Math Help - AP Exam Practice Packet : Free Response

1. 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.

2. 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.
...

3. 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.

4. Originally Posted by skeeter
c. Given that , write an equation of the line tangent to the graph of at the point where

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)?

5. Originally Posted by ione
Why would the slope of the line tangent to g at x=0 be g'(4)?
my mistake ... g'(0)

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

7. 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
$f'(0)=-3$