1. Given y=f(x) with f(1)=4 and f'(1)=3, find

a) g'(1) if g(x)=(f(x))^(1/2)

b) h'(1) if h(x)= f ((x)^(1/2))

For part a,

g'(1)=1/2 (f(x))^-1/2 (f'(x))

=(1/2)(4^-1/2)(3)=.75

Is this correct?

Mr F says: Yes.
For part b,

h'(x)=f(1/2)(x^-1/2)

Mr F says: Using the chain rule, $\displaystyle h'(x) = f'\left( x^{1/2} \right) \left( \frac{1}{2} x^{-1/2} \right) = f'\left( \sqrt{x} \right) \cdot \frac{1}{2} \frac{1}{\sqrt{x}}$.

So $\displaystyle \, h'(1) = f'\left( \sqrt{1} \right) \frac{1}{2 \sqrt{1}} = f'(1) \frac{1}{2} = \frac{3}{2}$.
I am unsure if part b's setup is even right...if it is what do I plug in for f? Please help me out soon!

2. Let f(v) be the gas consumption (in liters/km) of a car going at velocity v (i km/r). In other words, f(v) tells you how many liters of gas the car uses to go one km, if it is going at velociy v. You are told that...

f(80)=.05 and f'(80)=.0005

a) Let g(v) be the distance the same car goes on one liter of gas at velocity v. What is the relationship between f(v) and g(v)? Find g(80) and g'(80)

b) Let h(v) be te gas consumption in liters per hour. In other words, h(v) tells you how many liters of gas the car uses in one hour if it is going at velocity v. What is the relationship beween h(v) and f(v)? Find h(80) and h'(80).

c) How would you explain the practical meaning of the values of these functions and their derivatives to a driver who knows no calculus?

I have no idea where to start and I don't exactly understand the questions. Help/guidance/hints would be appreciated greatly! Thanks in advance