3 Differentiation

Question 1

What is a secant?

Answer 1

A line connecting two points on a function

Question 2

What can be said about f(x) if f’(x)>0?

Answer 2

F(x) is strictly increasing

Question 3

What can be said of f(x) if f’(x)<=0

Answer 3

F(x) is decreasing

Question 4

How can you find the percentage change in f at a?

Answer 4

f’(a)/f(a)

Question 5

If F(x)=f(x)+g(x) what can we say about the derivatives?

Answer 5

F’(x)=f’(x)+g’(x)

Question 6

If F(x)=f(x) x g(x) what can be said about the derivatives?

Answer 6

F’(x)= f’(x) x g(x) + f(x) x g’(x)

Question 7

If y=u/v what is dy/dx?

Answer 7

dy/dx= (u’v-uv’)/v^2

Question 8

When is the chain rule used?

Answer 8

If there is a composite function

Question 9

What is the chain rule?

Answer 9

dy/dx= dy/du x du/dx

Question 10

If f(x)= a^x what is f’(x)?

Answer 10

a^x x ln(a)

Question 11

If f(x)= ln(h(x)) what is f’(x)

Answer 11

h’(x)/h(x)

Question 12

If f(x)=loga(x) what is f’(x)?

Answer 12

1/ln(a)x

Question 13

When is f convex?

Answer 13

If f’’(x)>=0 for all x on domain

Question 14

When is f strictly concave?

Answer 14

If f’’(x)<0 for all x on domain

Question 15

What is true about the gradients of inverse functions

Answer 15

If f(x) gradient is a then it’s inverse has the gradient 1/a

Question 16

What is the linear approximation of f near x=x1

Answer 16

F(x)= f(x1) + f’(x1)(x-x1)

When x-x1 is small

Question 17

Equation for elasticity if y at x

Answer 17

Ey,x(x1)= f’(x1) x x1/f(x1)

Question 18

How can Ey,x be interpreted?

Answer 18

As the percentage change in y as a result in a 1% change in x

Question 19

Does continuity imply differentiability?

Answer 19

No a function can be continuous but not differentiable at certain points

Question 20

Does differentiability imply continuity?

Answer 20

Yes, if a function is differentiable it must be continuous

Question 21

What is the differential if e^x?

Answer 21

e^x