Calculus

Question 1

Chain Rule:

f(x) = C

Answer 1

f’(x) = 0

Question 2

Sum Rule:

f(x) = g(x) + h(x)

Answer 2

f’(x) = g’(x) + h’(x)

Question 3

Product Rule:

f(x) = g(x)*h(x)

Answer 3

f’(x) = g’(x)*h(x) + g(x)h’(x)

Question 4

Chain Rule:

f(x) = g(h(x))

Answer 4

f’(x) = g’(h(x)) * h’(x)

Question 5

Power Rule:

f(x) = x^t

Answer 5

f’(x) = t * x^(t-1)

Question 6

f(x) = ln(x)

Answer 6

f’(x) = 1/x

Question 7

f(x) = exp(x)

Answer 7

f’(x) = exp(x)

Question 8

f(x) = sin(x)

Answer 8

f’(x) = cos(x)

Question 9

f(x) = cos(x)

Answer 9

f’(x) = -sin(x)

Question 10

Quotient Rule:

f(x) = g(x)/h(x)

Answer 10

f’(x) = (g’(x)h(x) - g(x)h’(x)) / (h(x)*h(x))

Question 11

What does the first derivative show?

Answer 11

The gradient

Question 12

What can f’‘(x) show at f’(x)=0?

Answer 12

Whether point is minima or maxima

Question 13

What does f’‘(x) > 0 tell us about critical point at x?

Answer 13

It is a (local) minimum

Question 14

What does f’‘(x) < 0 tell us about critical point at x?

Answer 14

It is a (local) maximum

Question 15

What does f’‘(x) = 0 tell us about critical point at x?

Answer 15

No conclusion can be made

Question 16

How do you get partial derivatives of a function f(x,y)?

Answer 16

Differentiate assuming y is contant then differentiate assumming x is constant to get fx(x,y) and fy(x,y) respectively where each is a differentiation in respect to x and y respectively.

Question 17

What is fxx(x,y)?

Answer 17

The partial derivative of fx(x,y) with respect to x

Question 18

What is fxy(x,y)?

Answer 18

The partial derivative of fx(x,y) with respect to y

Question 19

What is fyy(x,y)?

Answer 19

The partial derivative of fy(x,y) with respect to y

Question 20

What is fyx(x,y)?

Answer 20

The partial derivative of fy(x,y) with respect to x

Question 21

What link do fxy(x,y) and fyx(x,y) usually have?

Answer 21

They are typically the same

Question 22

What is the precondition for second derivative testing with 2-variable function f(x,y)?

Answer 22

((fxx)(fyy) - (fxy)^2)(α,β) > 0

Question 23

What is F(x)?

Answer 23

∫f(x)dx

Question 24

general rule for integrating:

f(x) = x^t

Answer 24

F(x) = 1/(t+1) * x^(t+1)

Up the power then divide by it as a constant