MATH 215 MIDTERM 2 EXAM QUESTIONS AND ANSWERS

MATH 215 MIDTERM 2 EXAM
QUESTIONS AND ANSWERS
Directional Derivative Formula - ANSWERSDu = u (dot product) gradient

Largest directional derivative - ANSWERS= |gradient| in the direction of (gradient/|
gradient|)

normal line to a surface at (a, b, c) - ANSWERS(x-a)/Fx = (y-b)/Fy = (z-c)/Fz

Tangent plane to a surface at (a, b, c) - ANSWERSFx(x-a) + Fy(y-b) + Fz(z-c) = 0


sin^2(theta) = - ANSWERS1-cos(2*theta)/2

cos^2(theta) = - ANSWERS1+cos(2*theta)/2

Polar coordinates forming an integral - ANSWERSadd extra r
dr dtheta

Cylindrical coordinates - ANSWERS3d version of polar coordinates; z = z; still add the
extra r
dz dr dtheta
Second derivative test equation for D - ANSWERSD = FxxFyy - (Fxy)^2

Second derivative test:
if D >0 and Fxx>0 - ANSWERSlocal minimum

Second derivative test:
if D>0 and Fxx<0 - ANSWERSlocal maximum

Second derivative test:
if D<0 - ANSWERSsaddle point

Second derivative test:
if D = 0 - ANSWERSinconclusive

Spherical Coordinates change of variable equations - ANSWERSx = p * sin(phi) *
cos(theta)
y = p * sin(phi) * sin(theta)
z = p * cos(phi)
x^2 + y^2 + z^2 = p^2
lagrange multipliers equation - ANSWERSgradient of f = lambda * gradient of g