Math 10C WI24 Final Jeopardy

Midterm 1

100

Let A = (2,3,1), B = (3, 4, 2), P = (-2, -2, 4), Q = (0, 0, 6) . Are AB and PQ equivalent?

No. AB = (1,1,1), PQ = (2,2,2)

200

Find the equation of the plane passing through P = (2, 3, 4), Q = (3, -2, 1), R = (4, 4, 4)

3x - 6y + 11z - 32 = 0

300

Find the parametric equations of the line which passes through (2, 3, 1) and (0,8,-3)

x = 2 - 2t

y = 3 + 5t

z = 1 - 4t

400

Find a vector in the direction of (2,1,1) of magnitude 3

(√6 , √6 / 2, √6 / 2)

500

Find t such that (2, t, 3) is parallel to the plane 3x + 2y - z + 12 = 0

t = -3/2

Midterm 2

100

Let r(t) = (3t, t², e^-t) . Find the velocity and acceleration.

v = (3, 2t, 3^-t)

a = (0, 2, e^-t)

200

Find the tangent line to r(t) = (5t, t², sin(t)) at time t=0.

l(t) = (5,0,1)t

300

Find the limit as (x,y) -> (0,0) of (xy + x⁴)/(x²+y²) approaching on the line y = x.

1/2

400

Does z = x² - 2y satisfy z_xx + z_y = 0 ?

yes

500

Let r(t) = (-3sin(ct), 3cos(ct)) . Find the speed

3c

Partial Derivatives / Multivar. chain rule

100

f(x,y) = 2x + 3y . Find f_x and f_y

f_x = 2 , f_y = 3

200

Let f(x,y,z) = 3xyz + 2xz²-y³ . Find f_x + f_y

3yz + 3xz + 2z² - 3y²

300

f(x,y) = x² - xy² 

x(t) = 2t - 1     y(t) = t²

Calculate f '(t) in terms of t

-8t⁴ + 4t³ + 8t - 4

400

Given f_x = 2x ; f_y = 3xy² 

x'(t) = 3  ;  y'(t) = -t  ;  x(2) = 4  ;  y(2) = -3

Find df/dt evaluated at t = 2

-192

500

f(x,y,z) = x²cos(y) + sin(xz) - z 

x(u,v) = 3u²v      y(u,v) = u-2v      z(u,v) = 3u + 1

Find f_u (OK to leave x, y, z, u, v in answer)

12uvxcos(y) + 6uvzcos(xz) - x²sin(y) + 3xcos(xz) - 3

Tangent Planes / Lin. Approx.

100

Find the tangent plane to f(x,y) = 9 - x² - y² at the point (1,1)

z = 9

200

Find the tangent plane to f(x,y) = 9 - x² - y² at the point (1,1)

z = 11 - 2x - 2y

300

Using the tangent plane at (-π, π), approximate the efunction value of sin(x)cos(y) at (-3,3)

π - 3 

(AKA ~0.141592... )

400

Find the tangent plane to 3x² - y³ + 2 at (2,-2) and use it to approximate the function value at (2.1, -2.1)

24.4

500

Let f(x,y) = tan^-1(ye^x) . Find the tangent plane to the curve at (0,1)

π/4 + x/2 + (y-1)/2

Optimization

100

Find the critical point(s) of f(x,y) = 20 - x² - y²

(0,0)

200

f(x,y) = 2x² - 3y² + 4x. Find all critical points. Are they miinima? Maxima? Neither?

(-1, 0) is a saddle point

300

What point optimizes x²+ 2y² subject to x + y = 1

(2/3 , 1/3)

400

f(x,y) = sin(x) + y² - 2y. Find critical points and are they max/min/neither? 

Only consider trig values from 0 to 2π

(π/2 , 1) is a saddle point

(3π/2 , 1) is a minima

500

Find the max AND min values of xyz subject to the constraint x²+2y²+3z²=6

max: 2√3 / 3

min: -2√3 / 3