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Calculus II Sample Exam 1: Multivariable Calculus and Applications, Exams of Calculus

Korea University – SeoulCalculus

Sample Exam Calculus (MATH162-24)

Typology: Exams

2018/2019

Available from 10/29/2023

hartawan
hartawan 🇰🇷

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2019 Calculus II (MATH 162-22) Sample Exam 1
1. Find the curvature at each point of the ellipse
(Solution)
Let’s write a parametric equation of the ellipse as  cos 
  cossin
′ sin cos Then
′
sincos
′×″
sin cos
acossin
 .
So the curvature
′
′×
″
sincos

2. Find an equation for the level curve of the function through the given
point
.
passing through the given point .
(Solution)
KOREA UNIVERSITY
pf3

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2019 Calculus II (MATH 162-22) Sample Exam 1

  1. Find the curvature at each point  ^ of the ellipse

(Solution)

Let’s write a parametric equation of the ellipse as   cos   

      cos  sin  

 ′   sin  cos  Then   ′  

 sin

   

 cos

 

′ ×″ 

 sin  cos 

 acos    sin  

So the curvature    ′

′ ×″

 sin

   

 cos

 



  1. Find an equation for the level curve of the function through the given

point^ .

  

 

passing through the given point    .

(Solution)

KOREA UNIVERSITY

  1. (Thomas p.791) A surface  ^ 

  (^) 

 intersects the plane  ^ ^ .

Find parametric equations for the tangent line to this surface at the point

(Solution)   

  at   . Thus the equation of the tangent line

is  ^  ^  ^ , so  ^ ^  ^  ^ ^ ^  ^  ^  are the parameter

equation.

  1. (a) Find an equation of the tangent plane to the given surface at the

specific point.

  cos 

 

 (^)  , ^ ^ ^ 

(b) Use the linear approximation to approximate   .

(Solution)