MATH 215 MIDTERM 1 EXAM QUESTIONS AND ANSWERS 2026

MATH 215 MIDTERM 1 EXAM
QUESTIONS AND ANSWERS 2026
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)

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

or Ax + By + Cz + D = 0

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 - ANSWERSdz/dt = df/dx*dz/dt + df/dy*dy/dt

space curve intersection - ANSWERSset coordinates equal as functions of t

equation for sphere centered at origin and tangent to plane P - ANSWERSfind normal
vector using 3 points
use distance eq to find distance from origin = r
x^2+y^2+z^2=r^2

a·b = - ANSWERS|a||b|cosθ

a and b = perpendicular if - ANSWERSa·b = 0

a and b = parallel if - ANSWERSa×b = 0

a×b = - ANSWERSperpendicular to a and b

|a×b| = - ANSWERS|a||b|sinθ
General Equation of an Ellipsoid - ANSWERSx²/a² + y²/b² + z²/c² = 1

Cone - ANSWERSz² = x² + y²

Equation of line needs ___ & ___ - ANSWERSpoint P and direction vector V

equation of line - ANSWERS<x,y,z> = <x₀+ta, y₀+tb, z₀+tc>

find where lines intersect - ANSWERSset parametric equations equal and solve