Using a FunctionIn Exercises 67 and 68, (a) find the gradient of the function at P , (b) find a unit normal vector to the level curve f ( x , y ) = c at P , (c) find the tangent line to the level curve f ( x , y ) = c at P , and (d) sketch the level curve, the unit normal vector, and the tangent line in the x y -plane. f ( x , y ) = 4 y sin x − y c = 3 , P ( π 2 , 1 )
Using a FunctionIn Exercises 67 and 68, (a) find the gradient of the function at P , (b) find a unit normal vector to the level curve f ( x , y ) = c at P , (c) find the tangent line to the level curve f ( x , y ) = c at P , and (d) sketch the level curve, the unit normal vector, and the tangent line in the x y -plane. f ( x , y ) = 4 y sin x − y c = 3 , P ( π 2 , 1 )
Solution Summary: The author explains the formula for the gradient of a function f(x,y) at the point
Using a FunctionIn Exercises 67 and 68, (a) find the gradient of the function at
P
, (b) find a unit normal vector to the level curve
f
(
x
,
y
)
=
c
at
P
, (c) find the tangent line to the level curve
f
(
x
,
y
)
=
c
at
P
, and (d) sketch the level curve, the unit normal vector, and the tangent line in the
x
y
-plane.
f
(
x
,
y
)
=
4
y
sin
x
−
y
c
=
3
,
P
(
π
2
,
1
)
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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) =…
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