Calculus I (MAT 145) Dr. Day Mon day April 15, 2019

1. Calculus I (MAT 145)Dr. Day Monday April 15, 2019 Chapter 5: Integrals and Anti-Derivartives • Recovering a Function Knowing its Derivative • Riemann Sums • Riemann Sums with an Infinite Number of Subdivisions • Definite Integrals and Indefinite Integrals: Connecting Derivatives and Anti-Derivatives • The Fundamental Theorem of Calculus • Part 1 • Part 2 MAT 145

2. 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. Read this: “The antiderivative of 2x with respect to the variable x” MAT 145

3. If we know a rate function . . . A particle moves along the x-axis. It’s velocity is given by v(t) = 2t2-3t+1 If we know that the particle is at location s = 3 at time t = 0, that is, that s(0) = 3, determine the position function s(t). What is the particle’s location at time t = 10? MAT 145

4. If we know a rate function . . . Snow begins falling at midnight at a rate of 1 inch of snow per hour. It stops snowing at 6 am, 6 hours later. Write an accumulation function S(t), to describe the total amount of snow that had fallen by time t, where 0 ≤ t ≤ 6 hrs. MAT 145

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

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

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

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

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

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

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

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

13. Areas and Distances (5.1) • Use What You Know to Get at What You’re Looking For • Choosing Endpoints • Notation • Accumulations From Rates MAT 145

14. 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

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

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18. Riemann Sums Calculate the exact value of a Riemann Sum to approximate the area under the curve y = x2-4 for 1 ≤ x ≤ 4, using n = 3 rectangles and using midpoints. • Show a graph of the function that includes a sketch of your rectangles. • Show all calculations using exact values. • Clearly indicate the value of your Riemann Sum. MAT 145

19. Riemann Sums Suppose y = f(x) = x2-4 is an object’s velocity function. On 1 ≤ x ≤ 4: • What is the NET CHANGE IN POSITION? • What is the TOTAL DISTANCE TRAVELED? • If , What is ? MAT 145

20. upper limit of integration differential This is called a definite integral. It includes lower bounds and upper bounds that represent boundary values of an x-axis interval. integrand lower limit of integration MAT 145

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