009B Sample Midterm 2, Problem 2

This problem has three parts:

a) State the fundamental theorem of calculus.

b) Compute ${\frac {d}{dx}}\int _{0}^{\cos(x)}\sin(t)dt$

c) Evaluate $\int _{0}^{\frac {\pi }{4}}\sec ^{2}xdx$

Foundations:

Solution:

(a)

Step 1:
The Fundamental Theorem of Calculus has two parts.
The Fundamental Theorem of Calculus, Part 1
Let $f$ be continuous on $[a,b]$ and let $F(x)=\int _{a}^{x}f(t)dt$.
Then, $F$ is a differential function on $(a,b)$ and $F'(x)=f(x)$.

Step 2:
The Fundamental Theorem of Calculus, Part 2
Let $f$ be continuous on $[a,b]$ and let $F$ be any antiderivative of $f$.
Then, $\int _{a}^{b}f(x)dx=F(b)-F(a)$

(b)

Step 1:
The Fundamental Theorem of Calculus, Part 1

Step 2:

(c)

Step 1:
Using the Fundamental Theorem of Calculus, Part 2, we have
$\int _{0}^{\frac {\pi }{4}}\sec ^{2}xdx=\left.\tan(x)\right|_{0}^{\frac {\pi }{4}}$

Step 2:
So, we get
$\int _{0}^{\frac {\pi }{4}}\sec ^{2}xdx=\tan {\bigg (}{\frac {\pi }{4}}{\bigg )}-\tan(0)=1$

Final Answer:
(a)
The Fundamental Theorem of Calculus, Part 1
Let $f$ be continuous on $[a,b]$ and let $F(x)=\int _{a}^{x}f(t)dt$.
Then, $F$ is a differential function on $(a,b)$ and $F'(x)=f(x)$.
The Fundamental Theorem of Calculus, Part 2
Let $f$ be continuous on $[a,b]$ and let $F$ be any antiderivative of $f$.
Then, $\int _{a}^{b}f(x)dx=F(b)-F(a)$
(b)
(c) $1$

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