78
1.8. THE TANGENT LINE APPROXIMATION
Figure 1.44: Axes for plotting y = g(x) and its tangent line to the point (2, g(2)).
(e) On the axes provided in Figure 1.44, sketch an accurate, labeled graph of y = g(x)
along with its tangent line at the point (2, g(2)).
⊲⊳
The tangent line
Given a function f that is differentiable at x = a, we know that we can determine the slope
of the tangent line to y = f (x) at (a, f (a)) by computing f ′ (a). The resulting tangent line
through (a, f (a)) with slope m = f ′ (a) has its equation in point-slope form given by
y − f (a) = f
′ (a)(x − a),
which we can also express as y = f ′ (a)(x − a) + f (a). Note well: there is a major difference
between f (a) and f (x) in this context. The former is a constant that results from using
the given fixed value of a, while the latter is the general expression for the rule that defines
the function. The same is true for f ′ (a) and f ′ (x): we must carefully distinguish between
these expressions. Each time we find the tangent line, we need to evaluate the function
and its derivative at a fixed a-value.
In Figure 1.45, we see a labeled plot of the graph of a function f and its tangent line
at the point (a, f (a)). Notice how when we zoom in we see the local linearity of f more
clearly highlighted as the function and its tangent line are nearly indistinguishable up
close. This can also be seen dynamically in the java applet at http://gvsu.edu/s/6J.
the point-slope form of the equation.
1.8. THE TANGENT LINE APPROXIMATION
Figure 1.44: Axes for plotting y = g(x) and its tangent line to the point (2, g(2)).
(e) On the axes provided in Figure 1.44, sketch an accurate, labeled graph of y = g(x)
along with its tangent line at the point (2, g(2)).
⊲⊳
The tangent line
Given a function f that is differentiable at x = a, we know that we can determine the slope
of the tangent line to y = f (x) at (a, f (a)) by computing f ′ (a). The resulting tangent line
through (a, f (a)) with slope m = f ′ (a) has its equation in point-slope form given by
y − f (a) = f
′ (a)(x − a),
which we can also express as y = f ′ (a)(x − a) + f (a). Note well: there is a major difference
between f (a) and f (x) in this context. The former is a constant that results from using
the given fixed value of a, while the latter is the general expression for the rule that defines
the function. The same is true for f ′ (a) and f ′ (x): we must carefully distinguish between
these expressions. Each time we find the tangent line, we need to evaluate the function
and its derivative at a fixed a-value.
In Figure 1.45, we see a labeled plot of the graph of a function f and its tangent line
at the point (a, f (a)). Notice how when we zoom in we see the local linearity of f more
clearly highlighted as the function and its tangent line are nearly indistinguishable up
close. This can also be seen dynamically in the java applet at http://gvsu.edu/s/6J.
the point-slope form of the equation.