In this activity we will be trying to find areas under curves. We will start with a simple example. 1. Sketch the graph of 16 14 y = 2t + 3. 12 10 6 -6 5 2 3 3 5 6 t 2. Find the area under this curve between the lines t = 1, t = 4, and the t-axis using geometry. A= 5. 3 A = 6.3(15) A = 15 + A : 9 A- 15+ 9 = (24 = A 3. There is a point marked x on the axes above. It does not have a specific value but simply represents a generic location on the x axis. Find the area under this curve line between the lines t = 1, t = x, and the t-axis using geometry. A = W. (HI. Hz ) 4 = 24 + 3 -= 4=2 x+3 and 4 = 2+ + 3 4 = 2(1)+ 3 A = ( x - 1 ) (5+2x+3 4: 2 + 3 = 5 Your answer to question 3 should involve x, and we can think of this formula as representing a function which gives the area under the curve between t = 1 and t = x. We will call this area function A(x). 4. Evaluate A(4) and check that you get the same answer as you did to question 2. Next we will look at the rate of change of A with respect to x. 5. Indicate the region on your graph from question 1 whose area corresponds to A(x). 6. Pick a value h > 0, and indicate the region on your graph from question 1 whose area corresponds to A(x + h). 7. Use geometry to find an algebraic expression for the area of the region in your original A = (2*+ 3 + 26xth) graph which corresponds to A (x + h) - A(x). (Your answer will involve both x and h.) 2 8. Divide your answer to question 7 by h. This gives the average rate of change of A over a small interval. A(x+ h ) -A(x) h 9. Look at your answer to question 8 and let h - 0. 10. In the language of calculus describe what question 7, 8, and 9 accomplished. 11. Look at your answer to question 9 and the original function defined in question 1. How are they related? 12. Describe in the language of calculus how the function A you found in question 3 and the original function given in question 1 are related. Problem If the function you were given was not linear (such as g (t) = t3 + 2) -so that you could not find a formula for A(x) by geometry-can you think of another way to think of A(x + h) - A(x) that will make clear what the average rate of change converges to as h goes to 0? Write a detailed description of your method, using the language and notation of calculus