PH 6 PDF - Calculus Past Paper

# PH #6 ## 1 **X = 2 intersects** $z = f(x,y) = 4x + 3y^2$ **in curve C** **(a) Calculate** $\frac{df}{dy}$ **at (x,y) = (2,3)** $by (x,y) = 24x + 9y^2$ $2(3)(2) + 9(3)^2$ $= 12 + 81 = 93$ **(b) Find parameterization of tangent line to C where y = 3** We need a point and vector! $C = y^2 (...

# PH #6 ## 1 **X = 2 intersects** $z = f(x,y) = 4x + 3y^2$ **in curve C** **(a) Calculate** $\frac{df}{dy}$ **at (x,y) = (2,3)** $by (x,y) = 24x + 9y^2$ $2(3)(2) + 9(3)^2$ $= 12 + 81 = 93$ **(b) Find parameterization of tangent line to C where y = 3** We need a point and vector! $C = y^2 (2) + 34^3$ $C = 2y^2 + 34^3$ $C = 2(3)^2 + 3(3)$ $C = 18 + 81 = 99$ $z = 2y^2 + 34^3$ $z = 2(3)^2 + 34^3$ $z = 18 + 81 = 99$ $P(t) = (2, 3, 99) + t <0, 1, 93>$ $P(t) = p + t\vec{v}$ $P(t) = (2, t, 2t^2 + 3t^3>$ $P'(t) = <0, 1, 4t + 9t^2> $ (vector at y = 3 is tangent) $P'(3) = <0, 1, 4(3) + 9(3)^2>$ $P'(3) = <0, 1, 81> $ $P(3) = (2, 3, 99)$ - The slope is negative when moving along the **x-axis** and also negative when moving along the **y-axis** so $f_{xy} < 0$. ## 2 $U(x,y,z) = ( x^2 + y^2 + z^2)^{3/2}$ $U = (x^2 + Ý^2 + z^2)^{3/2}$ $4x = - \frac{1}{2} *(x^2 + Ý^2 + z^2)^{3/2} * 2x$ $4x = *(x^2 + Ý^2 + z^2)^{3/2} * \frac{2x}{-1*(x^2 + Ý^2 + z^2)^{3/2}}$ $4x = (x^2 + Ý^2 + z^2)^{3/2} * \frac{-2x}{(x^2 + Ý^2 + z^2)^{3/2}}$ $4x = (x^2 + Ý^2 + z^2)^{3/2-3/2} * (- 2x)$ $4x = (x^2 + Ý^2 + z^2)^{0} * (- 2x)$ $4x = (- 2x) * (x^2 + Ý^2 + z^2)^{0}$ $4x = (- 2x) * 1 $4x = - 2x$ $4xx^2 = 2x^2 * ( x^2 + Ý^2 + z^2)^{3/2}$ $4xx^2 * \frac{ -(x^2 + Ý^2 + z^2)^{3/2}}{(x^2 + Ý^2 + z^2)^{5/2}} = 2x^2 * ( x^2 + Ý^2 + z^2)^{3/2} * \frac{ -(x^2 + Ý^2 + z^2)^{3/2}}{(x^2 + Ý^2 + z^2)^{5/2}}$ $4xx^2 * ( x^2 + Ý^2 + z^2)^{-1} = -2x^2 * ( x^2 + Ý^2 + z^2)^{3/2 - 5/2} $ $Uxx = \frac{4x^2}{(x^2 + Ý^2 + z^2)} - \frac{2x^2}{(x^2 + Ý^2 + z^2)}$ $Uyy = \frac{3y^2(2y)}{(x^2 + Ý^2 + z^2)^{3/2}} - (x^2 + Ý^2 + z^2)^{3/2} * \frac{3y^2(2y)}{(x^2 + Ý^2 + z^2)^{5/2}}$ $Uyy = \frac{6y^3}{(x^2 + Ý^2 + z^2)^{3/2}} - \frac{6y^3}{(x^2 + Ý^2 + z^2)^{3/2}} $ $Uzz = \frac{3z^2(2z)}{(x^2 + Ý^2 + z^2)^{3/2}} - (x^2 + Ý^2 + z^2)^{3/2} * \frac{3z^2(2z)}{(x^2 + Ý^2 + z^2)^{5/2}}$ $Uzz = \frac{6z^3}{(x^2 + Ý^2 + z^2)^{3/2}} - \frac{6z^3}{(x^2 + Ý^2 + z^2)^{3/2}} $ $Uxx + Uyy + Uzz = \frac{4x^2}{(x^2 + Ý^2 + z^2)} - \frac{2x^2}{(x^2 + Ý^2 + z^2)} + \frac{6y^3}{(x^2 + Ý^2 + z^2)^{3/2}} - \frac{6y^3}{(x^2 + Ý^2 + z^2)^{3/2}} + \frac{6z^3}{(x^2 + Ý^2 + z^2)^{3/2}} - \frac{6z^3}{(x^2 + Ý^2 + z^2)^{3/2}} $ $Uxx + Uyy + Uzz = \frac{4x^2}{(x^2 + Ý^2 + z^2)} - \frac{2x^2}{(x^2 + Ý^2 + z^2)} = 0$ $(\frac{x^2}{(x^2 + Ý^2 + z^2)}) * \frac{ 2 }{(x^2 + Ý^2 + z^2)^{3/2}} = 0$ $\frac{2x^2}{(x^2 + Ý^2 + z^2)^{5/2}} =0$ ## 3 **(a) Equation of tangent plane at (3,-4,5)** $f(x,y) = \sqrt{x^2 + y^2} = 2$ - $z = f(a,b) + f_x (a,b) (x-a) + f_y (a,b)(y-b)$ - $f_x = \frac{1}{2} * (x^2 + y^2)^{3/2} * 2x = \frac{x}{(x^2 + y^2)^{1/2}}$ - $f_y = \frac{1}{2} *(x^2 + y^2)^{-1/2} * 2y = \frac{y}{(x^2 + y^2)^{1/2}}$ Plug in (3, -4) for both - $f_x (3, -4) = \frac{3}{5}$ - $f_y (3, -4) = -\frac{4}{5}$ $z = 5 + \frac{3}{5} *(x - 3) + -\frac{4}{5} *(y+4) $ $z = x - \frac{3}{5} + 5 + -\frac{4}{5}y - \frac{16}{5}$ $z = x - \frac{4}{5}y + \frac{14}{5}$ **(b)** $L(x,y) = 5 + \frac{3}{5}(3.1 - 3) - \frac{4}{5}(3.8 + 4) $ $L(x,y) = 5 + \frac{3}{5}(.1) - \frac{4}{5}(.2)$ $L(x,y) = 5 + .06 - .16 $ $L(x,y) = 4.9$ ## 4 $L(x,y) = 6 + 5(3.1 - 3) - 2(3.9-4) $ $L(x,y) = 6 + .5 - 1.2$ $L(x,y) = 6.7$ - Previous given $x$ and $y$ - $L(x,y) = f(x,y) + f_x(x-a) + f_y(y-b)$ - Plug in $(3.1, 3.9)$