AP Exam Practice Packet : Free Response

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March 18th 2010, 05:30 PM
DarkestEvil

AP Exam Practice Packet : Free Response

i cant seem to understand this concept, its very confusing, can anyone give me any tips?

4. Suppose that the function $f$ has a continuous second derivative for all $x$ , and that $f(0)=2$ , $f'(o)=-3$ , and $f''(0)=0$ . Let $g$ be a function whose derivative is given by $g'(x)=e^{-2x} (3f(x)+2f'(x))$ for all $x$ .

a. Write an equation of the line tangent to the graph of the $f$ at the point where $x=0$
b. Is there sufficient information to determine whether or not the graph of $f$ has a point of inflection when $x=0$ ? Explain your answer.
c. Given that $g(0)=4$ , write an equation of the line tangent to the graph of $g$ at the point where $x=0$
d. Show that $g''(x)=e^{-2x} (-6f(x)-f'(x)+2f''(x))$ . Does $g$ have the local maximum at $x=0$ ? Justify your answer.

Some help would very much be appreciated.
March 18th 2010, 05:47 PM
skeeter

Quote:

Originally Posted by DarkestEvil

i cant seem to understand this concept, its very confusing, can anyone give me any tips?

4. Suppose that the function $f$ has a continuous second derivative for all $x$ , and that $f(0)=2$ , $f'(o)=-3$ , and $f''(0)=0$ . Let $g$ be a function whose derivative is given by $g'(x)=e^{-2x} (3f(x)+2f'(x))$ for all $x$ .

a. Write an equation of the line tangent to the graph of the $f$ at the point where $x=0$

they gave you everything you need to do that ... f(0) = 2 and f'(0) = -3

b. Is there sufficient information to determine whether or not the graph of $f$ has a point of inflection when $x=0$ ? Explain your answer.

what do you need to tell you that f(x) has inflection point(s) ?

c. Given that $g(0)=4$ , write an equation of the line tangent to the graph of $g$ at the point where $x=0$

another tangent line problem ... they gave you the point, g(0) = 4 , all you need is the value of g'(0) for the slope of the line ... can you find that?

d. Show that $g''(x)=e^{-2x} (-6f(x)-f'(x)+2f''(x))$ . Does $g$ have the local maximum at $x=0$ ? Justify your answer.

fairly straightforward ... they gave you g'(x) and want you to find g''(x) simplified to the given form.

...
March 18th 2010, 06:01 PM
DarkestEvil

Quote:

Originally Posted by skeeter

...

a. But how would I take the derivative if I don't know f(x)? How do I use g'(x) to do this?
b. I know to find these, I must take the second derivative of f(x), and set it equal to zero, find the x-values, and plug them into f(x) to get the y-value for the points.
March 18th 2010, 06:10 PM
ione

Quote:

Originally Posted by skeeter

c. Given that http://www.mathhelpforum.com/math-he...0cadc770-1.gif, write an equation of the line tangent to the graph of http://www.mathhelpforum.com/math-he...3614845d-1.gif at the point where http://www.mathhelpforum.com/math-he...8a3883fc-1.gif

another tangent line problem ... they gave you the point, g(0) = 4 , all you need is the value of g'(4) for the slope of the line ... can you find that?

Why would the slope of the line tangent to g at x=0 be g'(4)?
March 18th 2010, 06:14 PM
skeeter

Quote:

Originally Posted by ione

Why would the slope of the line tangent to g at x=0 be g'(4)?

my mistake ... g'(0)
March 18th 2010, 06:44 PM
DarkestEvil

I don't see where you're getting at, how do I use g'(x) in order to differentiate f(x)?
March 18th 2010, 09:28 PM
ione

Quote:

Originally Posted by DarkestEvil

I don't see where you're getting at, how do I use g'(x) in order to differentiate f(x)?

You do not need to differentiate f(x) because you are given that
$f'(0)=-3$