2.3 curve sketching (introduction). we have four main steps for sketching curves: 1.starting with...

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2.3 Curve Sketching (Introduction)

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Page 1: 2.3 Curve Sketching (Introduction). We have four main steps for sketching curves: 1.Starting with f(x), compute f’(x) and f’’(x). 2.Locate all relative

2.3 Curve Sketching (Introduction)

Page 2: 2.3 Curve Sketching (Introduction). We have four main steps for sketching curves: 1.Starting with f(x), compute f’(x) and f’’(x). 2.Locate all relative

We have four main steps for sketching curves:

1. Starting with f(x), compute f’(x) and f’’(x).2. Locate all relative maximum and minimum

points and make a partial sketch.3. Examine concavity of f(x) and locate

inflection points.4. Consider other properties of the graph such

as intercepts and complete the sketch.

Page 3: 2.3 Curve Sketching (Introduction). We have four main steps for sketching curves: 1.Starting with f(x), compute f’(x) and f’’(x). 2.Locate all relative

Locating Relative Extreme Points

Page 4: 2.3 Curve Sketching (Introduction). We have four main steps for sketching curves: 1.Starting with f(x), compute f’(x) and f’’(x). 2.Locate all relative

The tangent line has a slope of zero at relative maximum and relative minimum points. So, to find relative extreme points, we find values of x so that f’(x) = 0.

Look for possible relative extreme points of f(x) by setting f’(x) = 0 and solving for x.

Page 5: 2.3 Curve Sketching (Introduction). We have four main steps for sketching curves: 1.Starting with f(x), compute f’(x) and f’’(x). 2.Locate all relative

Is the point a relative maximum point or a relative minimum point?

How can we tell?

• Check concavity at relative extreme point using second derivative.

• Examine slope of nearby points on either side using the first derivative.

Page 6: 2.3 Curve Sketching (Introduction). We have four main steps for sketching curves: 1.Starting with f(x), compute f’(x) and f’’(x). 2.Locate all relative

Locating Inflection Points

Page 7: 2.3 Curve Sketching (Introduction). We have four main steps for sketching curves: 1.Starting with f(x), compute f’(x) and f’’(x). 2.Locate all relative

An inflection point can only occur at a value of x for which f’’(x) = 0 because the curve is concave up when f’’(x) is positive and concave down when f’’(x) is negative.

Look for possible points of inflection by setting f’’(x) = 0 and solving for x.

Page 8: 2.3 Curve Sketching (Introduction). We have four main steps for sketching curves: 1.Starting with f(x), compute f’(x) and f’’(x). 2.Locate all relative

563)(Graph 2 xxxf1. Find relative extreme points…find x where f’(x) = 0

563)(' 2 xxdx

dxf

rule sum via563 2

dx

dx

dx

dx

dx

d

rule multipleconstant via563 2

dx

dx

dx

dx

dx

d

rulepower via0)1(6)2(3 x66)(' xxf

066 x1

66

x

x Relative extreme point at (-1, f(-1)=-8)

Page 9: 2.3 Curve Sketching (Introduction). We have four main steps for sketching curves: 1.Starting with f(x), compute f’(x) and f’’(x). 2.Locate all relative

2. Check concavity at relative extreme point, x = -1.

66)(' xxf

66)(')('' xdx

dxf

dx

dxf

rule sum via66dx

dx

dx

d

rule multipleconstant via66dx

dx

dx

d

rulepower via0)1(6

6)('' xf

So, the second derivative is 6 (concave up) for all x, including x = -1.

Page 10: 2.3 Curve Sketching (Introduction). We have four main steps for sketching curves: 1.Starting with f(x), compute f’(x) and f’’(x). 2.Locate all relative

563)(Graph 2 xxxf

(-1, -8)Concave up

Page 11: 2.3 Curve Sketching (Introduction). We have four main steps for sketching curves: 1.Starting with f(x), compute f’(x) and f’’(x). 2.Locate all relative

2436152)(Graph 23 xxxxf1. Find relative extreme points…find x where f’(x) = 0

2436152)(' 23 xxxdx

dxf

rule sum via24 36152 23

dx

dx

dx

dx

dx

dx

dx

d

rulemult const via24 36152 23

dx

dx

dx

dx

dx

dx

dx

d

rulepower via0 )1(36)2(15)3(2 2 xx

36306)(' 2 xxxf

Page 12: 2.3 Curve Sketching (Introduction). We have four main steps for sketching curves: 1.Starting with f(x), compute f’(x) and f’’(x). 2.Locate all relative

36306)(' 2 xxxf

0 36306 2 xx0 )65(6 2 xx

0 652 xx0 2)3)(x-x(

3x

0 3-x

2x

0 2x

Relative extreme point at (3, f(3)= 3)

Relative extreme point at (2, f(2)= 4)

Page 13: 2.3 Curve Sketching (Introduction). We have four main steps for sketching curves: 1.Starting with f(x), compute f’(x) and f’’(x). 2.Locate all relative

2. Check concavity at relative extreme points, x = 2, 3.

36306)(')('' 2 xxdx

dxf

dx

dxf

rule sum via36306 2

dx

dx

dx

dx

dx

d

rulemult const via36 306 2

dx

dx

dx

dx

dx

d

3012)('' xxf

36306)(' 2 xxxf

rulepower via0 )1(30)2(6 x

down concave , 630)2(12)2('' f

up concave , 630)3(12)3('' f

Page 14: 2.3 Curve Sketching (Introduction). We have four main steps for sketching curves: 1.Starting with f(x), compute f’(x) and f’’(x). 2.Locate all relative

3. Find inflection points, f’’(x) = 0

3012)('' xxf

03012 x

5.22

5

12

30

3012

x

x

Inflection point at (2.5, f(2.5)=3.5)

Page 15: 2.3 Curve Sketching (Introduction). We have four main steps for sketching curves: 1.Starting with f(x), compute f’(x) and f’’(x). 2.Locate all relative

(2, 4)Concave down

(3, 3)

Concave up

(2.5, 3.5)Inflection point

2436152)(Graph 23 xxxxf

Graphical Relationships Among f(x), f'(x), f''(x)

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