3.2 - 3.3: Rates of Change and the Derivative
The average rate of change (AROC) of a function f (x) with respect to x from a to b is the slope of the
secant line connecting (a, f (a)) and (b, f (b)). That is,
The instantaneous rate of change (IROC) of a function f (x) with respect to x at a is the slope of the
line tangent to f (x) at a.
Example 1. Given f (x) = x2 , find the AROC of f on the following intervals. Sketch the secant lines
on the graph.
a) [1, 2]
b) [1, 1.5]
c) [1, 1.2]
d) [1, 1.1]
e) [0, 1]
f ) [0.5, 1]
g) [0.8, 1]
h) [0.9, 1]
i) Estimate the instantaneous rate of change of f (x) at x = 1.
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Example 2. Between what consecutive points is the AROC postive? negative? zero?
Example 3. Using the figure in the previous example, determine whether the instantaneous rate of
change at each point is positive, negative or zero.
As stated earlier, the average rate of change can be interpreted as the slope of the secant line between
two points on a curve and the instantaneous rate of change can be interpreted as the slope of the line
tangent to the graph at a point. We can numerically approximate the instantaneous rate of change by
calculating the slopes of secant lines nearer and nearer to the point of tangency and determining what
value they approach. That is, find the slope of the line connecting (a, f (a)) and (a + h, f (a + h)) for
values of h getting closer and closer to 0 (h can be negative!). Thus,
provided the limit exists. Its units of measurement are units of f per unit of x.
NOTE: You may need to find the limit from the left and limit from the right! Especially if finding the
derivative of a piecewise-defined function at one of the break points.
Example 4. Let f (x) = 2x2 −3x−1. Find the IROC of f (x) at x = 3. What does this value represent?
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Tangent Lines:
If a curve C has equation y = f (x) and we want to find the tangent to C at the point (a, f (a)), then
we consider a nearby point (x, f (x)), where x 6= a, and compute the slope of the secant line between
these two points:
Then we let (x, f (x)) approach (a, f (a)) along the curve C by letting x approach a. If the slopes of the
secant lines approach a number m, then we define the tangent t to be the line through (a, f (a)) with
slope m.
The
(a, f (a)) with slope
to the curve y = f (x) at the point (a, f (a)) is the line through
provided that this limit exists.
Example 5. Find the equation of the tangent line to the function f (x) = 2x2 − 3x − 1 at x = 3.
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Example 6. Find the equation of the tangent line to the function g(x) =
√
2x + 5 at x = b.
Velocities:
Suppose an object moves along a straight line according to an equation s = f (t), called the
, where s is the displacement of the object from the origin at time
t in the time interval from t = a to t = a + h, the change in position is
The average velocity over this time is
which is the same as the slope of the secant line.
Now suppose we compute the average velocites over shorter and shorter time intervals [a, a + h]. We
define the
v(a) at time t = a to be the limit of these average
velocities:
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Example 7. A particle moves along a straight line with equation of motion s(t) = t2 − 4t + 7, where s
is measured in meters and t in seconds.
a) Find the average velocity of the particle on [1, 5].
b) Find the average velocity of the particle on [1, 5 + h], h > 0.
c) Find the instantaneous velocity when t = 3.
The Derivative:
The definiton of IROC, the slope of the tangent line, and instantaneous velocity at x = a are all
.
interpretations of
The
, denoted by f 0 (a), is
if this limit exists.
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Example 8. Find the derivative of the function f (x) =
x2 − 5
at x = a.
x−3
The Interpretation of the Derivative as the Slope of a Tangent: As we have already discussed, the tangent line to y = f (x) at (a, f (a)) is the line through (a, f (a)) whose slope is equal to f 0 (a), the derivative
of f at a.
If f 0 (a) exists, then an equation of the tangent line to the curve y = f (x) at the point (a, f (a)) is
Example 9. Let g(x) = 1 + x − x3 . Find an equation of the tangent line at x = 1.
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The Derivative as a Function:
Given a function f , we associate with it a new function f 0 , called the
defined by
Example 10. Given the graph of f , sketch the graph of the derivative f 0 .
Example 11. Given the graph of f , sketch the graph of the derivative f 0 .
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,
Example 12. Find the derivative of f (x) =
√
x + 7x. State the domain of f 0 .
Other Notations for the Derivative:
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A function f is
It is
if f 0 (a) exists.
(a, b) if it is differentiable at every number in the interval.
Example 13. Where is the function f (x) = |x| + x differentiable?
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If f is differentiable at a, then f is continuous at a.
How Can a Function Fail to be Differentiable?
1.
2.
3.
Example 14. Sketch a graph of a function that fails to be differentiable because of
a) a corner.
b) a discontinuity.
c) a vertical tangent.
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