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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 sheets of paper and staple this page to the front. 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 > 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. € 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 € 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 ) . ⎣ ⎦