MATH 1142
Section 1.2 & 1.3 Worksheet
NAME
Slope of a Curve at a Point
We want to extend the notion of of a slope of a line to the slope of more general curves. However, this is somewhat
problematic. Why?
One way to think of extending the notion of the slope of a line is to look for the slope of the line tangent to the
curve at a particular point.
Slope of a Curve
The slope of a curve at a point P is defined to be the slope of the tangent line
to the curve at P .
What does this mean?
The slope of a curve at a point P measures the instantaneous rate of change measures the rate of change of the
curve as it passes through P .
Will the slope of a curve be the same for any point on the curve? Why or why not?
What does this mean about the slope of a curve?
For most functions, there is a formula for finding the slope of a curve, f (x), this formula is called the called the
derivative (or sometimes the slope formula) and is denoted f 0 (x).
Recall that we already know the slope of a line g(x) = mx + b is
this means that the derivative of the line
is g 0 (x) =
. Calculating this derivative can be more complicated when dealing with more complicated curves
since we are trying to find the slope of a line for which we only know one point.
One way to approximate the derivative is by calculating the slope of the secant line as opposed to the tangent line
(since we will know two points). Calculating the slope of the secant line is the same as finding the average rate of
change of the function between two points.
1
How would you calculate the rate of change of a function f (x) between the points x = a and x = b?
Sketch a picture/graph that describes what is happening in the previous formula.
If f were a linear function, this value would calculate the slope of that line.
Since f may not be a linear function, we say that this value is calculating the slope of the secant
line through (a, f (a)) and (b, f (b)) of f .
However, what we want to find is the slope of the tangent line at one particular point (a, f (a)). This will give us
the instantaneous rate of change.
Definition
The tangent line to the curve y = f (x) at the point (a, f (a)) is the line through (a, f (a)) with slope
f 0 (a) (provided that this limit exists).
If f 0 (a) is the slope of the tangent line at a, what is an equation of the tangent line at x = a?
Equation for the Tangent Line
Explain why f 0 (a) models that instantaneous rate of change of f at (a, f (a)). (You may want to use your
graph/picture.)
2
Now, setting h = b − a x = a in the previous definition, rewrite the average rate of change formula in terms of x
and h.
Calculating the Derivative Informally
To calculate f 0 (x),
1. Write the quotient
f (x+h)−f (x)
h
for h 6= 0.
2. Simplify the quotient.
3. Let h approach 0 and the quantity
f (x+h)−f (x)
h
will approach f 0 (x).
Using this technique, find the derivative of f (x) = x2 .
Power Rule
Let r be any number and f (x) = xr then
f 0 (x) =
Specifically, this rule says that the derivative or slopes of the following curves are:
• f (x) = b then f 0 (x) =
• g(x) = 2x then g 0 (x) =
• h(x) = x2 then h0 (x) =
• k(x) = x3 then k 0 (x) =
√
• m(x) = x then m0 (x) =
3
Examples:
1. Approximate the slope of the given graph at each point:
2. Find the slope of the line tangent to y = x2 at (3, 9) and write a corresponding equation of the tangent line.
3. Apply the three-step method to compute the derivative of f (x) = 3x2 − 1.
4