The bar graph shows the amount, in billions of dollars, that the United States government spent on human resources and total budget outlays for six selected years. (Human resources include education, health. Medicare. Social Security and veterans benefits and services.) The function p ( x ) = 1.75 x 2 − 15.9 x + 160 models the amount. p(x), in billions of dollars, that the United States government spent on human resources x years after 1970. The function q ( x ) = 2.1 x 2 − 3.5 x + 296 models total budget expenditures, q ( x ), in billions of dollars, x years after 1970. a. Use p and q to write a rational function that models the fraction of total budget outlays spent on human resources x years after 1970. b. Use the data displayed by the bar graph to find the percentage of federal expenditures spent on human resources in 2010. Round to the nearest percent. c. Use the rational function from part (a) to find the percentage of federal expenditures spent on human resources in 2010. Round to the nearest percent. Does this underestimate or overestimate the actual percent that you found in part (b)? By how much? d. What is the equation of the horizontal asymptote associated with the rational function in part (a)? If trends modeled by the function continue, what percentage of the federal budget will be spent on human resources over time? Round to the nearest percent.
The bar graph shows the amount, in billions of dollars, that the United States government spent on human resources and total budget outlays for six selected years. (Human resources include education, health. Medicare. Social Security and veterans benefits and services.) The function p ( x ) = 1.75 x 2 − 15.9 x + 160 models the amount. p(x), in billions of dollars, that the United States government spent on human resources x years after 1970. The function q ( x ) = 2.1 x 2 − 3.5 x + 296 models total budget expenditures, q ( x ), in billions of dollars, x years after 1970. a. Use p and q to write a rational function that models the fraction of total budget outlays spent on human resources x years after 1970. b. Use the data displayed by the bar graph to find the percentage of federal expenditures spent on human resources in 2010. Round to the nearest percent. c. Use the rational function from part (a) to find the percentage of federal expenditures spent on human resources in 2010. Round to the nearest percent. Does this underestimate or overestimate the actual percent that you found in part (b)? By how much? d. What is the equation of the horizontal asymptote associated with the rational function in part (a)? If trends modeled by the function continue, what percentage of the federal budget will be spent on human resources over time? Round to the nearest percent.
Solution Summary: The author explains that the rational function of the provided model is p(x)=1.75x
The bar graph shows the amount, in billions of dollars, that the United States government spent on human resources and total budget outlays for six selected years. (Human resources include education, health. Medicare. Social Security and veterans benefits and services.)
The function
p
(
x
)
=
1.75
x
2
−
15.9
x
+
160
models the amount. p(x), in billions of dollars, that the United States government spent on human resources x years after 1970. The function
q
(
x
)
=
2.1
x
2
−
3.5
x
+
296
models total budget expenditures, q (x ), in billions of dollars, x years after 1970.
a. Use p and q to write a rational function that models the fraction of total budget outlays spent on human resources x years after 1970.
b. Use the data displayed by the bar graph to find the percentage of federal expenditures spent on human resources in 2010. Round to the nearest percent.
c. Use the rational function from part (a) to find the percentage of federal expenditures spent on human resources in 2010. Round to the nearest percent. Does this underestimate or overestimate the actual percent that you found in part (b)? By how much?
d. What is the equation of the horizontal asymptote associated with the rational function in part (a)? If trends modeled by the function continue, what percentage of the federal budget will be spent on human resources over time? Round to the nearest percent.
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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