ome History homework help

ome History homework help. Name:
• I will be checking for organization, conceptual understanding, and proper mathematical communication, as well as
completion of the problems.
• Show as much work as you can, draw sketches if necessary and clearly explain why you are doing what you are doing.
• Use correct mathematical notation.
• You may work with your classmates. However, please submit your own work!
•
1. Implicit Differentiation The Chain Rule discussed in Section 14.6 can be used to give a more complete
description of the process of implicit differentiation that was introduced in Calculus I!
(a) (2 points) Suppose that an equation of the form F(x, y) = 0 defines y implicitly as a differentiable function
of x, that is, y = f(x), where F(x, f(x)) = 0 for all x in the domain of f. Show that if F and f are
differentiable,
dy
dx = −
Fx
Fy
,
(1)
where Fx =
∂F
∂x and Fy =
∂F
∂y (6= 0). (FYI: In order to derive Equation (1) we assumed that F(x, y) = 0
defines y implicitly as a function of x. The Implicit Function Theorem (IFT for short), typically
proved in advanced calculus courses, gives conditions under which this assumption is valid: It basically
states that if F is defined on a disk containing (a, b), where F(a, b) = 0, Fy(a, b) 6= 0, and both Fx and Fy
are continuous on a disk, then the equation F(x, y) = 0 defines y as a function of x near the point (a, b)
and the derivative of this function is given by Equation (1). The proof of the IFT is a bit involved and
beyond the scope of this course so don’t worry about it too much!)
(b) (2 points) Use Equation (1) to find dy
dx of y
5 + x
2
y
3 = 1 + yex
2
.
(c) (2 points) Now suppose that z is given implicitly as a function z = f(x, y) by an equation of the form
F(x, y, z) = 0. (This means that F(x, y, f(x, y)) = 0 for all (x, y) in the domain of f.) Show that if F and
f are differentiable,
∂z
∂x = −
Fx
Fz
∂z
∂y = −
Fy
Fz
,
(2) and
where Fx =
∂F
∂x , Fy =
∂F
∂y , and Fz =
∂F
∂z (6= 0).
(d) (2 points) Use Equation (2) to find both ∂z
∂x AND ∂z
∂y of x − z = arctan(yz) .
1
2. Unconstrained vs. Constrained Optimization
(a) (3 points) Let f(x, y) = x
2 +y
2
. Find the critical point(s) of the function. Then use the Second Derivative
Test to determine whether they are local minima, local maxima, or saddle points (or state that the test
fails).
(b) (3 points) Now, let f(x, y) = x
2 + y
2 be subjected to the constraint 2x − y = 5. Use the method of
Lagrange Multipliers to find the minimum and/or maximum value of f subjected to the constraint. Also,
what is the value of the Lagrange multiplier λ? You must solve the resulting system of equations
by hand! State whether the value you found is minimum or maximum and explain how you
know.
(c) (1 point) Locate the point you found in part (b) on the contour map below. Also, graph of the constraint
function (i.e., 2x − y = 5).
(d) (1 point) Add the gradient vectors for both f (the objective function) and g (the constraint function) at the
point you located in the contour map above. Label clearly which is which. You do not need to correctly
scale the gradient vectors ∇f and ∇g, however, make sure that they are proportional to
each other.
(e) (2 points) Redo part (b) using constraint 2x − y = 6 instead (same objective function f(x, y) = x
2 + y
2
).
(f) (1 point) What is the difference between the minimum (or maximum) values obtained in parts (b) and
(e)? That is, subtract the minimum (or maximum) value in part (b) from the minimum (or maximum)
value in part (f).
(g) (1 point) What is the difference in part (f) (approximately) related to?
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ome History homework help