M ATH 1325 – B USINESS C ALCULUS
S ECTION 9.3 T HE D ERIVATIVE
Finding the rate of change of one quantity with respect to another is mathematically equivalent to finding the
slope of the tangent lines to a curve at a given point.
A secant line is a line that intersects a curve at (at least) two points.
A tangent line is a line that intersects a curve at a single point.
Recall: Slope formula:
Notice, we are taking the slope of the secant line between point P(a, f (a)) and Q(x, f (x)) and letting Q
approach P by letting x get closer and closer to a.
.
Slope of the secant line
represents average rate
of change.
.
Slope of the tangent line
represents instantaneous
rate of change.
To find the slope of the tangent line, we take the limit of the above formula as the distance between the points,
h, approaches zero.
Definition: The derivative of a function f with respect to x is the function f 0
lim
h→0
f (x + h) − f (x)
, provided this limit exists.
h
The domain of f 0 is the set of all x values for which the limit exists.
Notation:
Definition: A function f is differentiable at a if f 0 (a) exists. A function f is not differentiable (i.e. the
derivative does not exist) if the graph of f has a corner or a cusp or has a vertical tangent line.
Math 1325
Section 9.3 Continued
Theorem: If f is differentiable at a, then f is continuous at a.
Four-Step Process for Finding f 0 (x):
1. Find f (x + h)
2. Find the difference f (x + h) − f (x)
f (x + h) − f (x)
3. Find the quotient
h
f (x + h) − f (x)
4. Find the limit f 0 (x) = lim
h→0
h
Ex: Let f (x) = 2x2 − 8x.
(a) Find the derivative, f 0 , using the definition of the derivative as a limit. (Use the four-step process).
(b) Find an equation of the tangent line to the curve at the point (1, −6).
(c) Find the point on the graph of f where the tangent line to the curve is horizontal.
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Math 1325
Section 9.3 Continued
(d) Sketch the function and the equation of the tangent line at the point (1, −6).
Ex: The losses in millions of dollars due to bad loans extended chiefly in agriculture, real estate, shipping, and
energy by the Franklin Bank are estimated to be
A = f (t) = −t 2 + 10t + 30
(0 ≤ t ≤ 10)
where t is the time in years, with t = 0 corresponding to the beginning of 2007.
(a) What was the average rate of change of losses between 2010 and 2012?
(b) How fast were the losses mounting at the beginning of 2010?
(c) How fast were the losses mounting at the beginning of 2012?
(d) How fast were the losses mounting at the beginning of 2014?
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Math 1325
Section 9.3 Continued
Ex: Find an equation of the tangent line to the function f (x) =
4
2
at x = 2.
x+3