80
1.8. THE TANGENT LINE APPROXIMATION
above, we note that since L(x) = f (a) + f ′ (a)(x − a) and L(x) ≈ f (x) for x near a, it also
follows that we can write
f (x) ≈ f (a) + f
′ (a)(x − a) for x near a.
The next activity explores some additional important properties of the local linearization y = L(x) to a function f at given a-value.
Activity 1.21.
Suppose it is known that for a given differentiable function y = g(x), its local linearization at the point where a = −1 is given by L(x) = −2 + 3(x + 1).
(a) Compute the values of L(−1) and L ′ (−1).
(b) What must be the values of g(−1) and g ′ (−1)? Why?
(c) Do you expect the value of g(−1.03) to be greater than or less than the value
of g(−1)? Why?
(d) Use the local linearization to estimate the value of g(−1.03).
(e) Suppose that you also know that g ′′ (−1) = 2. What does this tell you about the
graph of y = g(x) at a = −1?
(f) For x near −1, sketch the graph of the local linearization y = L(x) as well as a
possible graph of y = g(x) on the axes provided in Figure 1.46.
Figure 1.46: Axes for plotting y = L(x) and y = g(x).
⊳
As we saw in the example provided by Activity 1.21, the local linearization y = L(x)
is a linear function that shares two important values with the function y = f (x) that it is
derived from. In particular, observe that since L(x) = f (a) + f ′ (a)(x − a), it follows that
L(a) = f (a). In addition, since L is a linear function, its derivative is its slope. Hence,
1.8. THE TANGENT LINE APPROXIMATION
above, we note that since L(x) = f (a) + f ′ (a)(x − a) and L(x) ≈ f (x) for x near a, it also
follows that we can write
f (x) ≈ f (a) + f
′ (a)(x − a) for x near a.
The next activity explores some additional important properties of the local linearization y = L(x) to a function f at given a-value.
Activity 1.21.
Suppose it is known that for a given differentiable function y = g(x), its local linearization at the point where a = −1 is given by L(x) = −2 + 3(x + 1).
(a) Compute the values of L(−1) and L ′ (−1).
(b) What must be the values of g(−1) and g ′ (−1)? Why?
(c) Do you expect the value of g(−1.03) to be greater than or less than the value
of g(−1)? Why?
(d) Use the local linearization to estimate the value of g(−1.03).
(e) Suppose that you also know that g ′′ (−1) = 2. What does this tell you about the
graph of y = g(x) at a = −1?
(f) For x near −1, sketch the graph of the local linearization y = L(x) as well as a
possible graph of y = g(x) on the axes provided in Figure 1.46.
Figure 1.46: Axes for plotting y = L(x) and y = g(x).
⊳
As we saw in the example provided by Activity 1.21, the local linearization y = L(x)
is a linear function that shares two important values with the function y = f (x) that it is
derived from. In particular, observe that since L(x) = f (a) + f ′ (a)(x − a), it follows that
L(a) = f (a). In addition, since L is a linear function, its derivative is its slope. Hence,