Post #30

AP questions from last week..

2. Find the slope of the tangent line to the graph of f at x=4, given that f(x) = -x^2 += 4(x)^1/2
A. -8 B. -10 C. -9 D. -5 E. -7

To find the slope of a tangent line, take the derivative of the function given, then plug in the x-value given.
-2x + 2x^-1/2 OR -2x + 2 / x^1/2
-2(4) +2 / (4) ^1/2
-8 + 2/2 = -8 + 1 which equals -7.
The answer is E.

If the problems was asking for the equation of the tangent line, you would have to plug x into the original to get a y. Then you would plug x, y, and the slope you just found into point slope form.
-(4)^2 + 4(4)^1/2
-16+8 = -8
y+8 = -7 (x-4)

9. Give the equation of the normal line to the graph of y= 2x (x^2+8)^1/2 + 2 at the point (0,2).

Normal line is the same steps as tangent line except instead of using the slope, you use the perpendicular slope, which is the negative reciprocal of the slope.
First take the derivative which will be product rule
2x ((1/2 (x^2+8)^-1/2 ) (2x)) + (x^2+8)^1/2 (2)
4x^2 / 2 (x^2 +8) ^1/2 + 2 (x^2+8)^1/2

Instead of trying to simplify this, you can just plug in your x value (0) to find the slope.
0 + 2 times the square root of 8
which simplifies to, 4 times the square root of 2.
That would be the slope if you was finding the tangent line, but since you are finding the normal line, the slope is -1/ 4 times the square root of 2.
Now that we have a point and a slope, plug into point slope form.
y-2 = -1/4 times the square root of 2 (x-0)
Multiple each side by 4 square root of 2 to get rid of the fraction
4 square root of 2 y - 8 square root of 2 = -x
OR x + 4 square root of 2 y = 8 square root of 2
Answer choice B.

11. Compute the integral of 4x^2 (x^3+4) ^1/2 dx

u = x^3 +4 du= 3x^2

You have to get rid of the 3 in du, which you do by dividing something by 3 and gain a 4 which you do by multiplying by 4 so the integral is
4/3 S u^1/2
4/3 (2/3 ) u ^3/2
8/9 ( x^3 +4) ^3/2 + c
The answer is C.

I have a few question on the calculator portion if anyone wants to help.
2. I'm not sure what invertible means.
11.
14.
and 16.