increasing/decreasing% concavity% …keshet/m102/2015/lect5.1.pdf · 2015-10-05 ·...
TRANSCRIPT
What deriva+ves tell us about a func+on
Increasing/decreasing Concavity
Cri+cal and inflec+on points
UBC Math 102
Course Calendar: Quiz this Friday
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Help with Homework
A space traveler is moving from leK to right along the curve y=x2. When she shuts off the engines, she will con+nue traveling along the tangent line at the point where she is at that +me. At what point (x,y) should she shut off the engines in order to reach the point (4,15)? Reworded: Find the tangent line to this curve that goes through the point (4, 15). See example we did in Lecture 4.1.
Help with Homework
Where is the func+on Increasing? Decreasing? Hint: Find cri+cal points. Determine which are min/max. draw rough sketch to see where fn is increasing or decreasing.
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Help with homework
Determine where the absolute extrema of the func+on on the interval [1,4] occur. Hint: Find cri+cal points and classify as local mins and maxes. Check f(x) at each of these as well as at the endpoints of the interval.
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Using calculus tools to understand func+ons
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• Be able to classify “special points”: cri+cal points (local maxima and minima), inflec+on points, etc.
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Using Calculus to sketch graphs: 1st deriva+ve
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Cri+cal points: f’(x)=0
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Checking the type of cri+cal point
• There are several ways to determine this, and we will see examples soon:
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Second deriva+ve, f’’(x)
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Concavity
• The second deriva+ve informa us about the curvature (concavity) of a func+on.
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Inflec+on points (IP):
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CHANGES SIGN
How to check for inflec+on points:
• (1) Reasoning algebraically:
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How to check for inflec+on points: (2) Reasoning usng the third deriva+ve f ‘’’(x): Ensure f ’’(x) is either increasing or decreasing at the poten+al inflec+on point (so that it changes sign there). f’’’(x)>0 f’’’(x)<0 (When f’’(x)=0 and f’’’(x) ≠0, we can conclude IP is present)
(1) Suppose f’(x)<0, f’’(x)<0.
Then we can conclude that (A) The func+on is constant and concave up. (B) The func+on is decreasing and concave
down. (C) The func+on is decreasing and concave up. (D) The func+on is increasing and concave down. (E) There is a local maximum.
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(1) Suppose f’(x)<0, f’’(x)<0.
Then we can conclude that f(x) is decreasing and concave down f(x) x
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(2) If f’(x0)=0 and f’’(x0)>0
Then we can conclude that (A) The func+on is increasing and has an
inflec+on point at x0 (B) The func+on is decreasing and concave up. (C) There is a local max at x0 (D) There is a local min at x0 (E) None of the above.
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(2) If f’(x0)=0 and f’’(x0)>0
• Cri+cal point concave up
x0 x0 x0
èLocal min at x0
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(3) Inflec+on points
Which of the following is true? (A) Both y=x4 and y=x3 have inflec+on points at
x=0. (B) Both y=x4 and y=x3 sa+sfy f’’(x)=0 at x=0. (C) Both (A) and (B) are true. (D) Neither (A) nor (B) is true.
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Inflec+on points
Graphs of y=x4 and y=x3 Both have f’’(0)=0!
Only one has an inflec+on point
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Sketching a graph using calculus tools
Consider the func+on y = f(x) = Use pen & paper to sketch the graph of this func+on, indica+ng the zeros, cri+cal points, inflec+on points of the func+on. Determine the absolute minimum and absolute maximum of the func+on on the interval [0,3]
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(4) Overall shape using simple reasoning
I expect the func+on to look like: (A) (B) (C)
(D) (E)
Overall shape of f
• Shape (from Week 1):
• ShiKed up y axis by 5
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Zeros of f(x)
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First deriva+ve and cri+cal points
• f(x)=
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Second deriva+ve and poten+al inflec+on points (to be checked)
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Overall sketch
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Actual graph
• Infl Pts
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Local max
Local mins
Next +me: Smallest Net Growth Rate
Given rate of preda+on And rate of growth Find the popula+on size x for which the Net Growth Rate F(x)= G(x)-‐P(x) is smallest.
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Answers
• 1 B • 2 D • 3 B • 4 D
Try this
• Sketch a graph of the func+on y = f(x) = x3 – a x Using calculus tools. (Find cri+cal points, inflec+on points, sketch f)
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Test problem (2008)
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Test problem
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Test problem
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Test problem
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Newton’s Method: Related test-‐like problem
Use the func+on and ini+al guess
to find an approxima+on for using Newton’s Method. Find x1 only.
Note: Last +me we used Linear Approxima+on to solve the same problem.
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Find √105
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Newton’s method
Solve x2-‐105=0
(4) Example Cont’d
• Using the func+on and ini+al guess
Newton’s method leads to which value for x1?
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Test problem (MT 1 2014)
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Solu+on to test problem
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