Understanding Instantaneous Rate of Change

The video tutorial explains how to estimate the instantaneous rate of change of a function at x=4 by using the graph of f(x). It describes the process of locating the point on the graph, sketching the tangent line, and calculating its slope. The slope of the tangent line is equivalent to the derivative of the function at that point. The tutorial demonstrates how to find the slope using both graphical methods and ordered pairs, emphasizing the interpretation of the slope as the rate of change in function values.

What is the instantaneous rate of change at a point equivalent to?

• The maximum value of the function

• The y-intercept of the function

• The slope of the tangent line at that point

• The average rate of change over an interval

What is the first step in estimating the instantaneous rate of change using a graph?

• Draw a secant line

• Locate the point of tangency

• Calculate the derivative

• Find the y-intercept

How do you determine the slope of the tangent line?

• By calculating the area under the curve

• By measuring the angle of the line

• By using the coordinates of two points on the tangent line

• By finding the average of all slopes

What is the change in y when moving from point (4,5) to (5,7) on the tangent line?

• +2

• +3

• +1

• +4

What is the change in x when moving from point (4,5) to (5,7) on the tangent line?

• +3

• +2

• +4

• +1

What is another method to find the slope of the tangent line besides using the graph?

• Using the slope formula with two points

• Using the integral of the function

• Using the derivative formula

• Using the midpoint formula

What is the result of the slope calculation using the points (4,5) and (5,7)?

• 4/1

• 1/2

• 2/1

• 3/1

What does the numerator in the fraction 2/1 represent in the context of instantaneous rate of change?

• The change in y

• The change in x

• The total distance

• The average rate

What does the denominator in the fraction 2/1 represent in the context of instantaneous rate of change?

• The total distance

• The change in y

• The average rate

• The change in x

What does the instantaneous rate of change tell us about the function at x = 4?

• The function value is constant

• The function value is decreasing

• The function value is changing at a rate of 2 units per 1 unit change in x

• The function value is changing at a rate of 1 unit per 2 unit change in x