Math 160 – Section 1.4 – The Slope of a curve at a Point

Shown below is the graph of and three lines which are tangent to the graph of at x = -3, -1, and 2.

a) From the graph, obtain the value of the slope and y-intercept of each of the lines. Then, complete the table and label each of the lines on the graph.

x-coordinate / Coordinates of the point on the graph / Slope of line tangent to the graph at the given point / y-intercept / Equation of the tangent line
x = -3 /
x = -1 /
x = 2 /

b) Use the information from the previous page to complete the following table:

x-coordinate of point / -3 / -1 / 2
Slope of the tangent

- Describe in words the pattern that lets you find the slope of the line tangent to the graph at a point in terms of the x-coordinate of the point.

- Now describe this relationship with an expression in terms of x

m =

c) Use the answer to part (b) to give the value of the slope of the line tangent to atthe point when x = 3. Then write the equation of that line (show all work). Sketch the graph of this line on the grid from the previous page to check your answer.

d) Slope of the tangent line as a rate of change

The slope of the tangent line at a given point is indicating the rate at which y is changing with respect to x. It’s the instantaneous rate of change of the function at the given x.

- For the function, at the point A( _____, _____ ), y is decreasing at a rate of 6 units per unit.

- For the function, at the point when x = -1 and y = ____ , we can say that y is increasing/decreasing at a rate of ______units per unit.

- For the function, at the point B( 4, _____ ) we can say that ______

Math 160 – Sections 1.4–Definition of the Derivative

Slope of Tangent Lines – Equation of the line tangent to the graph at a point

1) Given the function

a) Find the difference quotient

b) Find

c) The derivative of is

d) Complete the table:

The value of the derivative at x = -1 is / /

e) Complete the table with the values of the slope of the line tangent to the graph of at the given point

At the point when x = -1 / At the point (1, -1) / At the point (2, ...... )

f) Find the equation of the line which is tangent to the graph of at the point when x = -1. Check by using a graph in the calculator.

g) Find the equation of the line which is tangent to the graph of at the point (1, -1). Check by using a graph in the calculator.

h) Find the equation of the line which is tangent to the graph of at the point (2, ...... ). Check by using a graph in the calculator.

i) According to the results obtained on this page what is your conjecture about the value of the derivative and the direction of the function at the given point?

If / The function is ______at the point when .
(Select from the following choices:
Increasing, decreasing, it has a horizontal tangent line)
> 0
= 0
< 0

1