Calculus II Practice (1)

2017-07-11 22:14:58 +0900 -
Calculus 2

Find all the polar coordinates of the point.

Q1)

A1)

일때,


Find the polar coordinates, and , of the point given in Cartesian coordinates.

Q2)

A2)

  • r의 값을 구할때는 피타고라스의 법칙을 사용합니다. ( )
  • 이제 를 구합니다. ()
  • 하지만 은 답이 아니다. 범위가 이기 때문이다. 의 값인 를 더하면

Find the slope of the polar curve at the indicated point.

Q3) ,

A3)


Q4) ,

A4)

  • ( 라고 생각한다. )
  • ( , )

Solve the problem.

Q7) Find the point on the curve , , closest to the point

(Hint: Minimize the square of the distance as a function of .)

A7)


Find the value of at the point defined by the given value of .

Q8) , ,

A8)


Find the area.

Q9) Find the area of the region between the curve , and the x-axis,

A9)


Find the length of the curve.

Q10) , ,

A10)


The eccentricity is given of a conic section with one focus at the origin, along with the directrix corresponding to that focus. Find a polar equation for the conic section.

Q11) ,

A11)


Q12) ,

A12)


Find the distance between points and .

Q13) and

A13)


Find an equation for the sphere with the given center and radius.

Q14) Center , radius =

A14)

Center , Radius = r

Equation:


Find the magnitude.

Q15) Let and . Find the magnitude (length) of the vector: .

A15)

magnitude (length) :


Solve the problem.

Q16) Find a vector of magnitude in the direction of .

A16)


Find .

Q17) and

A17)


Find the vector .

Q18) ,

A18)


Find the length and direction (when defined) of .

Q19) ,

A19)

,

Length :

Direction :

Length is , Direction is


Find the triple scalar product of the given vectors.

Q20) , ,

A20)


Find parametric equations for the line described below.

Q21) The line through the point parallel to the vector

A21)

, , , any real number.


Write the equation for the plane.

Q22) The plane through the point and normal to .

A22)


Calculate the requested distance.

Q23) The distance from the point to the plane .

A23)

, ,

  • 위에서 구한 , , 에 넣습니다.

Find the specific function value.

Q27) Find when

A27)


Q28) Find when

A28)


Provide an appropriate response.

Q29) Find the extrema of on the line defined by , , and . Classify each extremum as a minimum or maximum. (Hint: is a differentiable function of .)

A29)


Find the limit.

Q30)

A30)


Q31)

A31)


Q32)

A32)


Find and .

Q33)

A33)


Find , , and .

Q34)

A34)


Q35)

A35)


Solve the problem.

Q36) Evaluate at for the function , , , .

A36)

  • ,


Q37) Evaluate at for the function , , , .

A37)


Provide an appropriate answer.

Q38) Find when and if , , , and .

A38)


Find the derivative of the function at in the direction of u.

Q40) , ,

A40)

at =

  • 에서 얻는다.


Q41) , ,

A41)

at =


Find all the second order partial derivatives of the given function.

Q42)

A42)


Q43)

A43)


Provide an appropriate response.

Q45) Find any local extrema (maxima, minima, or saddle points) of given that and .

A45)

, ,

  • & LMAX at
  • & LMIN at
  • neither max nor min at
  • test tells as nothing.

LMAX at =

  • 부분은 를 사용하여 , 를 구하여 좌표를 알아낼 수 있다.


Q46) Determine whether the function has a maximum, a minimum, or neither at the origin.

A46)

,

,

test tells as nothing


Solve the problem.

Q47) Find the point on that is closest to the origin.

A47)

,

, ,

위에서 구한 , , 에 대입하면,


Q48) The planes and intersect in a line. Find the point on that line closest to the origin.

A48)

, ,

  • 나머지는 Q47 를 참고하여, 를 구한뒤, 대입하여 좌표를 알아내면 된다.