Maths 208

Question 1

f(x) = kg(x) then f’(x)

Answer 1

f’(x) = k*g’(x)

Question 2

d/dx sin(x) =

Answer 2

cos(x)

Question 3

d/dx cos(x) =

Answer 3

-sin(x)

Question 4

d/dx tan(x) =

Answer 4

1/cos^2 (x) = sec^2 (x)

Question 5

d/dx e^x =

Answer 5

e^x

Question 6

d/dx ln(x) =

Answer 6

1/x

Question 7

power rules

Answer 7

1/x = x^-1

square root of x = x^ 1/2

Question 8

product rule

Answer 8

d/dx f(x)g(x) = f’(x)g(x) + f(x)g’(x)

Question 9

chain rule

Answer 9

d/dx f(g(x)) = f’(g(x))g’(x)

Question 10

quotient rule

Answer 10

(f/g)’ = (f’g-g’f)/g^2

Question 11

d/dx ln(2x+3)

Answer 11

1/2x+3

Question 12

implicit differentiation formula

Answer 12

Dz/Dx = - (Fx/Fz)

Question 13

the tangent to the level curve and the gradient vector of F are orthogonal

Answer 13

gradient f * U = 0

Question 14

Duf(x0,y0) = gradient f (x0,y0) * U

Answer 14

measures the rate of change of f at x0,y0 as x,y moves from x0,y0 in the direction of u

note: U must be a unit vector

Question 15

min point if

Answer 15

det Hf (x0,y0) > 0 and fxx (x0,y0) > 0

Question 16

max point if

Answer 16

det Hf (x0,y0) > 0 and fxx (x0,y0) < 0

Question 17

saddle point if

Answer 17

det Hf (x0,y0) < 0

Question 18

Lagrange multipliers

Answer 18

fx(x,y) = lamda gx(x,y)
fy(x,y) = lamda gy(x,y)
g(x,y) = 0 (ie contraint)

Question 19

sin differentiated = cos

Answer 19

cos differentiated = -sin