AP Exam Practice Packet : Free Response

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• Mar 18th 2010, 04:30 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 \$\displaystyle f\$ has a continuous second derivative for all \$\displaystyle x\$, and that \$\displaystyle f(0)=2\$, \$\displaystyle f'(o)=-3\$, and \$\displaystyle f''(0)=0\$. Let \$\displaystyle g\$ be a function whose derivative is given by \$\displaystyle g'(x)=e^{-2x} (3f(x)+2f'(x))\$ for all \$\displaystyle x\$.

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

Some help would very much be appreciated.
• Mar 18th 2010, 04:47 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 \$\displaystyle f\$ has a continuous second derivative for all \$\displaystyle x\$, and that \$\displaystyle f(0)=2\$, \$\displaystyle f'(o)=-3\$, and \$\displaystyle f''(0)=0\$. Let \$\displaystyle g\$ be a function whose derivative is given by \$\displaystyle g'(x)=e^{-2x} (3f(x)+2f'(x))\$ for all \$\displaystyle x\$.

a. Write an equation of the line tangent to the graph of the \$\displaystyle f\$ at the point where \$\displaystyle 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 \$\displaystyle f\$ has a point of inflection when \$\displaystyle x=0\$? Explain your answer.

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

c. Given that \$\displaystyle g(0)=4\$, write an equation of the line tangent to the graph of \$\displaystyle g\$ at the point where \$\displaystyle 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 \$\displaystyle g''(x)=e^{-2x} (-6f(x)-f'(x)+2f''(x))\$. Does \$\displaystyle g\$ have the local maximum at \$\displaystyle x=0\$? Justify your answer.

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

...
• Mar 18th 2010, 05:01 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.
• Mar 18th 2010, 05:10 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)?
• Mar 18th 2010, 05:14 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)
• Mar 18th 2010, 05:44 PM
DarkestEvil
I don't see where you're getting at, how do I use g'(x) in order to differentiate f(x)?
• Mar 18th 2010, 08:28 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
\$\displaystyle f'(0)=-3\$