MATH 215 @ UMICH MIDTERM 1 EXAM QUESTIONS WITH COMPLETE SOLUTIONS

MATH 215 @ UMICH MIDTERM 1
EXAM QUESTIONS WITH COMPLETE
SOLUTIONS

General Equation of an Ellipsoid - ANSWERSx²/a² + y²/b² + z²/c² = 1

Cone - ANSWERSz² = x² + y²

Elliptic Paraboloid - ANSWERSz = x² + y²

Hyperboloid of One Sheet - ANSWERSx² + y² - z² = 1

Hyperbolic Paraboloid - ANSWERSz = x² - y²

Hyperboloid of 2 sheets - ANSWERS-x² - y² + z² = 1
Vector Equation of a Line - ANSWERSr = r₀ + t*v

Symmetric Equations - ANSWERS(x-x₀)/a = (y-y₀)/b = (z-z₀)/c :: (solve for t, equate)

Tangent Plane - ANSWERSz - z₀ = fx(x₀, y₀)(x - x₀) + fy(x₀, y₀)(y - y₀)

Linear Approximation - ANSWERSf(x, y) ≈ f(a, b) + fx(a, b)(x - a) + fy(a, b)(y - b)

Differential - ANSWERSdz = fx(x, y)dx + fy(x, y)dy

Chain Rule - case 1 - ANSWERSdz/dt = df/dx*dz/dt + df/dy*dy/dt

Chain Rule - case 2 - ANSWERSmake a tree bitch

Duf(x, y) - ANSWERS= ∀f(X, y) * u

"*" = dot product
where u is a unit vector in the direction of given vector

Max of Directional Derivative - ANSWERS|∀f(x)|

Volume of Parallelpiped - ANSWERS|(a×b)*c|

Equation of a Plane (Dot Product form) - ANSWERSn * (r - r₀) = 0

Equation of a Plane (expanded) - ANSWERSa(x - x₀) + b(y - y₀) + c(z - z₀) = 0