Bridges Math 115BAnswers to DIFFERENTIATION, INTEGRATION AND THE FUNDAMENTAL THEOREM OF CALCULUS1. MR(q)2.F(x)=1−e−x/ αReasoning: f (x) is that part of the pdf of the exponential random variable for x ≥ 0. If we integrate a pdf we get its cdf.3.f(x)=1b−aReasoning: F(x ) is that part of the cdf of the uniform random variable for a ≤ x ≤ b. If we differentiate a cdf we get its pdf.4. MP(q)5.f(x)=1b−a6. MC(q) This is the same kind of problem as #5, but it’s written in symbols. It says: If we integrate MC(q) and then we differentiate the result, we get what?7.f (x)8.4 x3;4 x3This is our “short cut” (The Power Rule) for differentiation.9.3 q210.x411.q312. The Fundamental Theorem of Calculus says: If F'(x)= f (x ), then∫abf(x)dx=F(b)−F(a). ∫134 x3dx=F(3)−F (1)In problem #10, we see that F(x)=x4, so ∫134 x3dx=(3)4−(1)4 ∫134 x3dx=81−1 ∫134 x3dx=80.13.4 x314.4 u3Bridges Math 115B15.3 q216.4 x3I will spend class time on Monday explaining 16, 17, and 18. The result of problem18 gets to the heart of the matter (possible final exam question). The result states:17.4 x3ddx∫axf(u)du=f (x ). See slide #156 in the file “MBD 2 Proj 2.ppt” in your e-text. This theorem can also be found on slides #194 and #195 in the file “MBD 2 Proj 1.ppt” in your e-text. This result is the second part of the Fundamental Theorem of Calculus.18.f
View Full Document
Unlocking...