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1.4. THE DERIVATIVE FUNCTION
• Given the graph of a function y = f (x), we can sketch an approximate graph of its
derivative y = f ′ (x) by observing that heights on the derivative’s graph correspond to
slopes on the original function’s graph.
• In Activity 1.10, we encountered some functions that had sharp corners on their graphs,
such as the shifted absolute value function. At such points, the derivative fails to exist,
and we say that f is not differentiable there. For now, it suffices to understand this as a
consequence of the jump that must occur in the derivative function at a sharp corner
on the graph of the original function.
Exercises
1. Let f be a function with the following properties: f is differentiable at every value of x
(that is, f has a derivative at every point), f (−2) = 1, and f ′ (−2) = −2, f ′ (−1) = −1,
f ′ (0) = 0, f ′ (1) = 1, and f ′ (2) = 2.
(a) On the axes provided at left in Figure 1.19, sketch a possible graph of y = f (x).
Explain why your graph meets the stated criteria.
(b) On the axes at right in Figure 1.19, sketch a possible graph of y = f ′ (x). What
type of curve does the provided data suggest for the graph of y = f ′ (x)?
(c) Conjecture a formula for the function y = f (x). Use the limit definition of the
derivative to determine the corresponding formula for y = f ′ (x). Discuss both
graphical and algebraic evidence for whether or not your conjecture is correct.
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Figure 1.19: Axes for plotting y = f (x) in (a) and y = f ′ (x) in (b).
2. Consider the function g(x) = x 2 − x + 3.
(a) Use the limit definition of the derivative to determine a formula for g ′ (x).
1.4. THE DERIVATIVE FUNCTION
• Given the graph of a function y = f (x), we can sketch an approximate graph of its
derivative y = f ′ (x) by observing that heights on the derivative’s graph correspond to
slopes on the original function’s graph.
• In Activity 1.10, we encountered some functions that had sharp corners on their graphs,
such as the shifted absolute value function. At such points, the derivative fails to exist,
and we say that f is not differentiable there. For now, it suffices to understand this as a
consequence of the jump that must occur in the derivative function at a sharp corner
on the graph of the original function.
Exercises
1. Let f be a function with the following properties: f is differentiable at every value of x
(that is, f has a derivative at every point), f (−2) = 1, and f ′ (−2) = −2, f ′ (−1) = −1,
f ′ (0) = 0, f ′ (1) = 1, and f ′ (2) = 2.
(a) On the axes provided at left in Figure 1.19, sketch a possible graph of y = f (x).
Explain why your graph meets the stated criteria.
(b) On the axes at right in Figure 1.19, sketch a possible graph of y = f ′ (x). What
type of curve does the provided data suggest for the graph of y = f ′ (x)?
(c) Conjecture a formula for the function y = f (x). Use the limit definition of the
derivative to determine the corresponding formula for y = f ′ (x). Discuss both
graphical and algebraic evidence for whether or not your conjecture is correct.
-3
3
-3
3
-3
3
-3
3
Figure 1.19: Axes for plotting y = f (x) in (a) and y = f ′ (x) in (b).
2. Consider the function g(x) = x 2 − x + 3.
(a) Use the limit definition of the derivative to determine a formula for g ′ (x).