Chapter 1: Introduction To Calculus

1. Chapter 1:Introduction To Calculus Jenn Huynh & Yvonne Huynh

2. 1.1: Radical Expression - Rationalizing • To re-write a radical expression with one term in the denominator √a / √b = √a / √b * √b / √b = √ab/ b • To rationalize the denominator, multiply with the conjugate 1 / √a-b = 1/√a-b * √a+b / √a+b = √a+b / √ a-b One Term Radical Two Term Expression

3. Example 1

4. 1.1: Con’t • Conjugate: • RECALL Difference of squares • x2 – 4 = (x – 2)(x + 2) • Apply to rationalization

5. 1.2: The Slope of a Tangent • What is slope of tangent? Is the limit of slopes of secants PQ as Q moves closer to P • To find the slope of a tangent at a point P(a, f(a)) • Find the value of f(a) • Find the value of f(a+h) • Evaluate lim h->0 f (a+h) – f(a) h When looking for the slope of the secant, use: m = y2 – y1because this slope has NO limit X2 – x1

6. Example 2 • Little Reminder: • When finding a perpendicular slope  use the negative reciprocal of the given slope

7. 1.3: Rates of change • ROC: Looking at how rapidly a Dependent variable can CHANGE when the independent variable CHANGE • Average Rate of Change (aka slope of secant) • Instantaneous Rate of Change Average Velocity is

8. Example 3 • Consider f(x) = (x+1)2. Find the rate of change in the y variable over the interval [-1, 2].

9. 1.4: The limit of a function • The limit of a function y=f(x) at x=a is written as: • This tells us the value of (y) as (x) gets infinity closer to (a) from either side […we don’t care about (a) but rather what happens near (a)] To have a limit: there must be the same limit value on both sides of (a) …or else it d.n.e

10. 1.4: Con’t • The limit MAY EXIST even when f(a) is not defined (when there is an asymptote or a hole) • The limit EQUAL to F(A) if the graph of f(x) passes the pt [a, f(a)]  Direct Substitution (as long as the graph is smooth/continuous on both sides) • One side limit: looking at the left OR the right side of the limit • Two sided limit: looking at both sides of the limit

11. 1.5: properties of limits • The following methods can be used to evaluate limits: • Direct substitution • Factoring • Rationalizing • One-sided limits • Change of variables • Always use direct substitution first to see if you can get a limit value • If you get 0/0, try other methods • If you get 0/1, the limit is 0 • If you get 1/0, there is an asymptote or a hole

12. 1.5: Con’t

13. 1.5: Con’t • HINT: if you’re dealing with a absolution function, remember to change it to a piecewise • Differences of Cube: 1. (A3 – B3) = (A – B)*(A2 + AB + B2) 2. (A3 + B3) = (A + B)*(A2 - AB + B2)

14. 1.6: Continuity • Remember that: A function is continuous when the graph doesn’t have any breaks. • Three types of discontinuity: 1. Point/ Removable Discontinuity 2. Jump Discontinuity 3. Infinite Discontinuity

15. 1.6: Con’t • The Definition of a Limit has 3 parts: • f(a) is defined which means that ‘a’ is in domain of f(x) • The limit exists (Right Hand limit = Left Hand limit) • lim x->a f(x) = f(a) • All polynomial functions are continuous (unless there are restrictions) • A rational function in simplified form is discontinue in the denominator at the zeros