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

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

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

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

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

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

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

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

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

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

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

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

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

Monday, Nov 27, 2017 MAT 145

Monday, Nov 27, 2017 MAT 145

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

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

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

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

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

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

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

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