Calculus Notes 4.3: How Derivatives Affect the Shape of a Graph4.4: Limits at infinity; Horizontal Asymptotes.4.5: Summary of Curve Sketching4.6: Graphing with Calculus and Calculators

Answers:2. 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.

Start up: 1. Compute

2. Why is it true that if f is concave upward, then all the tangent lines to f lie below the curve?

3 2

3 2

2 4 1lim

4 4x

x x

x x x

Calculus Notes 4.3: How Derivatives Affect the Shape of a Graph4.4: Limits at infinity; Horizontal Asymptotes.4.5: Summary of Curve Sketching4.6: Graphing with Calculus and Calculators

1. .

Start up: 1. Compute

3 2

3 2

2 4 1lim

4 4x

x x

x x x

2 4 0 0

4 4 0 0

3 2

3

3 2

3

2 4 1

lim4 4x

x xx

x x xx

3

2

4 1lim 2

4 1lim 4

x

x

x x

x x

3

2

1 12 4 lim lim

1 14 4 lim lim

x x

x x

x x

x x

3

2

4 1lim 2 lim lim

4 1lim 4 lim lim

x x x

x x x

x x

x x

Limit Law #5 Limit Law #1 & #2

Limit Law #7 & #3 Theorem 4

2

4 1

2

Increasing/Decreasing Test:(a) If f ‘ (x) > 0 on an interval, then f is increasing on that interval.(b) If f ‘ (x) < 0 on an interval, then f is decreasing on that interval.

Calculus Notes 4.3: How Derivatives Affect the Shape of a Graph.

The First Derivative Test: Suppose that c is a critical number of a continuous function f.

(a) If f ‘ changes from positive to negative at c, then f has a local maximum at c.(b) If f ‘ changes from negative to positive at c, then f has a local minimum at c.(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: (a) If f “(x) > 0 for all x in I, then the graph of f is concave upward on I.(b) If f “(x) < 0 for all x in I, then the graph of f is concave downward on I.

Calculus Notes 4.5: Summary of Curve Sketching

Example 1:(a) On what interval(s) is the above function concave upward (CU)?

(b) On what interval(s) is the above function concave downward (CD)?

, , , , ,CU on b c d e e p

, , , , ,CD on a b c d p q

Calculus Notes 4.3: How Derivatives Affect the Shape of a Graph.

The Second Derivative Test: Suppose that f “ is continuous near c.(a) If f ‘ (c) = 0 and f “ (c) > 0, then f has a local minimum at c.(b) If f ‘ (c) = 0 and f “ (c) < 0, then f has a local maximum at c.

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.

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.

, . limx

a Then f x L

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.

, . limx

a Then f x L

Definition: The line y=L is called a horizontal asymptote of the curve y=f(x) if either: lim

xf x L

lim

xor f x L

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

1lim 0rx x

1

lim 0rx x

Definition: Let f be a function defined on some interval means that for every there is a corresponding number N such that

, . limx

a Then f x L

f x L whenever x N

0

Definition: Let f be a function defined on some interval means that for every there is a corresponding number N such that

, . limx

a Then f x L

f x L whenever x N

0

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

, . limx

a Then f x

f x M whenever x N

Example 2: given:(a) Find the intervals of increase or decrease.

(b) Find the local maximum and minimum values.

(c) Find the intervals of concavity and the inflection points.

(d) Use the information from parts (a)-(c) to sketch the graph.Check your work with a graphing device if you have one.

32 3f x x x

' 3 1 1f x x x 2' 3 3f x x

1,1

" 6f x x

, 1 & 1,

" 1 6 0

" 1 6 0

f

f

min 1,0 max 1,4

" 0 0f x on x

,0 0,CU on and CD on

Calculus Notes 4.3: How Derivatives Affect the Shape of a Graph.

Interval (1-x) (1+x) f ‘ f " 0 0f x on x

. 0,2Inflection pt at

1,1 , 1 sinDecrea g

1,

sinIncrea g

sinDecrea g

Calculus Notes 4.5: Summary of Curve Sketching

Guidelines for Sketching a Curve: A. DomainB. InterceptsC. SymmetryD. AsymptotesE. Intervals of Increase or DecreaseF. Local Maximum and Minimum valuesG. Concavity and points of InflectionH. 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)