Calculus 3 Final Exam Review: Prac ce Problems from
Exam 3 Material With complete solu on RATED A+
2025/2026 NEW!!
distance formula in R^3 -
magnitude of a vector -
unit vector -
standard basis vectors - i=<1,0,0>, k=<0,1,0>, j=<0,0,1>
two vectors are orthogonal if - their dot product = 0
two vectors are parallel if - their cross product = 0
angle between two vectors v and w -
projec"on of u onto v -
comp of b along a - (a•b)/ ll a ll
area of a parallelogram with lengths a, b - ll a x b ll
volume of parallelepiped determined by a, b, c - ll a • (b x c) ll
, torque - magnitude of
unit tangent vector -
arc length formula - integral from a to b of |r'(t)| dt
curvature -
radius of oscula"ng circle - 1/K where K is the curvature
unit normal vector -
shortest distance from a point to a plane - use comp
lineariza"on - choose some f(x,y) to es"mate what you want. then take both par"al deriva"ves
and plug in (a,b) and then the point that was originally in the func"on and use that as (x,y).
Write the equa"on of the tangent plane and then plug in the points and evaluate.
direc"onal deriva"ve -
where is the value of the direc"onal deriva"ve the largest? - when the unit vector and the
gradient are in the same direc"on. The max value is the magnitude of the gradient, or |▽f|
if f has a local max or min at (a,b) and the first order par"al deriva"ves of f exist there, then... -
fx(a,b)=0 and fy(a,b)=0 OR ▽f(a,b)=<0,0,0>
Exam 3 Material With complete solu on RATED A+
2025/2026 NEW!!
distance formula in R^3 -
magnitude of a vector -
unit vector -
standard basis vectors - i=<1,0,0>, k=<0,1,0>, j=<0,0,1>
two vectors are orthogonal if - their dot product = 0
two vectors are parallel if - their cross product = 0
angle between two vectors v and w -
projec"on of u onto v -
comp of b along a - (a•b)/ ll a ll
area of a parallelogram with lengths a, b - ll a x b ll
volume of parallelepiped determined by a, b, c - ll a • (b x c) ll
, torque - magnitude of
unit tangent vector -
arc length formula - integral from a to b of |r'(t)| dt
curvature -
radius of oscula"ng circle - 1/K where K is the curvature
unit normal vector -
shortest distance from a point to a plane - use comp
lineariza"on - choose some f(x,y) to es"mate what you want. then take both par"al deriva"ves
and plug in (a,b) and then the point that was originally in the func"on and use that as (x,y).
Write the equa"on of the tangent plane and then plug in the points and evaluate.
direc"onal deriva"ve -
where is the value of the direc"onal deriva"ve the largest? - when the unit vector and the
gradient are in the same direc"on. The max value is the magnitude of the gradient, or |▽f|
if f has a local max or min at (a,b) and the first order par"al deriva"ves of f exist there, then... -
fx(a,b)=0 and fy(a,b)=0 OR ▽f(a,b)=<0,0,0>
