian, the derivative is the limit o the rate o change o y with respect to as the change in goes to zero. Suppose y, then the derivative is deined as the ollowing: d d 0 lim 9

Equivalency o Derivative Notation There are many ways that are used to represent the derivative. Suppose that y, then the derivative can be represented in the ollowing ways: ' dy d d d 30

3 Eample o Using Limits or a linear Equation m m d dy m d dy b m b m m d dy d dy b m m b m b m y lim lim lim lim 0 0 0 0

3 Eample o Using Limits or a Quadratic d dy d dy d dy d dy y lim lim lim lim 0 0 0 0

33 Eample o Using Limits or a Cubic 0 3 0 3 3 3 0 0 3 3 3 3 3 3 lim3 3 3 lim 3 3 lim lim * 3* * 3* d dy d dy d dy d dy y

Dierentiation Rules Constant Rule Power Rule Constant Times a Function Rule Sum and Dierence Rule Product and Quotient Rule The Chain Rule Generalized Power Rule Eponential Rule 3

Constant Rule The constant rule states that the derivative o a constant unction is zero. y c ' 0 y 5 ' 0 35

36 Power Rule Suppose y n, then the derivative o is the ollowing: n n- 3 * ' 3 ' ' ' y y y y

Constant Times a unction Rule Suppose yag, then the derivative o is the ollowing: a*g y ' y ' ** 8 37

Sum and Dierence Rule Suppose yk ± g, then the derivative o K is: K ± g K 7 K ' 7* 38

The Product Rule Suppose yk * g, then the derivative o K is: K * g * g K 5 K' 5 5 K' 0 5 K' 5 39

0 Quotient Rule Suppose yk / g, then the derivative o K is: K [ * g - * g ]/ g ' 8 ' 3* ' 3 3 K K K K

Chain Rule Suppose ykg, then the derivative o K is: K g*g K K' K' 3

Generalized Power Rule Suppose yk[] n, then the derivative o K is: K n[] n- K 5 K' 5 K' 3 0

Eponential Rule Suppose y K e, then the derivative o K is: K e K e ; K e K e ; K e 3

The Second Derivative It is useul in calculus to look at the derivative o a unction that has already been dierentiating. This is known as taking the second derivative. The second derivative is usually represented by.

Second Derivative Cont. Suppose that y. The second derivative can also be represented as the ollowing: '' y'' d y d d d d d D 5

6 Partial Derivative Suppose that y,. The partial derivative o y is deined as the ollowing: 0 0,, lim,,, lim,

7 Eample o Taking a Partial Derivative Using Limits Suppose that y, 5 3. The partial derivative o y w.r.t. is deined as the ollowing: 0 0 0 0, 5 lim 0, 5 0 lim,,, lim, 3 5 0 5, 3 5, 3 5,

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 ecept the one that you are changing. 8