Study Guides (248,356)
Canada (121,501)
Mathematics (559)
MAT237Y1 (48)
all (20)

Additional Exercises 1

5 Pages
83 Views
Unlock Document

Department
Mathematics
Course
MAT237Y1
Professor
All Professors
Semester
Fall

Description
1. (a) Verify whether or not the following statements are correct. No marks for guessing. 2 2 (i) [3 marks] If S = {( , )∈R :x − ≤1y ≤ 1}\{( ,0): x > 0}, then int 2 2 S ={(x, y)∈R : x −1< y <1}\{(0,0)}. int 2 2 NO. S ={(x, y)∈R : x −1< y <1}\{(x,0): x ≥ 0} (ii) [4 marks] Every continuous function f S ⊂:R → R 2 attains its absolute minimum value ∞ and its absolute maximum value on the set S = U Li, where L ienotes the line segment in i=1 1 1 R from the origin (0,0) to the point ( , 1 − ) on the circular arc y = 1− x . 2 i i2 NO. By the Extreme Value Theorem, continuous function is sure to attain its absolute minimum value and its absolute maximum value on the set S if S is compact. 1 1 Consider the sequence of points x = k ( , 1− 2 ) ⊂ S , k→∞x =k(0,1)∉S so by the k k Bolzano-Weierstrass S is not compact. Hence the statement is false. 2 1.(c) [5 marks] Consider the area A of the parallelogram generated by the vectors u = (2,3,0), = ( 1,5,0) Use differentials to answer the following question: “To which non-zero component of the vectors, v is the value of the area A most sensitive?” (That is a small change of that component causes the biggest change of the value of A) 2 3 HINT: Show first that A = −1 5 2 2 2 2 3 2 3 2 3 2 3 A = u× v =| 0 +0 + − 1 5 = | − 1 5 | = −1 5 since − 1 5 =13 > 0 x y Consider the functionA( , . , , ) z w = xw− yz Then dA = wdx − zdy − ydz + xdw. At the point (2,3,−1,5) we get dA = 5dx + dy −3dz + 2dw. Hence a small change of dx causes the largest change in A, and consequently A is most sensitive to the entry 2 of the vectoru. 3 2. (a) [4 marks] The direction of the greatest increase in height at some po0nt0P 0x 0 y ,z ) on a hill, 20 per 100m, is towards the southwest. At this point, in the direction toward the west, how steep is the hill? Remark: Give the answer in meters per 100m, positive direction of y-axis points north. Suppose that the hill is in the shape of the graph of the differentiable fuz = f x y) The greatest increase in height is in the direction of a gradient and this direction is given by the vector ∇ f = 1 (−1,−1) which is in southwest direction, and the value of the change is || f || 2 also given to be20 = || || Hence∇f = 20 ( 1, 1) (−10 2 ,−10 2 ). 100 100 2 100 100 The west is in the direction of= (−1,0), so 10 2 10 2 10 2 ∂u= (∇f )⋅u = −( ,− )⋅ −1,0)= . Hence the rate of change in the west 100 100 100 direction is 102 m per 100m (approximately 14m per 100m) π 20 2 10 2 Remark: The shortest way would be ∂ fW= ∇f (u =)∇f || || cos = = 4 100 2 100 (b) [4 marks] Suppose that the hill of part (a) in some neighborhood of the point P0(400,200,1000) has the shape of a graph of the differentiable function z = f (x, y). Can the curve C : ( ) (400 2cos ,200 sin2 ) t ,0 ≤ t ≤ 2π, be a level curve of f ? Justify your answer. This can happen only when the tangent to the curve at the point (400,200) is orthogonal 1 to the direction of the gradient u = (−1,−1). The point (400,200) corresponds to 2 π 3π the values t = and t = on the curve. The tangent vector to the curve at the point 2 2 π ′ t = 2 is v = g (π /2) = (2sint,2cos2tt=π / 2(2,− 2). Sinceu⋅ v = 0, the curve C can be the level curve of f locally near t =. The tangent vector to the curve at the point 2 3π 3π t = is v = (−2,−2) so C cannot be a level curve neart = . So the curve C would 2 2 not be a level curve of f globally (it intersects itself). 4 2 2 3. [4 marks each part] Let f (x, y) = x(3x − 2y ) if (x, y) ≠ (0.0) and f (0,0) = 0. x + y 2 (a) Prove, using the “ ε −δ ” definition of continuity, that f is continuous at t
More Less

Related notes for MAT237Y1

Log In


OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Sign up

Join to view


OR

By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.


Submit