32
1.3. THE DERIVATIVE OF A FUNCTION AT A POINT
-3
3
-3
3
-3
3
-3
3
Figure 1.17: Axes for plotting y = f (x) in (a) and y = g(x) in (b).
greater or less than the instantaneous rate of change of the population on
January 1, 2000? Explain and justify, being sure to include proper units on all
your answers.
(b) According to the model, what is the average rate of change of the population
of China in the ten-year period starting on January 1, 2012?
(c) Write an expression involving limits that, if evaluated, would give the exact
instantaneous rate of change of the population on today’s date. Then estimate
the value of this limit (discuss how you chose to do so) and explain the meaning
(including units) of the value you have found.
(d) Find an equation for the tangent line to the function y = P(t) at the point
where the t-value is given by today’s date.
4. The goal of this problem is to compute the value of the derivative at a point for several
different functions, where for each one we do so in three different ways, and then to
compare the results to see that each produces the same value.
For each of the following functions, use the limit definition of the derivative to compute
the value of f ′ (a) using three different approaches: strive to use the algebraic approach
first (to compute the limit exactly), then test your result using numerical evidence (with
small values of h), and finally plot the graph of y = f (x) near (a, f (a)) along with the
appropriate tangent line to estimate the value of f ′ (a) visually. Compare your findings
among all three approaches; if you are unable to complete the algebraic approach, still
work numerically and graphically.
(a) f (x) = x 2 − 3x, a = 2
(b) f (x) =
1
x , a = 1
(c) f (x) =
√
x, a = 1
1.3. THE DERIVATIVE OF A FUNCTION AT A POINT
-3
3
-3
3
-3
3
-3
3
Figure 1.17: Axes for plotting y = f (x) in (a) and y = g(x) in (b).
greater or less than the instantaneous rate of change of the population on
January 1, 2000? Explain and justify, being sure to include proper units on all
your answers.
(b) According to the model, what is the average rate of change of the population
of China in the ten-year period starting on January 1, 2012?
(c) Write an expression involving limits that, if evaluated, would give the exact
instantaneous rate of change of the population on today’s date. Then estimate
the value of this limit (discuss how you chose to do so) and explain the meaning
(including units) of the value you have found.
(d) Find an equation for the tangent line to the function y = P(t) at the point
where the t-value is given by today’s date.
4. The goal of this problem is to compute the value of the derivative at a point for several
different functions, where for each one we do so in three different ways, and then to
compare the results to see that each produces the same value.
For each of the following functions, use the limit definition of the derivative to compute
the value of f ′ (a) using three different approaches: strive to use the algebraic approach
first (to compute the limit exactly), then test your result using numerical evidence (with
small values of h), and finally plot the graph of y = f (x) near (a, f (a)) along with the
appropriate tangent line to estimate the value of f ′ (a) visually. Compare your findings
among all three approaches; if you are unable to complete the algebraic approach, still
work numerically and graphically.
(a) f (x) = x 2 − 3x, a = 2
(b) f (x) =
1
x , a = 1
(c) f (x) =
√
x, a = 1