MATH 215 MIDTERM EXAMINATION TEST
PRACTICE SOLUTION EXAM QUESTIONS AND
SOLUTIONS 2026 GRADED A+.
◍ Largest directional derivative - = |gradient| in the direction of
(gradient/|gradient|)
◍ normal line to a surface at (a, b, c) - (x-a)/Fx = (y-b)/Fy = (z-c)/Fz
◍ Tangent plane to a surface at (a, b, c) - Fx(x-a) + Fy(y-b) + Fz(z-
c) = 0
◍ Second derivative test equation for D - D = FxxFyy - (Fxy)^2
◍ Second derivative test:
if D >0 and Fxx>0 - local minimum
◍ Second derivative test:
if D>0 and Fxx<0 - local maximum
◍ Second derivative test:
if D<0 - saddle point
PRACTICE SOLUTION EXAM QUESTIONS AND
SOLUTIONS 2026 GRADED A+.
◍ Largest directional derivative - = |gradient| in the direction of
(gradient/|gradient|)
◍ normal line to a surface at (a, b, c) - (x-a)/Fx = (y-b)/Fy = (z-c)/Fz
◍ Tangent plane to a surface at (a, b, c) - Fx(x-a) + Fy(y-b) + Fz(z-
c) = 0
◍ Second derivative test equation for D - D = FxxFyy - (Fxy)^2
◍ Second derivative test:
if D >0 and Fxx>0 - local minimum
◍ Second derivative test:
if D>0 and Fxx<0 - local maximum
◍ Second derivative test:
if D<0 - saddle point
