Applied Calculus (MAT 121) Dr. Day Wednesday, March 28, 2012 Calculating Antiderivatives (6.1) and Area Under a Curve (6.3) Wednesday, March 28, 2012 MAT 121

Integration Big Ideas What is an antiderivative, how do we determine one, and how do we represent it with symbols? How can we determine the area under a curve or trapped between two curves, and how does the integral relate to that? The definite integral of a rate function gives use an accumulation. The Fundamental Theorem of Calculus: Connecting derivatives and integrals. Wednesday, March 28, 2012 MAT 121

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. Wednesday, March 28, 2012 MAT 121

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. Wednesday, March 28, 2012 MAT 121

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. Wednesday, March 28, 2012 MAT 121

Approximating Area: Riemann Sums Riemann Sum Applet Wednesday, March 28, 2012 MAT 121

Assignments WebAssign 6.1 due tonight 6.3 due tomorrow night Test #4: Check your scores, look for errors, revisit every problem you missed! Wednesday, March 28, 2012 MAT 121