Suppose \( f(\theta) \) is the function that relates the angular turn, \( \theta \) of a steering wheel in a car to the angular position \( f(\theta) \) of the front wheels.
(a) What does it mean if \( f'(\theta) \lt 0 \)?
(b) What does it mean if \( f'(\theta) = 0 \)?
(c) Is \( f'(\theta) = 1 \) a reasonable value?
(d) What would be a reasonable value for \( f'(\theta) \)?
(a) A function is even if
\[ f(-x) = f(x) \]
Examine some typical even functions and differentiate them. Sketch some slightly more complicated even functions and check their derivatives. What can you say about the symmetry of f'(x)?
(b) A function is odd if
\[ g(-x) = -g(x) \]
Examine some typical odd functions and differentiate them. Sketch some slightly more complicated odd functions and check their derivatives. What can you say about the symmetry of g'(x)?