AP Calculous
posted by Yoona on .
let f be the function defined by
x1+2 for X<1
f(x)=
ax^2Bx, for X>or equal to 1. where a and b are constants
a)if a=2 and b=3 is f continious for all x? justify your answer
b)describe all the values of a and b for which f is a continious function
c) For what values of a and b is f both continious and differentiable?

f(x) is both functions.. I don't know why the system didn't let me keep the spaces to show that

First off, stop misspelling "calculus".
Now, if a=2 and b=3, we have
f(x) = 2x^2  3x
f(1) = 1
But lim x>1 = 11 + 2 = 2
So, f is not continuous at x=1.
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If f is to be continuous, it needs to be continuous at x=1. It is continuous everywhere else already.
So, ax^2  bx must = 2 at x=1
a  b = 2
So, there are any number of parabolas which will make f continuous at x=1.
6x^2  4x
3x^2 + 5x
etc.
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Now, for f to be also differentiable, the slopes must match at x=1
The slope of x1 is 1 when approaching from the left.
ax^2  (a2)x must also have slope 1 at x=1
f'(x) = 2ax  (a2)
f'(1) = 2a  a + 2 = 1
a = 3
So,
f(x) = x1 + 2 for x<1
f(x) = 3x^2 + 5x for x>=1
is both continuous and differentiable everywhere.