Questions on the applications of the derivative are presented. These questions have been designed to help you gain deep understanding of the applications of derivatives in calculus. Answers to the questions are also presented.

Questions with Solutions

Question 1

True or False. An absolute maximum or minimum must occur at a critical point or at an endpoint.
Answer :
True.

Question 2

True or False. To find the linear approximation to a function at \( x = a \) you need to know the first derivative of that function.
Answer :
True. The linear approximation \( L(x) \) to a function \( f(x) \) is given by

\( L(x) = f(a) + (x - a) f '(a) \), where \( f ' \) is the derivative of \( f \)

Question 3

True or False. Newton's method is used to approximate the extrema of a function.
Answer :
False. Newton's method is used to approximate the zeros of a function.
Note: If the first derivative of \( f \) is known, then yes Newton's method can be used to find the zero of the derivative which is the location of an extremum of function \( f \).

Question 4

True or False. Newton's methods fails if \( f '(a) \) is equal to zero and \( a \) is used as an initial value in the iteration process.
Answer :
True.

Question 5

True or False. L'Hopital's rule states that the limit of a function of the form \( f(x) / g(x) \) is equal to the limit of the derivative of \( f(x) / g(x) \).
Answer:
False. L'Hopital's rule states that the limit of a function of the form \( f(x) / g(x) \) is equal to the limit of \( f '(x) / g '(x) \).