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AP Calculus AB

An interactive log for students and parents in my AP Calculus class. This ongoing dialogue is as rich as YOU make it. Visit often and post your comments freely.

Sunday, April 23, 2006

Blog Assignment 1

A differentiable function f defined on -7 < 0< 7 has f(0)=0 and f'(x) = 2x sin x- e^(-x^2) +1
a) describe the symmetry of f.
b) On what intervals is f decreasing?
c) For what values of x does f have a relative maximum? Justify your answer.
d) How many points of inflection does f have? Justify your answer.

a) f is an even function.
b) f'(x) = 2x sinx - e^(-x^2) + 1
2x sinx- e^(-x^2) +1 =0
x=-6.2024, -3.294, 0, 3.294, 6.2024
+ - + + - +
--------1----------1----------1-----------1-----------1-----------
-6.2024 , -3.294 , 0 , 3.294 , 6.2024
If f'(x) less than zero,f(x) is decreasing, therefore, f(x) is decreasing at (-6.2024, -3.294) and (3.294, 6.0224).'
c) If f'(x) larger than zero, f(x) is increasing; If f'(x) less than zero, f is decreasing.
from increasing to decreasing, f reaches a maximum. f has relative maximum at x= -6.2024, and x= 6.2024

d) f"(x)=2x cos x+ 2 sinx + 2x e^(-x^2)
2x cosx+2 sin x +2x e^(-x^2) = 0
x= -4.9136, -2.0405, 0, 2.0405, -4,9136
f has inflection points when f"(x)= o, so, f has 5 inflection points

• At 8:50 AM CDT,  Mr. Kuropatwa said…

(a) f' is even and f is odd. Can you figure out why? (Do you know any points on f?)

(b) The second interval of decrease should be (3.294, 6.202). Can you figure out why or was it a typo?

(c) The positive relative maximum is at x = 3.294 not x = 6.202. Can you figure out why?

(d) Correct!