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What deriva+ves tell us about a func+on Increasing/decreasing Concavity Cri+cal and inflec+on points UBC Math 102
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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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.
UBC Math 102
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.
UBC Math 102
Using calculus tools to understand func+ons
UBC Math 102
• Be able to classify “special points”: cri+cal points (local maxima and minima), inflec+on points, etc.
UBC Math 102
Using Calculus to sketch graphs: 1st deriva+ve
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Cri+cal points: f’(x)=0
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UBC Math 102
Checking the type of cri+cal point
• There are several ways to determine this, and we will see examples soon:
UBC Math 102
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.
UBC Math 102
(1) Suppose f’(x)<0, f’’(x)<0.
Then we can conclude that f(x) is decreasing and concave down f(x) x
UBC Math 102
(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.
UBC Math 102
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]
UBC Math 102
(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
UBC Math 102
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)
UBC Math 102
Test problem (2008)
UBC Math 102
Test problem
UBC Math 102
Test problem
UBC Math 102
Test problem
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UBC Math 102
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.
UBC Math 102 solu+on
Find √105
UBC Math 102
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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UBC Math 102 solu+on
Solu+on to test problem
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UBC Math 102
