THS Step By Step Calculus Chapter 1

Transcription

1 Name: Class Period: Throughout this packet there will be blanks you are epected to fill in prior to coming to class. This packet follows your Larson Tetbook. Do NOT throw away! Keep in 3 ring binder until the end of the course. Chapter 1.1 Tangent Line/Area Problems How would you approimate: Tangent Line? Area Under Curve? 1 P a g e

2 Chapter 1.2 Finding Limits Graphically and Numerically: 1.2 Finding Limits: What happens NEAR the value a? Possible solutions: Eists give value or Does not eist (DNE) and reason a. Graphically Draw a graph and observe behavior on left and right a.1 Eists: Graph approaches same value on right and left. NOT have to equal the limit or even eist at the point! a.2 Does not eist: 3 Reasons a Limit does not eist: Left limit does not equal right limit Becomes unbounded Oscillates 1.2.b Numerically Make a chart of values EXISTS: If numbers approach a value: DNE: Functional values do no approach one value: 2 P a g e

3 Formal Definition of Limit: To solve Epsilon-Delta problems for Delta: Problem format: Given the limit: lim ( ) find δ such that ( ) ε whenever Eample: 3 f L c lim(2 5) 1 find δ such that ( ) whenever To find delta: Step 1: Simplify (2-5)-1 = 2-6 = 2(-3) Step 2: Use inequality properties to solve for so / Step 3: This will be the delta: δ= P a g e

4 Basic Limits: (I filled in the first one you fill in the rest) Chapter 1.3 Finding Limits Analytically Properties of Limits: Limits of Polynomials and Rational Functions Limits of Radical Functions Limits of Composite Functions Limits of Trig Functions Limits of Functions that agree at all but one point: Steps for solving limits Analytically Step 1: Determine if function can be evaluated by direct substitution: 2 lim 3 5 = 2 Step 2: Function cannot be evaluated by direct substitution (zero in denominator) Find a function g() that agrees with f() at every point ecept the point c. ( ) 2.1 If function is a ratio of two polynomials: ( ) ( ) ( )( ) Step 1: Factor both numerator and denominator ( ) ( ) Step 2: Divide out like factors and evaluate ( ) ( ) 2.2 If the function has a radical being added in numerator or denominator: Step 1: Rationalize 4 P a g e 1 2 ( 1 2) ( 1 2) ( 1) 4 lim lim lim ( 3) ( 1 2) 3 ( 3)( 1 2) Step 2: Algebraically Simplify ( 3) lim 3 ( 3)( 1 2) Step 3: Evaluate

5 1 1 lim 3 ( 1 2) 4 Special Trig Limits: 1) sin lim 0 Graph on calculator and sketch here: Make a table of Values: ?? f() Does the function eist at =0? What is the limit? sin lim 0 2) 1 cos lim 0 Graph on calculator and sketch here: Make a table of Values: ?? f() Does the function eist at =0? What is the limit? 1 cos lim 0 3) If argument of sine or cosine is not, do a change of variables. sin 2 lim?? 0 Step 1: Identify argument: y=2 Step 2: Solve for : =y/2 Step 3: Identify how the two limits are related: Step 4: Change variables lim y lim 2 0. So you can conclude 0 0 sin 2 sin(2 y / 2) sin(2 y / 2) 2 sin( y) lim lim lim lim 0 y 0 ( y / 2) y 0 ( y / 2) 1 y 0 y y 1 y 0 y Step 5: Take Limit: 2 sin( ) lim P a g e

6 Squeeze Theorem: Eample: CLUE that you need this: a bounded trig function with a comple argument like ( ). 1 limsin( ) 0 2 Step 1: Identify the bounded function: Since ( ) you know Step 2: Write the rest of the bounded function: Step 3: Multiply through by the rest of the function: 2 Step 4: Take the limit. 1 sin( ) lim( ) lim sin( ) lim sin( ) sin( ) 1 Step 5: Since the limit on the LEFT = limit on the RIGHT = 0, then 6 P a g e

7 Chapter 1.4 Continuity and One Sided Limits Are these continuous on the open interval? If not what can you say about the nature of the discontinuity? 3 Criterion for Continuity at a Point: Continuity on an Open Interval 2 Types of Discontinuities: Steps for Analyzing continuity. If any condition is violated the function is not continuous. Step 1. Identify the domain Step 2. Check: Is f(c) defined for all c in domain? Step 3. Does Step 4. Does One Sided Limits: 1. Limit from left means: 2. Limit from right means: Give one eample for why you might need a one sided limit. 7 P a g e

8 Eistence of Limit: Definition of Continuity on Closed Interval Properties of Continuity What 4 functions are always continuous on their domain? Continuity of a Composite Function 8 P a g e

9 Intermediate Value Theorem To use IVT To find Zeroes: Step 1: Show f() is continuous on [a,b] Step 2: Pick a closed interval [a,b] such that f(a)>0 and f(b)<0 {or f(a)<0 and f(b)>0} Step 3: Evaluate f(a) and f(b) Step 4: Let k = 0 Step 5: Use IVT to state there must be at least one zero 9 P a g e

10 Chapter 1.5 Infinite Limits Definition of Infinite Limit: Does the = sign mean the limit eists? 10 P a g e

11 Vertical Asymptotes: To Find Vertical Asymptotes: Step 1: Factor numerator and denominator Step 2: Reduce Step 3: Set denominator = 0 Properties of Infinite Limits: 11 P a g e