Derivatives of Inverse Functions

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Jessem

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Let f(v) be the gas consumption (L/km) of a car going at velocity v (km/h). In other words, f(v) tells you how many liters of gas the car uses to go one kilometer, if it is going at velocity v. You are told that: f(80) = 0.05 and f`(80) = 0.0005.

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).

Okay, I'm pretty sure their relationship is: g(v) = f^-1(v). I tried to find the value of g(80) and came out with the pretty ridiculous value of 159980km/L. I am almost certain I did this wrong, but I can't find any relevant info in my textbook. For g`(80) I know I have to use the inverse rule to find the derivative (which I know), but I can't find this until I have found the value of g(80).

b) Let h(v) be the gas consumption in L/h at velocity v. What is the relationship between h(v) and f(v)? Find h(80) and h`(80).

Okay, now I'm just lost. I don't even know where to start here.

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

I think I could get this on my own once I get help with the previous problems.
 
Jessem said:
Let f(v) be the gas consumption (L/km) of a car going at velocity v (km/h). In other words, f(v) tells you how many liters of gas the car uses to go one kilometer, if it is going at velocity v. You are told that: f(80) = 0.05 and f`(80) = 0.0005.

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).

Okay, I'm pretty sure their relationship is: g(v) = f^-1(v). I tried to find the value of g(80) and came out with the pretty ridiculous value of 159980km/L. I am almost certain I did this wrong, but I can't find any relevant info in my textbook. For g`(80) I know I have to use the inverse rule to find the derivative (which I know), but I can't find this until I have found the value of g(80).

b) Let h(v) be the gas consumption in L/h at velocity v. What is the relationship between h(v) and f(v)? Find h(80) and h`(80).

Okay, now I'm just lost. I don't even know where to start here.

L/h = L/km * km/h

From the relationship above - see what relationship you can come up between f(v) and g(v)

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


I think I could get this on my own once I get help with the previous problems.
Click to expand...
 
They are inverses as you suspected. \(\displaystyle f(v)=\frac{1}{g(v)}\)

So, \(\displaystyle g(80)=\frac{1}{f(80)}\)

\(\displaystyle g(80)=\frac{1}{.05}=20 \;\ \text{km/L}\)

To find g'(80), use the quotient rule when \(\displaystyle g(v)=\frac{1}{f(v)}\)

\(\displaystyle g'(v)=\frac{f(v)(0)-(1)f'(v)}{(f(v))^{2}}\)

This will eventually give \(\displaystyle g'(80)=\frac{-f'(80)}{(f(80))^{2}}=\frac{-.0005}{(.05)^{2}}=-.2\)

This means that the km/L is decreasing about .2 for each km/hr increase in velocity when the car is traveling at 80 km/h.

See now a wee bit?.
 
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