Calculus I (MAT 145) Dr. Day Thursday, April 25, 2013

1. Calculus I (MAT 145)Dr. Day Thursday, April 25, 2013 • The Definite Integral (4.9,5.1-5.4) This Week • Complete remaining Ch 5 WA • Gateway Quiz #7: Today! • Chapter 5 Test: Tomorrow! MAT 145

2. Accumulate, Accumulate, Accumulate! How much snow fell? MAT 145

3. Approximating Area: Riemann Sums To generate a way to calculate the area under the curve of a rate function, in order to determine an accumulation, we begin with AREA APPROXIMATIONS. We create something called a Riemann Sum and use better and better area approximations that will lead to exact area. MAT 145

4. Approximating Area: Riemann Sums Riemann Sum Applet MAT 145

5. If we know a rate function . . . Carlota Music Company estimates that the marginal cost of manufacturing its Professional Series guitars is given by the following in dollars/month when the level of production is x guitars/month: C'(x) = 0.004x + 50. The fixed costs incurred by Carlota are $9000/month. Determine a model for the total monthly cost C(x) incurred by Carlota in manufacturing x guitars/month. MAT 145

6. Antiderivatives, Integrals, and Initial Value Problems • Knowing f’, can we determine f ? General and specific solutions: The antiderivative. • The integral symbol: Representing antiderivatives • Initial Value Problems: Transforming a general antiderivative into a specific function that satisfies the given information. MAT 145

7. The Fundamental Theorem of Calculus (Part I) For f continuous on [a, b], let the function gbe Then g(x) is an antiderivative of f: MAT 145

8. The Fundamental Theorem of Calculus (Part II) Let f be continuous on [a, b]. Then, where F is any antiderivative of f; that is, F′(x)= f(x). MAT 145

9. Assignments WebAssign • 5.3 and 5.4 due tonight. Also • Chapter 5 Test (5.1-5.4 and 4.9): Friday April 26 • Calc I Review: Next Week! • Semester Exam: Monday May 6, 8 pm MAT 145