# Calculus II Practice (1)

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

Find all the polar coordinates of the point.

Q1) $(6, \frac{ \pi }{ 4 })$

A1)

$k=1,2,3, \cdots$ 일때,

Find the polar coordinates, $0 \leq \theta \leq 2 \pi$ and $r \geq 0$, of the point given in Cartesian coordinates.

Q2) $(\frac{ 1 }{ 6 }, \frac{ - \sqrt{ 3 } }{ 6 })$

A2)

• r의 값을 구할때는 피타고라스의 법칙을 사용합니다. ( $r= \sqrt{ x^{2} + y^{2} }$ )
• 이제 $\theta$를 구합니다. ($tan^{-1} (\frac{y}{x})$)
• 하지만 $( \frac{1}{3} , - \frac{ \pi }{3} )$은 답이 아니다. 범위가 $0 \leq \theta \leq 2 \pi$ 이기 때문이다. $\theta$의 값인 $- \frac{ \pi }{3}$$2\pi$를 더하면 $(- \frac{ \pi }{3})+\frac{ 6\pi }{3} = \frac{ 5 \pi }{3}$

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

Q3) $r=4cos3\theta$, $\theta=\frac{ 5 \pi }{6}$

A3)

Q4) $r= \frac{6}{ \theta }$, $\theta=3 \pi$

A4)

• ( $\theta$$x$라고 생각한다. )
• ( $sin 3 \pi = 0$, $cos 3 \pi = -1$ )

Solve the problem.

Q7) Find the point on the curve $x=3sin(t)$, $y=cos(t)$, $- \frac{\pi}{2} \leq t \leq \frac{\pi}{2}$ closest to the point $( \frac{4}{3} , 0 )$

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

A7)

$sin (t) = \frac{1}{2}$ $이므로,$ $t = \frac{\pi}{6}$

Find the value of $d^{2}y/dx^{2}$ at the point defined by the given value of $t$.

Q8) $x = t + cos(t)$, $y = 2 - sin(t)$, $t = \frac{\pi}{6}$

A8)

Find the area.

Q9) Find the area of the region between the curve $x=e^{4t}$, $y= \frac{1}{4} e^{-3t}$ and the x-axis, $0 \leq t \leq ln^{9}$

A9)

Find the length of the curve.

Q10) $x=2sin(t)-2t \ cos(t)$, $y=2cos(t)+2t \ sin(t)$, $0 \leq t \leq \frac{\pi}{4}$

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) $e=5$, $x=4$

A11)

Q12) $e=\frac{1}{2}$, $x=9$

A12)

Find the distance between points $P_{1}$ and $P_{2}$.

Q13) $P_{1} (5, -9, 10)$ and $P_{2} (15, 1, 15)$

A13)

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

Q14) Center $(-3, 10, 0)$, radius = $5$

A14)

Center $(h, k, l)$, Radius = r

Equation: $(x-h)^{2} + (y-k)^{2} + (z-l)^{2} = r^{2}$

Find the magnitude.

Q15) Let $% %]]>$ and $% %]]>$. Find the magnitude (length) of the vector: $-2u - v$.

A15)

magnitude (length) $-2u - v$ : $\sqrt{(-8)^{2}+(-1)^{2}} = \sqrt{ 65 }$

Solve the problem.

Q16) Find a vector of magnitude $9$ in the direction of $v = 12i - 5k$.

A16)

Find $v \cdot u$.

Q17) $v=( \frac{1}{ \sqrt{3} }, \frac{1}{ \sqrt{11} } )$ and $u=( \frac{1}{ \sqrt{3} }, \frac{-1}{ \sqrt{11} } )$

A17)

Find the vector $proj_{v}u$.

Q18) $v = 3i - j + 3k$, $u = 11i + 2j + 10k$

A18)

Find the length and direction (when defined) of $u \times v$.

Q19) $u= - \frac{1}{2} i + \frac{3}{2} j+k$, $v=i+j+2k$

A19)

$% %]]>$, $% %]]>$

Length : $\sqrt{2^{2}+2^{2}+(-2)^{2}} = \sqrt{12} = 2 \sqrt{3}$

Direction : $\frac{2(i+j-k)}{2 \sqrt{3} } = \frac{i+j-k}{\sqrt{3} }$

$\therefore$ Length is $2 \sqrt{3}$ , Direction is $\frac{i+j-k}{\sqrt{3} }$

Find the triple scalar product $( u \times v ) \cdot w$ of the given vectors.

Q20) $u = -5i - 3j + 4k$ , $v = 2i - 6j + 6k$, $w = 3i - 7j - 9k$

A20)

$( u \times v ) =$ $% $

Find parametric equations for the line described below.

Q21) The line through the point $P(-4,5,-1)$ parallel to the vector $-7i + 8j - 7k$

A21)

$\therefore$ $x=-4-7t$, $y=5+8t$, $z=-1-7t$, $t =$ any real number.

Write the equation for the plane.

Q22) The plane through the point $P(-3,7,3)$ and normal to $n = 5i + 8j + 8k$.

A22)

Calculate the requested distance.

Q23) The distance from the point $S(2,2,-5)$ to the plane $2x + 2y + z = -4$.

A23)

$a=2+2t$, $b=2+2t$, $c=-5+t$

• 위에서 구한 $a$, $b$, $c$$2x + 2y + z = -4$에 넣습니다.

Find the specific function value.

Q27) Find $f(3, 6)$ when $f(x, y) = \sqrt{3x+y^{2}}$

A27)

Q28) Find $f(4, 0, 9)$ when $f(x, y, z) = 4x^{2} + 4y^{2} - z^{2}$

A28)

Provide an appropriate response.

Q29) Find the extrema of $f(x, y, z) = x + yz$ on the line defined by $x = 4(4 + t)$, $y = t - 4$, and $z = t + 4$. Classify each extremum as a minimum or maximum. (Hint: $w = f(x, y, z)$ is a differentiable function of $t$.)

A29)

Find the limit.

Q30) $\lim_{(x,y) \rightarrow (0,0)} \frac{9x^{2} + 9y^{2} + 9}{9x^{2}-9y^{2}+6}$

A30) $\lim_{(x,y) \rightarrow (0,0)} \frac{9x^{2} + 9y^{2} + 9}{9x^{2}-9y^{2}+6} = \frac{9(0)^{2} + 9(0)^{2} + 9}{9(0)^{2}-9(0)^{2}+6} = \frac{9}{6}$

Q31) $\lim_{(x,y) \rightarrow (3,5)} \sqrt{ \frac{1}{xy} }$

A31) $\lim_{(x,y) \rightarrow (3,5)} \sqrt{ \frac{1}{xy} } = \sqrt{ \frac{1}{(3)(5)} }$

Q32) $\lim_{(x,y) \rightarrow (0,1)} \frac{y^{4}sinx}{x}$

A32) $\lim_{(x,y) \rightarrow (0,1)} \frac{y^{4}sinx}{x} = \lim_{x \rightarrow 0} \frac{sinx}{x} \times \lim_{y \rightarrow 1} y^{4} = 1$

Find $\frac{ \delta f}{ \delta x }$ and $\frac{ \delta f}{ \delta y }$.

Q33) $f(x,y) = \frac{x}{x+y}$

A33)

Find $f_{x}$, $f_{y}$, and $f_{z}$.

Q34) $f(x,y,z) = \frac{z}{ \sqrt{x+y^{2}}}$

A34)

Q35) $f(x,y,z) = \frac{cos \ y}{ xz^{2}}$

A35)

Solve the problem.

Q36) Evaluate $\frac{dw}{dt}$ at $t = \frac{7}{2} \pi$ for the function $w = \frac{xy}{z}$, $x = sin \ t$, $y = cos \ t$, $z = t^{2}$.

A36)

• $sin \frac{7}{2} \pi = -1$, $cos \frac{7}{2} \pi = 0$

Q37) Evaluate $\frac{ \delta u} { \delta x}$ at $(x,y,z)=(4,5,2)$ for the function $u = p^{2} - q^{2} - r$, $p = xy$, $q = y^{2}$, $r = xz$.

A37)

Q38) Find $\frac{ \delta w} { \delta r}$ when $r = -1$ and $s = -3$ if $w(x, y, z) = xz + y^2$, $x = 2r + 1$, $y = r + s$, and $z = r - s$.

A38)

Find the derivative of the function at $P_{0}$ in the direction of u.

Q40) $f(x, y) = 9x^{2} + 2y$, $P_{0} (-7, -9)$, $u = 3i - 4j$

A40)

$\triangledown f$ at $( -7, -9 )$ = $% \ \Rightarrow \ < -126, 2 > %]]>$

• $% %]]>$$u = 3i - 4j$에서 얻는다.

Q41) $f(x, y, z) = -10x - 9y + 8z$, $P_{0} (-8, 7, 3)$, $u = 3i - 6j - 2k$

A41)

$\triangledown f$ at $(-8, 7, 3)$ = $% %]]>$

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

Q42) $f(x, y) = x \ ln^{(y - x)}$

A42)

Q43) $f(x, y) = \frac{x}{x+y}$

A43)

Provide an appropriate response.

Q45) Find any local extrema (maxima, minima, or saddle points) of $f(x, y)$ given that $f_{x} = -10x + 7y$ and $f_{y} = 7x - 5y$.

A45)

$f_{xx} = -10$, $f_{yy} = -5$, $f_{xy} = 7$

• $D > 0$ & $% $ $\rightarrow$ LMAX at $(a,b)$
• $D > 0$ & $f_{xx} > 0$ $\rightarrow$ LMIN at $(a,b)$
• $% $ $\rightarrow$ neither max nor min at $(a,b)$
• $D = 0$ $\rightarrow$ test tells as nothing.

$\therefore$ LMAX at $(x,y)$ = $(0,0)$

• $(0,0)$ 부분은 $f_{x}$$f_{y}$를 사용하여 $x$, $y$를 구하여 좌표를 알아낼 수 있다.

Q46) Determine whether the function $f(x,y) = 4x^{2}y^{2}+7x^{4}y^{4}$ has a maximum, a minimum, or neither at the origin.

A46)

$f_{x} = 8xy^{2}+28^{3}y^{4}$, $f_{y} = 8x^{2}y+28x^{4}y^{3}$

$f_{xx} = 8y^{2} + 84x^{2}y^{4}$, $f_{yy} = 8x^{2}+84x^{4}y^{2}$

$\therefore$ test tells as nothing

Solve the problem.

Q47) Find the point on $3x+5y+7z=1$ that is closest to the origin.

A47)

$f(x,y,z) = x^{2} + y^{2} + z^{2}$, $g(x,y,z) = 3x+5y+7z-1=0$

$x = \frac{3}{2} \lambda$, $y = \frac{5}{2} \lambda$, $z = \frac{7}{2} \lambda$

위에서 구한 $x$, $y$, $z$$3x+5y+7z=1$에 대입하면,

Q48) The planes $3x + 5y = 1$ and $5y + 7z = 1$ intersect in a line. Find the point on that line closest to the origin.

A48)

$f(x,y,z) = x^{2} + y^{2} + z^{2}$, $g(x,y,z) = 3x + 5y - 1=0$, $h(x,y,z) = 5y + 7z - 1=0$

• 나머지는 Q47 를 참고하여, $\lambda$$\mu$를 구한뒤, 대입하여 좌표를 알아내면 된다.