Math 2204 Test 3 Practice Test Version B
.pdf
School
Indus International School, Bangalore**We aren't endorsed by this school
Course
AC 1504
Subject
Mathematics
Date
Dec 17, 2024
Pages
6
Uploaded by ChefSandpiperPerson697
MATH 2204 Test 3 Version BNAME:CRN:Honor Pledge:I have neither given nor received aid on this exam. Signature:1
Multiple Choice•No partial credit will be given.•Clearly circleyour answer.•No calculator!1. Which of the following statements is true about the osculating plane of a curve at a given point?(A) The osculating plane contains only the tangent vector at the point.(B) The osculating plane is perpendicular to the tangent vector at the point.(C) The osculating plane is defined by the tangent and normal vectors at the point and contains the curve’s velocityvector.(D) The osculating plane contains the tangent and normal vectors at the point and is tangent to the curve at thatpoint.2. Given a functionf(x, y)with continuous second-order partial derivatives, and a critical point(x0, y0), which of thefollowing conditions determines the nature of the critical point using the second derivative test?(A) If the discriminantD=∂2f∂x2∂2f∂y2−(∂2f∂x∂y)2>0and∂2f∂x2>0, the critical point is a local minimum.(B) If the discriminantD=∂2f∂x2∂2f∂y2−(∂2f∂x∂y)2<0, the critical point is a saddle point.(C) If the discriminantD=∂2f∂x2∂2f∂y2−(∂2f∂x∂y)2= 0, the second derivative test is inconclusive.(D) All of the above.3. Given the functionf(x, y) =x3−3xy2, calculate the second derivative test at the critical point(0,0).(A) The discriminantD= 6, and since∂2f∂x2>0, the critical point is a local minimum.(B) The discriminantD=−36, and sinceD <0, the critical point is a saddle point.(C) The discriminantD= 0, and the second derivative test is inconclusive.(D) The discriminantD= 6, and since∂2f∂x2<0, the critical point is a local maximum.4. Given the functionf(x, y) =x2+y2, calculate the directional derivative offat the point(1,2)in the direction of thevectorv=⟨3,4⟩.(A) The directional derivative isDvf(1,2) =225.(B) The directional derivative isDvf(1,2) =195.(C) The directional derivative isDvf(1,2) = 10.(D) The directional derivative isDvf(1,2) = 0.2
Free Response•Show reasoning that is complete and correct by the standards of this course.•Whenever using theorems, you should explicitly check that all hypotheses are satisfied.•Improper use of (or the absence of) proper notation will be penalized.•No calculator!5. Find the curvature of the curve with the following parametric equations:x=∫t0sin(12πθ2)dθ,y=∫t0cos(12πθ2)dθ3
6. Assumingx, y >0, find the point that maximizesxayb(1−x−y)cfor constantsa, b, c.4
7. Show that the curve with vector equationr(t) =⟨a1t2+b1t+c1, a2t2+b2t+c2, a3t2+b3t+c3⟩lies in a plane and find an equation of the plane.5
8. A farmer wants to create an enclosure in the shape of a right triangle with the following restrictions.• The farmer only has 24 miles of fence to surround the enclosure• The hypotenuse of the triangle needs to have a length of 10 milesDetermine the dimensions of the triangle that maximize the area of the enclosure.hi6