Difference between revisions of "009B Sample Final 2, Problem 1"

Revision as of 20:07, 17 February 2017

a) State both parts of the Fundamental Theorem of Calculus.

b) Evaluate the integral

\int _{0}^{1}{\frac {d}{dx}}{\bigg (}e^{\tan ^{-1}(x)}{\bigg )}dx

c) Compute

{\frac {d}{dx}}\int _{1}^{\frac {1}{x}}\sin t~dt

Solution:

(a)

(b)

Step 1:

Step 2:

(c)

Step 1:

Step 2:

@@ Line 1: / Line 1: @@
-<span class="exam">Consider the region bounded by the following two functions:
+::<span class="exam">a) State '''both parts''' of the Fundamental Theorem of Calculus.
-::::::::<span class="exam"> <math style="vertical-align: -5px">y=2(-x^2+9)</math> and <math style="vertical-align: -4px">y=0</math>.
-<span class="exam">a) Using the lower sum with three rectangles having equal width, approximate the area.
+::<span class="exam">b) Evaluate the integral
-<span class="exam">b) Using the upper sum with three rectangles having equal width, approximate the area.
+::::<math>\int_0^1 \frac{d}{dx} \bigg(e^{\tan^{-1}(x)}\bigg)dx</math>
-<span class="exam">c) Find the actual area of the region.
+::<span class="exam">c) Compute
+::::<math>\frac{d}{dx}\int_1^{\frac{1}{x}} \sin t~dt</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"