The document provides a summary and explanation of practice problems from an AP Calculus exam review session. It outlines solutions to questions on finding derivatives up to the fourth order, using the Fundamental Theorem of Calculus, finding tangent lines and intersections, optimizing an area problem using derivatives, applying the Second Fundamental Theorem of Calculus, and using separation of variables and derivative rules. It reminds students of upcoming exam deadlines and assignments.

3. Differentiate Me Four Times As shown, this question is asking for the fourth derivative of f(x). From the work shown here by Mr. K, you can see the common pattern as you make your way to the fourth derivative. That pattern being the fact that the exponent on the “thing” in the brackets is being decreased by 1 each time and the fact that you’re multiplying by 3 each time as well.

4. Differentiate Me Four Times Also take notice of the fact that the exponents from the previous derivatives are being brought down due to chain law and such as well. That would be where the 4! part is coming from in the answer.

5. Finding the Unknown “k” Mr. K said to just use your calculator for this question as it can get pretty ugly, kind of. But for those of you who wanted to know how to do it numerically, I’ve included my solution as to how to do that. Its basically the Fundamental Theorem of Calculus and for those who don’t remember what that was, it basically states that the integral of a derivative is equal to the change in the parent function along the given interval.

6. Tangent and Intersecting?! Looking back at this question, it wasn’t so hard afterall. I just failed at my reading comprehension. Anywho, the first part of this question is asking for the line tangent to the parent function between the two endpoints. To find that specific equation for that line, you’ll first have to find the slope of that line. So do your ∆y/∆x for the parent function and then you should find that the slope is equal to -2.

7. Tangent and Intersecting?! Now what about the specific (x,y) coordinates for that line? Well, that’s easy, you’ve found the slope haven’t you? And what’s one of the definitions of a derivative? Yes, it’s a rate of change or tangent line. Huzzah! Now substitute all the values into that handy dandy line equation given points (x,y) thing and solve! Now you can solve for the last part of the question which was to find the intersection. Look at your line equation, y-value is 31! No other choice has a 31 in it but choice c, question done.

8. Optimize That Area! There are a few other ways to solve this question, but I will explain using Mr. K’s method. First off, we should find the equation of the semi-circle. Basically just solve for ‘y’ since doing that will get you the function equation for the circle, which is the semi-circle. From there, write out the equation for the area of the rectangle.

9. Optimize That Area! Once you’ve gotten that, you can substitute in the value for ‘y’ and then massage the equation so that the 2x is within the square root. Moving onto the blue colored part, we are going to just take the part within the square root of A(x) since whatever is maximized within there is going to maximize the whole thing anyways.

10. Optimize That Area! Now, we find the derivative of a(x) so that we can use the 1 st Derivative rule on it in order to find our x-coordinate which has the maximum y-value. In this case, its +√32. Knowing that, we can now go back to A(x) and substitute that value in for ‘x’ and solve. You should end up with 64. Yay!

11. Accumulation Time This question follows the Second Fundamental Theorem of Calculus. “What’s that again?” you ask? Well let me refresh your memory, it basically states that if f is a continuous function on some interval [a,b] and an accumulation function is defined by the integral of parent function from [a,x] then the parent function is equal to the derivative, or accumulation function of that parent function. So as you can see in this question, all you’d have to do is arcsin(0.4) since the derivative is equal to the parent function anyways.

12. Separation of Variables Mr. K didn’t show how get the answer, but I’ve shown how to get it just in case there were some people in class who were too shy to say that they didn’t get it. So here it is. This question is just basic separation of variables, such as when we were working with Newton’s Law of Cooling. Basically, after you’ve found the general solution and such, just input the given values and solve. Voila!

13. Derivative Rules Galore Like the title implies, we are going to use the first and second derivative to solve for all these parts. First find where the derivative has zeroes and then use your 1 st Derivative analysis line and find where the minimum and maximum is. Parts a and c are finished with just that.

14. First Derivative Rule Galore As for the inflection points, first find the second derivative and then use the second derivative rule analysis to find where the signs change. This is because an inflection point is where the graph changes concavity. Now that you’ve got that done, question solved.

15. There you have it. Our first mini exam review, which I’m sure we’re going to do some more of in the (few) days to come. Also reminders, the Wiki Questions are due tonight at midnight. After that time, Mr. K will close down the site so that no one else will be able to edit for the time being, or until we are given our next task of editing each other’s significant contributions. Oh yeah, and remember that our DEV projects are due soon as well. So get a move on with that if you haven’t already got it moving. Aaaand…happy milkshake chooses Paul as next scribe. That is all. Ciao =D