1. Calculus I (MAT 145) Dr. Day Monday November 27, 2017
Recovering Functions From Derivatives (4.9)
Procedures and Methods
Need to be a Derivatives Expert!
Symbols and Definitions
Applications
Riemann Sums! (5.1 & 5.2)
Using areas of rectangles to approximate area under a curve Using something we KNOW in order to uncover something we need!
Monday, Nov 27, 2017
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.
Monday, Nov 27, 2017
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?
Monday, Nov 27, 2017
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.
Monday, Nov 27, 2017
MAT 145

5. Accumulate, Accumulate, Accumulate!
How much snow fell?
Monday, Nov 27, 2017
MAT 145

6. Accumulate, Accumulate, Accumulate!
How much snow fell?
Monday, Nov 27, 2017
MAT 145

7. Accumulate, Accumulate, Accumulate!
How much snow fell?
Monday, Nov 27, 2017
MAT 145

8. Accumulate, Accumulate, Accumulate!
How much snow fell?
Monday, Nov 27, 2017
MAT 145

9. Accumulate, Accumulate, Accumulate!
How much snow fell?
Monday, Nov 27, 2017
MAT 145

10. Accumulate, Accumulate, Accumulate!
How much snow fell?
Monday, Nov 27, 2017
MAT 145

11. Accumulate, Accumulate, Accumulate!
How much snow fell?
Monday, Nov 27, 2017
MAT 145

12. Accumulate, Accumulate, Accumulate!
How much snow fell?
Monday, Nov 27, 2017
MAT 145

13. 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.
Monday, Nov 27, 2017
MAT 145

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MAT 145

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MAT 145

16. Approximating Area: Riemann Sums
Riemann Sum Applet
Monday, Nov 27, 2017
MAT 145

17. Areas and Distances (5.1)
Use What You Know to Get at What You’re Looking For
Choosing Endpoints
Notation
Accumulations From Rates
Monday, Nov 27, 2017
MAT 145

18. 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).
Monday, Nov 27, 2017
MAT 145

19. The Fundamental Theorem of Calculus (Part I)
For f continuous on [a, b], let the function g be
Then g(x) is an antiderivative of f:
Monday, Nov 27, 2017
MAT 145

20. Derivative Patterns You Must Know
The Derivative of a Constant Function
The Derivative of a Power Function
The Derivative of a Function Multiplied by a Constant
The Derivative of a Sum or Difference of Functions
The Derivative of a Polynomial Function
The Derivative of an Exponential Function
The Derivative of a Logarithmic Function
The Derivative of a Product of Functions
The Derivative of a Quotient of Functions
The Derivatives of Trig Functions
Derivatives of Composite Functions (Chain Rule)
Implicit Differentiation
Monday, Nov 27, 2017
MAT 145

21. Position, Velocity, Acceleration
An object is moving in a positive direction when ….
An object is moving in a negative direction when ….
An object speeds up when ….
An object slows down when ….
An object changes directions when ….
The average velocity over a time interval is found by ….
The instantaneous velocity at a specific point in time is found by ….
The net change in position over a time interval is found by ….
The total distance traveled over a time interval is found by ….
Monday, Nov 27, 2017
MAT 145

22. Position, Velocity, Acceleration
An object is moving in a positive direction when v(t) > 0.
An object is moving in a negative direction when v(t) < 0.
An object speeds up when v(t) and a(t) share same sign.
An object slows down when v(t) and a(t) have opposite signs.
An object changes directions when v(t) = 0 and v(t) changes sign.
The average velocity over a time interval is found by comparing net change in position to length of time interval (SLOPE!).
The instantaneous velocity at a specific point in time is found by calculating v(t) for the specified point in time.
The net change in position over a time interval is found by calculating the difference in the positions at the start and end of the interval.
The total distance traveled over a time interval is found by first determining the times when the object changes direction, then calculating the displacement for each time interval when no direction change occurs, and then summing these displacements.
Monday, Nov 27, 2017
MAT 145

23. Position, velocity, acceleration
Velocity: rate of change of position
Acceleration: rate of change of velocity
Velocity and acceleration are signed numbers. Sign of velocity (pos./neg.) indicates direction of motion (right/left or up/down)
When velocity and acceleration have the same sign (both pos. or both neg.), then object is speeding up. This is because object is accelerating the same direction that the object is moving.
When velocity and acceleration have opposite sign (one positive and one negative), then object is slowing down. This is because object is accelerating the opposite direction that the object is moving.
Monday, Nov 27, 2017
MAT 145