MAT 1221 Survey of Calculus Section 2.3 Rates of Change

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Presentation transcript:

MAT 1221 Survey of Calculus Section 2.3 Rates of Change

Expectations Use equal signs Show formula steps Show individual derivatives steps Double check the algebra

HW WebAssign HW 2.3 There is a hint on problem 1 at the end of your HO. Additional HW listed at the end of the handout (need to get done, but no need to turn in)

Fact: Slope of Tangent Line

What is “Rate of Change”? We are going to look at how to understand and how to find the “ rate of change ” in terms of functions. (The connection between derivatives, slope of tangent lines and the rates of change.)

Two Worlds and Two Problems ?

The Velocity Problem y = distance dropped (ft) t = time (s) Displacement Function (Positive Downward) Find the velocity of the ball at t=2.

The Velocity Problem Again, we are going to use a limiting process. Find the average velocity of the ball from t=2 to t=2+h by the formula

The Velocity Problem thAverage Velocity (ft/s) 2 to 31 2 to to to

The Velocity Problem We “see” from the table that velocity of the ball at t=2 should be ____ft/s.

The Velocity Problem We “see” from the table that velocity of the ball at t=2 should be ____ft/s. The instantaneous velocity at t=2 is _____ ft/s. (The ball is traveling at____ ft/s 2 seconds after it dropped.)

Limit Notations When h is approaching 0, is approaching 64. We say as h  0, Or,

Definition For the displacement function, the instantaneous velocity at time t is if it exists.

Two Worlds and Two Problems

Remarks In the context of moving objects, the independent variable is time t. We use the following notations Distance function Velocity function Acceleration function : rate of change of the velocity function

Example 2 Given where s is in meters and t is in seconds, find (a) v(t) (b) a(t) (c) The velocity and acceleration at t=2s (d) The time when the velocity is 5m/s.

Remarks When units are given, you answers in (c) and (d) should have units. The wonderful design of the notations helps you to get the units easily.

Example 2 Suppose we model the amount of certain drug inside a patient’s body by mg after t hours of injection. (a) Find (b) Explain the meaning of the answer in (a)

Definition For y=f(t), the (instantaneous) rate of change at t is

Expectations Show the substitution step. Units are required for some of the answers. Use equal signs