Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible.
- what it means to say that y is an implicit function of x, and the meaning of implicit differentiation
It means that the relation between is not described by expressing in terms of .
It is mentioned that is an implicit function of .
It means that relation between and is not described by expressing explicitly in terms of .
In Implicit differentiation, we differentiate each side of the equation with two variables by treating one of the variables as a function of the other.
Velocity is the derivative of position . It is also true that acceleration (the rate of change of velocity) is the derivative of velocity. If a race car’s position in miles t hours after the start of a race is given by the function , what are the units of ? What are the units and real-world interpretation of ? What are the units and real-world interpretations of ?
Suppose h(t) represents the average height, in feet, of a person who is t years old.
(a) In real-world terms, what does h(12) represent and what are its units? What does h' (12) represent, and what are its units?
(b) Is h(12) positive or negative, and why? Is h'(12) positive or negative, and why?
(c) At approximately what value of t would h(t) have a maximum, and why? At approximately what value of t would h' (t) have a maximum, and why?
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