Transcript Lesson 6-2

Lesson 4-6
Summary of Curve Sketching
With Calculators
Quiz
• Homework Problem: MVT & Rolle’s 4-2
Verify Rolle’s Theorem applies and find all c’s
f(x) = -x³- 3x² + 2x + 5
on [0,2]
• Reading questions:
– What is an oblique asymptote called?
– What is f(x) called if f(-x) = -f(x) for all x?
Objectives
• Sketch or graph a given function using your
calculator to help you
Vocabulary
• None new
Graphing Checklist
Domain – for which values is f(x) defined?
Division by 0 or negatives under even roots
x -intercepts – where is f(x) = 0?
y -intercepts – what is f(0)?
Type in solve(f(x)=0,x)
Type in f(x) | x = 0
Symmetry
Even functions
y-axis – is f(-x) = f(x)?
Origin – is f(-x) = -f(x)? Odd functions
Period – is there a number p such that f(x + p) = f(x)?
Trig functions
Asymptotes
Horizontal – does or exist? Limit as x→±∞
Vertical – for what is ? for what is ?
F3, limits
Type in Lim(f(x),x,a)
Division by 0 (and not removed by canceling)
Graphing Checklist (cont)
Derivative Information:
F3 dif(f(x),x)
Copy derivative and paste into solve(f’(x)=0,x)
Critical numbers – where does f’(x) = 0 or DNE?
Increasing – on what intervals is f’(x) ≥ 0? Type in f’(x) | x = value
Decreasing – on what intervals is f’(x) ≤ 0?
Local extrema – what are the local max/min?
Use f’ or f’’ test. 2nd DT: Type in f’’(x) | x = critical #
F3 dif(f’(x),x)
Copy derivative and paste into solve(f’’(x)=0,x)
Concavity
Up – where is f’’(x) > 0?
Type in f’’(x) | x = value
Down – where is f’’(x) < 0?
Inflection points – where does f change concavity?
Use calculator to check your info by graphing the function.
Be Careful: the small screen can lead to some tricky views
Example 1
1
Graph -------------x² – 4
f’(x) = -2x/(x² - 4)²
f’’(x) = 2(3x² +4)/(x² - 4)³
x≠±2
Domain:
x –intercepts:
None, y ≠ 0
y –intercepts:
y = -1/4
Symmetry: Y-axis: Yes Origin: No
Asymptotes H: y = 0
Critical numbers:
Periodic: No
V: x = -2, 2
x=0
Increasing:
x<0
Decreasing:
x>0
Max/Min:
At x = 0, y = -1/4 is a relative max
Concavity
Up: |x|>2
Down: |x| < 2
Example 1 Graph
y
x
Summary & Homework
• Summary:
– Calculator is a great tool that can help you with
many things
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Derivatives
Solutions to Equations
Function zeros
Functions evaluated at specific values
– Because of its small screen it can trick us to
seeing something that isn’t really there
• Homework:
– pg 330-331: 1, 4, 12, 15, 16