The populations of two countries are given for January 1, 2000, and for January 1, 2010. a. Write a function of the form P t = P 0 e k t to model each population P t (in millions) t years after January 1, 2000. (See Example 3) b. Use the models from part (a) to approximate the population on January 1, 2020, for each country. Round to the nearest hundred thousand. c. Australia had fewer people than Taiwan in the year 2000, yet from the result of part (b), Australia would have more people in the year 2020? Why? d. Use the models from part (a) to predict the year during which each population would reach 30 million if this trend continues.
The populations of two countries are given for January 1, 2000, and for January 1, 2010. a. Write a function of the form P t = P 0 e k t to model each population P t (in millions) t years after January 1, 2000. (See Example 3) b. Use the models from part (a) to approximate the population on January 1, 2020, for each country. Round to the nearest hundred thousand. c. Australia had fewer people than Taiwan in the year 2000, yet from the result of part (b), Australia would have more people in the year 2020? Why? d. Use the models from part (a) to predict the year during which each population would reach 30 million if this trend continues.
Solution Summary: The author calculates the population of Australia and Taiwan by using the exponential growth function P(t)=P_oekt
Consider the function f(x) = x²-1.
(a) Find the instantaneous rate of change of f(x) at x=1 using the definition of the derivative.
Show all your steps clearly.
(b) Sketch the graph of f(x) around x = 1. Draw the secant line passing through the points on the
graph where x 1 and x->
1+h (for a small positive value of h, illustrate conceptually). Then,
draw the tangent line to the graph at x=1. Explain how the slope of the tangent line relates to the
value you found in part (a).
(c) In a few sentences, explain what the instantaneous rate of change of f(x) at x = 1 represents in
the context of the graph of f(x). How does the rate of change of this function vary at different
points?
1. The graph of ƒ is given. Use the graph to evaluate each of the following values. If a value does not exist,
state that fact.
и
(a) f'(-5)
(b) f'(-3)
(c) f'(0)
(d) f'(5)
2. Find an equation of the tangent line to the graph of y = g(x) at x = 5 if g(5) = −3 and g'(5)
=
4.
-
3. If an equation of the tangent line to the graph of y = f(x) at the point where x 2 is y = 4x — 5, find ƒ(2)
and f'(2).
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