Introduction to Calculus

1. Introduction to Calculus Harmanpreet, Richelle, Umar

2. 1.1 Radical Expressions: Rationalizing Denominators • A rational number is a number that can be expressed as a fraction (quotient) containing integers • The process of changing the denominator from a radical (square root) to a rational number (integer) is called rationalizing the denominator • In certain situations, it may be more appropriate to rationalize the numerator

3. Rewrite a radical expression with a one-term radical in denominator: For a two-term expression, rationalize denominator by multiplying numerator and denominator by the conjugate, and then simplify :

4. 1.2 The Slope of a Tangent • The slope of the tangent to a curve at point P is the limiting slope of the secant PQ as the point Q slides along the curve toward P • Slope of the tangent is said to be the limit of the slope of the secant as Q approaches P along the curve The slope of the tangent to the graph y=f(x) at point P(a, f(a)) is:

5. Average rate of Change • Instantaneous Rate of Change

6. 1.3 Rates of Change • Dependent variable, y, can represent quantities such as volume, air temperature, and area • Independent variable, x, can represent quantities such as height, elevation, and length • Rate of change describes how rapidly the dependent variable changes when there is a change in the independent variable

7. Average velocity = change in position change in time • The instantaneous velocity of an object with position function s(t) at time t=a, is:

8. 1.4 The Limit of a Function • The limit may exist if f(a) is not defined • The limit can be equal to f(a) (graph of f(x) passes through the point (a, f(a))

9. Limit exists: • Otherwise, does not exist

10. 1.5 Properties of Limits

11. Substituting x=a into can yield indeterminant form 0/0. You may then be able to find an equivalent function that is the same as the function f for all values except at x=a. Then use substitution to find limit • To evaluate a limit algebraically, you can use the following techniques:-direct substitution -factoring-rationalizing-one-sided limits-change of variable

12. 1.6 Continuity Continuous Point Discontinuity Jump Discontinuity Infinite Discontinuity

13. A function is continuous at x=a if • f(a) is defined • exists Remember! • All polynomial functions are continuous for all real numbers • A rational function (h(x)=f(x)/g(x)) is continuous at x=a if g(a) does not equal 0 • When one sided limits are not equal to each other, the limit does not exist on this point and is not continuous