(a) Use a CAS to graph f x , y = x 2 + 3 y 2 e − x 2 + y 2 . (b) At how many points do you think it is true that D u f x , y = 0 for all unit vectors u? (c) Use a CAS to find ∇ f . (d) Use a CAS to solve the equation ∇ f x , y = 0 for x and y . (e) Use the result in part (d) together with Theorem 13.6.5 to check your conjecture in part (b).
(a) Use a CAS to graph f x , y = x 2 + 3 y 2 e − x 2 + y 2 . (b) At how many points do you think it is true that D u f x , y = 0 for all unit vectors u? (c) Use a CAS to find ∇ f . (d) Use a CAS to solve the equation ∇ f x , y = 0 for x and y . (e) Use the result in part (d) together with Theorem 13.6.5 to check your conjecture in part (b).
(a) Use a CAS to graph
f
x
,
y
=
x
2
+
3
y
2
e
−
x
2
+
y
2
.
(b) At how many points do you think it is true that
D
u
f
x
,
y
=
0
for all unit vectors u?
(c) Use a CAS to find
∇
f
.
(d) Use a CAS to solve the equation
∇
f
x
,
y
=
0
for
x
and
y
.
(e) Use the result in part (d) together with Theorem 13.6.5 to check your conjecture in part (b).
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.
After a great deal of experimentation, two college senior physics majors determined that when a bottle of French champagne is shaken several times, held upright, and uncorked,
its cork travels according to the function below, where s is its height (in feet) above the ground t seconds after being released.
s(t)=-16t² + 30t+3
a. How high will it go?
b. How long is it in the air?
+6x²+135x+1) (0≤x≤10). a) Find the number of units
The total profit P(x) (in thousands of dollars) from a sale of x thousand units of a new product is given by P(x) = In (-x²+6x² + 135x+
that should be sold in order to maximize the total profit. b) What is the maximum profit?
The fox population in a certain region has an annual growth rate of 8 percent per year. It is estimated that the
population in the year 2000 was 22600.
(a) Find a function that models the population t years after 2000 (t = 0 for 2000).
Your answer is P(t)
=
(b) Use the function from part (a) to estimate the fox population in the year 2008.
Your answer is (the answer should be an integer)
Chapter 13 Solutions
Calculus Early Transcendentals, Binder Ready Version
University Calculus: Early Transcendentals (4th Edition)
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