100
2.2. THE SINE AND COSINE FUNCTIONS
⊳
Summary
In this section, we encountered the following important ideas:
• If we consider the graph of an exponential function f (x) = a x (where a > 1), the graph
of f ′ (x) behaves similarly, appearing exponential and as a possibly scaled version of the
original function a x . For f (x) = 2 x , careful analysis of the graph and its slopes suggests
that
d
dx [2 x ] = 2 x ln(2), which is a special case of the rule we stated in Section 2.1.
• By carefully analyzing the graphs of y = sin(x) and y = cos(x), plus using the limit
definition of the derivative at select points, we found that
d
dx [sin(x)] = cos(x) and
d
dx [cos(x)] = − sin(x).
• We note that all previously encountered derivative rules still hold, but now may also be
applied to functions involving the sine and cosine, plus all of the established meaning
of the derivative applies to these trigonometric functions as well.
Exercises
1. Suppose that V (t) = 24 · 1.07 t + 6 sin(t) represents the value of a person’s investment
portfolio in thousands of dollars in year t, where t = 0 corresponds to January 1, 2010.
(a) At what instantaneous rate is the portfolio’s value changing on January 1, 2012?
Include units on your answer.
(b) Determine the value of V ′′ (2). What are the units on this quantity and what
does it tell you about how the portfolio’s value is changing?
(c) On the interval 0 ≤ t ≤ 20, graph the function V (t) = 24 · 1.07 t + 6 sin(t) and
describe its behavior in the context of the problem. Then, compare the graphs
of the functions A(t) = 24 · 1.07 t and V (t) = 24 · 1.07 t + 6 sin(t), as well as
the graphs of their derivatives A ′ (t) and V ′ (t). What is the impact of the term
6 sin(t) on the behavior of the function V (t)?
2. Let f (x) = 3 cos(x) − 2 sin(x) + 6.
(a) Determine the exact slope of the tangent line to y = f (x) at the point where
a =
π
4 .
(b) Determine the tangent line approximation to y = f (x) at the point where
a = π.
(c) At the point where a =
π
2 , is f increasing, decreasing, or neither?
(d) At the point where a =
3π
2 , does the tangent line to y = f (x) lie above the
curve, below the curve, or neither? How can you answer this question without
even graphing the function or the tangent line?
2.2. THE SINE AND COSINE FUNCTIONS
⊳
Summary
In this section, we encountered the following important ideas:
• If we consider the graph of an exponential function f (x) = a x (where a > 1), the graph
of f ′ (x) behaves similarly, appearing exponential and as a possibly scaled version of the
original function a x . For f (x) = 2 x , careful analysis of the graph and its slopes suggests
that
d
dx [2 x ] = 2 x ln(2), which is a special case of the rule we stated in Section 2.1.
• By carefully analyzing the graphs of y = sin(x) and y = cos(x), plus using the limit
definition of the derivative at select points, we found that
d
dx [sin(x)] = cos(x) and
d
dx [cos(x)] = − sin(x).
• We note that all previously encountered derivative rules still hold, but now may also be
applied to functions involving the sine and cosine, plus all of the established meaning
of the derivative applies to these trigonometric functions as well.
Exercises
1. Suppose that V (t) = 24 · 1.07 t + 6 sin(t) represents the value of a person’s investment
portfolio in thousands of dollars in year t, where t = 0 corresponds to January 1, 2010.
(a) At what instantaneous rate is the portfolio’s value changing on January 1, 2012?
Include units on your answer.
(b) Determine the value of V ′′ (2). What are the units on this quantity and what
does it tell you about how the portfolio’s value is changing?
(c) On the interval 0 ≤ t ≤ 20, graph the function V (t) = 24 · 1.07 t + 6 sin(t) and
describe its behavior in the context of the problem. Then, compare the graphs
of the functions A(t) = 24 · 1.07 t and V (t) = 24 · 1.07 t + 6 sin(t), as well as
the graphs of their derivatives A ′ (t) and V ′ (t). What is the impact of the term
6 sin(t) on the behavior of the function V (t)?
2. Let f (x) = 3 cos(x) − 2 sin(x) + 6.
(a) Determine the exact slope of the tangent line to y = f (x) at the point where
a =
π
4 .
(b) Determine the tangent line approximation to y = f (x) at the point where
a = π.
(c) At the point where a =
π
2 , is f increasing, decreasing, or neither?
(d) At the point where a =
3π
2 , does the tangent line to y = f (x) lie above the
curve, below the curve, or neither? How can you answer this question without
even graphing the function or the tangent line?