1. No calculators or other electronic devices are allowed during this exam.

Transcription

1 Version A Math 2E Spring 24 Midterm Exam Instructions. No calculators or other electronic devices are allowed during this exam. 2. You may use one page of notes, but no books or other assistance during this exam. 3. Write your Name, PI, and Section on the front of your Blue Book. 4. Write the Version of your exam at the top of the page on the front of your Blue Book. 5. Write your solutions clearly in your Blue Book (a) Carefully indicate the number and letter of each question and question part. (b) Present your answers in the same order they appear in the exam. (c) Start each question on a new side of a page. 6. ead each question carefully, and answer each question completely. 7. Show all of your work; no credit will be given for unsupported answers. f(u, v) (cos u, v + sin u) and g(x, y, z) (x 2 + πy 2, xz) 2. Find the second-order Taylor approximation to the function f(x, y) ln(xy) at (, e). 3. Evaluate dx dy, where is the interior of the triangle with vertices (, ), (2, 2), and (2, 6). 4. Let be region bounded by x, y,, 5. Let T : be the mapping given by T (u, v) (u uv, uv). escribe the region and calculate 5. Verify that c(t) (t 3, e t, t ) is a flow line for the vector field F(x, y, z) ( 3z 4, y, z 2 ).

2 Version A - Solutions f(u, v) (cos u, v + sin u) and g(x, y, z) (x 2 + πy 2, xz) f(u, v) g(x, y, z) f g f g sin u cos u g 2x 2πy z x We have x (,, ) and y g(x ) (π, ). Then 2π (f g)(,, ) f(y )g(x ) 2π 2. Find the second-order Taylor approximation to the function f(x, y) ln(xy) at (, e). Note that r(, e) ln(e). We compute, f x (x, y) x, f x(, e) f y (x, y) y, f y(, e) e f xx (x, y) x 2, f xx(, e) f xy (x, y) f yy (x, y) y 2, f xx(, e) e 2 Thus, the second-order Taylor approximation is f(x, y) + (x ) + e (y e) 2 (x )2 (y e)2 2e2

3 3. Evaluate (2, 2), and (2, 6). dx dy, where is the interior of the triangle with vertices (, ), The upper bound of the region is y 6x and the lower bound is y 2x. Thus, the integral is dx dy 2 3x 2 x dy dx 6x 2 dx x xy + 2 y2 3x 4. Let be region bounded by x, y,, 5. Let T : be the mapping given by T (u, v) (u uv, uv). escribe the region and calculate We have x u uv and y uv. Since u, then the preimage of is the line u and the preimage of 4 is the line u 4. On the other hand, u uv u( v) and uv imply that v is bounded between and. Thus, is a rectangular region. The Jacobian of the transformation is J v v u u u. x dx Thus, dy dx u du dv u 4 du dv Verify that c(t) (t 3, e t, t ) is a flow line for the vector field F(x, y, z) ( 3z 4, y, z 2 ). We have c (t) ( 3t 4, e t, t 2 ) and F(c(t)) ( 3(t ) 4, e t, (t ) 2) ( 3t 4, e t, t 2).

4 Version B Math 2E Spring 24 Midterm Exam Instructions. No calculators or other electronic devices are allowed during this exam. 2. You may use one page of notes, but no books or other assistance during this exam. 3. Write your Name, PI, and Section on the front of your Blue Book. 4. Write the Version of your exam at the top of the page on the front of your Blue Book. 5. Write your solutions clearly in your Blue Book (a) Carefully indicate the number and letter of each question and question part. (b) Present your answers in the same order they appear in the exam. (c) Start each question on a new side of a page. 6. ead each question carefully, and answer each question completely. 7. Show all of your work; no credit will be given for unsupported answers. f(u, v) (v + sin u, cos u) and g(x, y, z) (xz, x 2 + πy 2 ) 2. Find the second-order Taylor approximation to the function f(x, y) ln(xy) at (e, ). 3. Evaluate dx dy, where is the interior of the triangle with vertices (, ), (3, 2), and (3, 6). 4. Let be region bounded by x, y,, 4. Let T : be the mapping given by T (u, v) (u uv, uv). escribe the region and calculate 5. Verify that c(t) (e t, t, t 3 ) is a flow line for the vector field F(x, y, z) (x, y 2, 3y 4 ).

5 Version B - Solutions f(u, v) (v + sin u, cos u) and g(x, y, z) (xz, x 2 + πy 2 ) f(u, v) g(x, y, z) f g f g cos u sin u g z x 2x 2πy We have x (,, ) and y g(x ) (π, ). Then (f g)(,, ) f(y )g(x ) 2π 2π 2. Find the second-order Taylor approximation to the function f(x, y) ln(xy) at (e, ). Note that r(e, ) ln(e). We compute, f x (x, y) x, f x(e, ) e f y (x, y) y, f y(e, ) f xx (x, y) x 2, f xx(e, ) e 2 f xy (x, y) f yy (x, y) y 2, f xx(e, ) Thus, the second-order Taylor approximation is f(x, y) + e (x e) + (y ) 2 e 2 (x e) 2 (y )2 2

6 3. Evaluate (3, 2), and (3, 6). dx dy, where is the interior of the triangle with vertices (, ), The upper bound of the region is y 6x and the lower bound is y (2/3)x. Thus, the integral is dx dy 3 2x 3 (2/3)x dy dx 28 9 x2 dx x xy + 2 y2 2x (2/3)x 4. Let be region bounded by x, y,, 4. Let T : be the mapping given by T (u, v) (u uv, uv). escribe the region and calculate We have x u uv and y uv. Since u, then the preimage of is the line u and the preimage of 4 is the line u 4. On the other hand, u uv u( v) and uv imply that v is bounded between and. Thus, is a rectangular region. The Jacobian of the transformation is J v v u u u. dx Thus, dy dx u du dv u 4 du dv Verify that c(t) (e t, t, t 3 ) is a flow line for the vector field F(x, y, z) (x, y 2, 3y 4 ). We have c (t) (e t, t 2, 3t 4 ) and F(c(t)) ( e t, (t ) 2, e t, 3(t ) 4) ( e t, t 2, 3t 4).