1. Calculus Notes 4.3: How Derivatives Affect the Shape of a Graph
4.4: Limits at infinity; Horizontal Asymptotes.
4.5: Summary of Curve Sketching
4.6: Graphing with Calculus and Calculators
Start up:
Compute
Why is it true that if f is concave upward, then all the tangent lines to f lie below the curve?
Answers:
If f is concave up, the derivative of f is always increasing. Therefore, at a given point, the function is always steeper than its tangent line to the right, and shallower than its tangent line to the left.

2. Start up: Compute 1. . Limit Law #1 & #2 Limit Law #5
Calculus Notes 4.3: How Derivatives Affect the Shape of a Graph
4.4: Limits at infinity; Horizontal Asymptotes.
4.5: Summary of Curve Sketching
4.6: Graphing with Calculus and Calculators
Start up:
Compute
1. .
Limit Law #1 & #2
Limit Law #5
Limit Law #7 & #3
Theorem 4

3. Increasing/Decreasing Test:
Calculus Notes 4.3: How Derivatives Affect the Shape of a Graph.
Increasing/Decreasing Test:
If f ‘ (x) > 0 on an interval, then f is increasing on that interval.
If f ‘ (x) < 0 on an interval, then f is decreasing on that interval.
The First Derivative Test: Suppose that c is a critical number of a continuous function f.
If f ‘ changes from positive to negative at c, then f has a local maximum at c.
If f ‘ changes from negative to positive at c, then f has a local minimum at c.
If f ‘ does not change sign at c, then f has no local maximum or minimum at c.
(for example, if f ‘ is positive on both sides of c or negative on both sides).
Definition: If the graph of f lies above all of its tangents on an interval I, then it is called concave upward on I. If the graph of f lies below all of its tangents on I, it is called concave downward on I.
Concavity Test:
If f “(x) > 0 for all x in I, then the graph of f is concave upward on I.
If f “(x) < 0 for all x in I, then the graph of f is concave downward on I.

4. On what interval(s) is the above function concave upward (CU)?
Calculus Notes 4.5: Summary of Curve Sketching
Example 1:
On what interval(s) is the above function concave upward (CU)?
On what interval(s) is the above function concave downward (CD)?

5. Calculus Notes 4.4: Limits at infinity; Horizontal Asymptotes.
Calculus Notes 4.3: How Derivatives Affect the Shape of a Graph.
Definition: A point P on a curve y=f(x) is called an inflection point if f is continuous there and the curve changes from concave upward to concave downward or from concave downward to concave upward at P.
The Second Derivative Test: Suppose that f “ is continuous near c.
If f ‘ (c) = 0 and f “ (c) > 0, then f has a local minimum at c.
If f ‘ (c) = 0 and f “ (c) < 0, then f has a local maximum at c.
Calculus Notes 4.4: Limits at infinity; Horizontal Asymptotes.
Definition: Let f be a function defined on some interval
means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large.
Definition: Let f be a function defined on some interval
means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large negative.
Definition: The line y=L is called a horizontal asymptote of the curve y=f(x) if either:

6. Calculus Notes 4.4: Limits at infinity; Horizontal Asymptotes.
Theorem: If r > 0 is a rational number, then
If r > 0 is a rational number such that xr is defined for all x, then
Definition: Let f be a function defined on some interval
means that for every there is a corresponding number N such that
Definition: Let f be a function defined on some interval
means that for every there is a corresponding number N such that
Definition: Let f be a function defined on some interval
means that for every positive number M there is a corresponding positive number N such that

7. Find the intervals of increase or decrease.
Calculus Notes 4.3: How Derivatives Affect the Shape of a Graph.
Example 2: given:
Find the intervals of increase or decrease.
Find the local maximum and minimum values.
Find the intervals of concavity and the inflection points.
Use the information from parts (a)-(c) to sketch the graph.
Check your work with a graphing device if you have one.
Interval
(1-x)
(1+x)
f ‘ f

8. Guidelines for Sketching a Curve: Domain Intercepts Symmetry
Calculus Notes 4.5: Summary of Curve Sketching
Guidelines for Sketching a Curve:
Domain
Intercepts
Symmetry
Asymptotes
Intervals of Increase or Decrease
Local Maximum and Minimum values
Concavity and points of Inflection
Sketch the curve.
For next class, detail how you would find and/or do the above list of items. Include things from Algebra 2; College Algebra; Trigonometry; and Calculus
PS 4.3 pg.247 #1, 3, 6, 7, 8, 11, 12, 17, 27, 29, 40, 48, 51 (13)
PS 4.4 pg.260 #3, 7, 9, 10, 13, 24, 25, 42, 43, 47, 48, 65a (12)