Use the approach of Exercise 77 to show that d d x f ( x ) = d d x ( f ( x ) + c ) For any costant c .[Hint: Compare the tangent lines to the graph of f ( x ) and f ( x ) + c ] Draw two graphs of your choice that represent a function y = f ( x ) and its vertical shift y = f ( x ) + 3 Pick a value of x and consider the points ( x , f ( x ) ) and ( x , f ( x ) + 3 ) . Draw the tangent lines to the curves at these points and describe what you observe about the tangent lines. Based on your observation in part (b), explain why d d x f ( x ) = d d x ( f ( x ) + 3 )
Use the approach of Exercise 77 to show that d d x f ( x ) = d d x ( f ( x ) + c ) For any costant c .[Hint: Compare the tangent lines to the graph of f ( x ) and f ( x ) + c ] Draw two graphs of your choice that represent a function y = f ( x ) and its vertical shift y = f ( x ) + 3 Pick a value of x and consider the points ( x , f ( x ) ) and ( x , f ( x ) + 3 ) . Draw the tangent lines to the curves at these points and describe what you observe about the tangent lines. Based on your observation in part (b), explain why d d x f ( x ) = d d x ( f ( x ) + 3 )
Solution Summary: The author analyzes the function representing y=f(x) and its vertical shift. The tangent lines for the curves are parallel and the slopes of the parallel lines are equal
For any costant
c
.[Hint: Compare the tangent lines to the graph of
f
(
x
)
and
f
(
x
)
+
c
]
Draw two graphs of your choice that represent a function
y
=
f
(
x
)
and its vertical shift
y
=
f
(
x
)
+
3
Pick a value of
x
and consider the points
(
x
,
f
(
x
)
)
and
(
x
,
f
(
x
)
+
3
)
. Draw the tangent lines to the curves at these points and describe what you observe about the tangent lines.
Based on your observation in part (b), explain why
d
d
x
f
(
x
)
=
d
d
x
(
f
(
x
)
+
3
)
Can you answer this question and give step by step and why and how to get it. Can you write it (numerical method)
Can you answer this question and give step by step and why and how to get it. Can you write it (numerical method)
There are three options for investing $1150. The first earns 10% compounded annually, the second earns 10% compounded quarterly, and the third earns 10% compounded continuously. Find equations that model each investment growth and
use a graphing utility to graph each model in the same viewing window over a 20-year period. Use the graph to determine which investment yields the highest return after 20 years. What are the differences in earnings among the three
investment?
STEP 1: The formula for compound interest is
A =
nt
= P(1 + − − ) n²,
where n is the number of compoundings per year, t is the number of years, r is the interest rate, P is the principal, and A is the amount (balance) after t years. For continuous compounding, the formula reduces to
A = Pert
Find r and n for each model, and use these values to write A in terms of t for each case.
Annual Model
r=0.10
A = Y(t) = 1150 (1.10)*
n = 1
Quarterly Model
r = 0.10
n = 4
A = Q(t) = 1150(1.025) 4t
Continuous Model
r=0.10
A = C(t) =…
Chapter 1 Solutions
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