Calc IV 14.6 & 14.7

Gradient

100

Find the gradient at given point: 

f(x,y)=xy² , (2,-1)

(1,-4)

200

f(x,y)= ln(x²+y²)  ,  (1,1)

(1,1)

300

f(x,y)= square root( 2x+3y)  ,  (-1,2)

(-1,2)

400

f(x,y,z)= x²+y²-2z²+zlnx  ,  (1,1,1)

(3,2,-4)

Directional Dervatives

100

Find the derivative of the function at P_o in the direction of vector v. 

f(x,y) = 2xy-3y²  ,  P₀(5,5)  ,  v=(4,3)

(D_uf)_Po=-4

200

Find the derivative of the function at Po in the direction of vector v. 

f(x,y) = 2x² + y²    ,  P_o(-1,1)  ,  v = (3,-4)

(D_uf)_Po = -4

300

Find the derivative of the function at Po in the direction of vector v.

g(x,y) = (x-y)/(xy+2)    ,  P_o(1,-1)  ,  v = (12,5)

(D_ug)_Po = 21/13 

400

Find the derivative of the function at Po in the direction of vector v. 

f(x,y,z) = xy + yz + zx    ,  P_o(1,-1,2)  ,  v = (3,6,-2)

(D_uf)_Po = 3

500

Find the derivative of the function at Po in the direction of vector v. 

f(x,y,z) =  3e^x cos(yz)   ,  P_o(0,0,0)  ,  v = (2,1,-2)

(D_uf)_Po = 2

Direction of Max Increase and Decrease

100

Find the functions direction of max increase and decrease. Answer should be a unit vector. 

f(x,y)=x²+xy+y²          P_o(-1,1)

Max Increase: (-1/sqrt(2), 1/sqrt(2))

Max Decrease: (1/sqrt(2), -1/sqrt(2))

200

Find the functions direction of max increase and decrease. Answer should be a unit vector. 

f(x,y)=x²y+e^xysin(y)          Po(1,0)

Max Increase: (0,1)

Max Decrease: (0,-1)

300

Find the functions direction of max increase and decrease. Answer should be a unit vector. 

f(x,y,z)= (x/y) - yz          P_o(4,1,1)

Max Increase: (1/sqrt(27), -5/sqrt(27), -1/sqrt(27))

Max Decrease: (-1/sqrt(27), 5/sqrt(27), 1/sqrt(27))

400

Find the functions direction of max increase and decrease. Answer should be a unit vector. 

f(x,y,z)= xe^y + z²          P_o(1, ln2, 1/2)

Max Increase: (2/3, 2/3, 1/3)

Max Decrease: (-2/3, -2/3, -1/3) 

Extrema

100

Find the local maxima, local minima, and/or saddle points:

f(x,y) = x² - y² - 2x + 4y + 6

(1,2) is a saddle point

200

Find the local maxima, local minima, and/or saddle points:

f(x,y) = x² + xy + y² + 3x -3y + 4

Local minimum at (-3,3) 

f(-3,3) = -5

300

Find the local maxima, local minima, and/or saddle points:

f(x,y) = 5xy - 7x² +3x - 6y + 2

Saddle point at (6/5, 69/25)

400

Find the local maxima, local minima, and/or saddle points:

f(x,y) = 6x² - 2x³ + 3y² + 6xy

(0,0) is local min f(0,0)=0

(1,-1) is saddle point

500

Find the local maxima, local minima, and/or saddle points:

f(x,y) = x³ + y³ +3x² - 3y² - 8

(0,0) and (-2,2) are saddle points 

(0,2) is local min f(0,2) = -12 

(-2,0) is local max f(-2,0) = -4