# Document 13685653

```Math 1B
Project #1
20 Points
Due Date: Monday 8 February
NAMES
Area Functions
The purpose of this project is to introduce and apply the Fundamental Theorem of Calculus.
This project may be done in groups of 2 – 4 students. Each group will submit only one set of
solutions but each group member is expected to contribute. Please show all work on separate
1. a) Draw the line y = 2t + 1 and use geometry to find the area under this line, above the t-axis,
and between the vertical lines t = 1 and t = 3.
b) If x &gt; 1, let A( x ) be the area of the region that lies under the line y = 2t + 1 between t = 1
€
€ region and
€ use geometry to find an expression for A( x ) .
and t = x . Sketch this
€
€
€
c) Differentiate the area function A( x ) . What do you notice?
€
2. a) If x ≥ −1 , let
A( x ) =
€
x
∫ (1+ t ) dt
2
−1
A( x ) represents the area of a region. Sketch that region.
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b) Integrate to find an expression for A( x ) .
c) Find A′ ( x ) . What do you notice?
3. Suppose f is a continuous function on the interval [ a,b] and we define a new function g by
the equation
x
g ( x ) = ∫ f ( t ) dt
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a
Based on your results in problems 1 and 2, conjecture an expression for g ′ ( x ) .
4. If f ( x ) =
g(x)
∫
0
x
1
1+ t
3
dt , where g ( x ) =
( )
5. If f ( x ) = ∫ x 2 sin t 2 dt , find f ′ ( x ) .
0
cos x
∫
0
( )
⎡1+ sin t 2 ⎤ dt , find f ! ( π2 ) .
⎣
⎦
```