<p>Do we have to know proofs for the AP Calc AB Exam? </p>
<p>Ex. The Proof for the the Product Rule for derivatives </p>
<p>Thanks :)</p>
<p>No proofs of that kind are on the AP Calculus exams (either AB or BC).</p>
<p>The closest thing to a proof that they might ask you to do on the AP exams is to use rules or theorems to show why something must be true (i.e. using the Mean Value Theorem to show that two expressions are equal, or using the Intermediate Value Theorem to explain why f(c) = 0 must be true on some interval from [a, b]).</p>
<p>^How exactly would you do that? Can you clarify with an example?</p>
<p>There are many in the old AP tests that you can find on AP Central, but let me show you a quick one.</p>
<p>Question: Let f(x) = x^3 + 3x^2 + 4x - 9. Show that f(x) = 2 must be true for some value c on the interval [0, 2].</p>
<p>Answer: Since f(x) is cubic, f(x) is continuous for all real numbers, and specifically, f(x) is continuous on the interval [0, 2]. f(0) = -9. f(2) = 8 + 12 + 8 - 9 = 19. Since f(0) = -9 < 2 < 19 = f(2) and f(x) is continuous, the Intermediate Value Theorem (IVT) guarantees that f(c) = 2 for some c on the interval [0, 2].</p>
<p>Commentary: There’s a lot of mathematical verbiage there that they would love to see on the AP test, but for scoring, they usually just want to see three things:</p>
<p>(1) That you’ve noted that f(x) is continuous on the appropriate interval.
(2) That the y-value they’ve given you (in this case 2) is indeed between the two y-values that correspond to the 2 x-values they’ve given you.
(3) That these facts require f(c) = 2 for some c between the two x-values.</p>
<p>You don’t usually need to cite the Intermediate Value Theorem by name. You just need to show that the hypothesis is true, and the conclusion that therefore follows.</p>