Find the average rate of change of r⃗(t) on the given interval. r⃗(t)=⟨t, t^2⟩ on [-2,2]

Name: Problem 31, Chapter 11: APEX Calculus
Uploaded: 2020-07-22T22:14:43-08:00
Duration: 4 min 13 s
Channel: Cinsy Krehbiel
Description: Find the average rate of change of r⃗(t) on the given interval. r⃗(t)=⟨t, t^2⟩ on [-2,2]

Find the average rate of change of r⃗(t) on the given...

Key Concepts

Parametric Vector Functions

Parametric vector functions represent curves in space by defining each coordinate as a function of a parameter, usually denoted t. They allow us to describe motion and paths by assigning to each value of t a position vector, combining multiple coordinate functions into one vector-valued function.

Average Rate of Change

The average rate of change of a function over an interval is the overall change in the function divided by the change in the independent variable. For vector functions, this involves subtracting the vector values at the endpoints of the interval and then dividing by the difference in the parameter values, providing a measure of the vector's overall change per unit change in the parameter.

Difference Quotient

The difference quotient is a fundamental concept in calculus, representing the ratio of the change in the function’s value to the change in the independent variable over an interval. It is used to compute the average rate of change and serves as the basis for defining the derivative as the limit of this quotient as the interval shrinks to zero.

Interval Endpoints Evaluation

Evaluating a function at the endpoints of a given interval is crucial for computing its average rate of change. This step involves substituting the endpoint values into the function to obtain the corresponding positions, which are then used in the difference quotient to determine the overall change on that interval.

Transcript

00:02 We're going to calculate the average rate of change for the function h on the interval, negative root 2 to 2 root 2.

00:14 The average rate of change is the change in the function's value.

00:18 So delta, that little triangle is delta, and it means change in.

00:21 The change in the value of the function divided by the change in the variable or in x.

00:29 In this case, the variable is t, so it's the change in t.

00:33 One might want to take the shortcut and say, oh, look, this is a line and the slope of this line is negative seven.

00:41 So no matter what two points you're connecting, that slope is going to be minus seven, and the average rate of change of the function will be negative seven.

00:49 The rise over the run's always going to be negative seven.

00:53 And that would be true.

00:55 We're going to go ahead and use the average rate of change process and formula to prove it.

01:01 So delta h would be, change in h and that means h of one of the t coordinates, so h of 2 root 2, minus h of negative root 2.

01:15 So that's going to be the change in the value of the function, and that's going to be divided in the width of the, by the width of the interval.

01:23 So that's going to be 2 root 2, one of the t values, minus negative root 2.

01:31 That's the other one.

01:32 All right.

01:33 So the denounce.

01:34 It's easy.

01:35 Let's tackle that first.

01:36 2 root 2 minus negative.

01:38 So plus root 2 gives us 3 times a square to 2 on the bottom.

01:44 Now we need to find h of 2 root 2.

01:49 So h of 2 times a square to 2.

01:51 That equals 16.

01:54 Ooh, that doesn't look like a 6...