Sketch the graph of a continuous function

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December 12th 2009, 08:00 AM
ughintegrals

Sketch the graph of a continuous function

Hi again I'm tackling a question that is very likely to be on a future exam, and I'm a bit stuck.

The question:

g(x){ 0 if x<2
2 if 2<x<4
-1 if x>4

Sketch the graph of a continuous function f(x) knowing that f(0)=1, and f'(x) = g(x)

Graphing g(x) is not a problem, but understanding how to formulate a seperate function with just the info given is a major roadblock.

Any suggestions would be helpful.
December 12th 2009, 09:16 AM
Danny

Quote:

Originally Posted by ughintegrals

Hi again I'm tackling a question that is very likely to be on a future exam, and I'm a bit stuck.

The question:

g(x){ 0 if x<2
2 if 2<x<4
-1 if x>4

Sketch the graph of a continuous function f(x) knowing that f(0)=1, and f'(x) = g(x)

Graphing g(x) is not a problem, but understanding how to formulate a seperate function with just the info given is a major roadblock.

Any suggestions would be helpful.

If $ g(x) = \left\{ \begin{array}{cl} 0 & \text{if}\; x < 2\\ 2 & \text{if}\; 2 < x < 4\\ -1 & \text{if}\; x > 4 \end{array}\right. $

then

$ f(x) = \left\{ \begin{array}{cl} c_1 & \text{if}\; x \le 2\\ 2x + c_2 & \text{if}\; 2 < x \le 4\\ -x + c_3 & \text{if}\; x > 4 \end{array}\right. $ .

Then use the fact that $f(0) = 1$ (this gives you $c_1$ ) and that $f(x)$ is continuous at $x = 2$ and $x = 4$ (this gives the remaining constants.)
December 12th 2009, 09:50 AM
ughintegrals

ugh, im almost there.

c1 was no sweat, but how am i supposed to use the x= 2 and x = 4 to solve the remaining constants? plug 2 into the 2x+c, or equate it to 2? or neither?

im just a bit stuck there....sorry for being difficult.
thanks for your help.