Computation and Properties of the Derivative in Calculus

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

Question 1:

True or False. If a function is continuous at x = a, then it has a tangent line at x = a.

Answer :

False. Function f(x) = | x |, for example, is continuous at x = 0 but has no tangent line at x = 0.

Question 2:

True or False. The derivative of a function at a given point gives the slope of the tangent line at that point.

Answer :

True. From the definition of the derivative.



Question 3:

True or False. If f ' is the derivative of f, then the derivative of the inverse of f is the inverse of f '.

Answer :

False. If g(x) is the inverse of f(x) then its derivative g '(x) is given by.

g '(x) = 1 / f ' (g(x)).


Question 4:

True or False. The derivative of ln a x, where a is a constant, is equal to 1 / x.

Answer :

True.

Question 5:

True or False. Rolle's theorem is a special case of the mean value theorem.

Answer:

True.

Question 6:

If f(x) = x 3 -3x 2 + x and g is the inverse of f, then g '(3) is equal to

(A) 10
(B) 1 / 10
(C) 1
(D) None of the above

Answer :

(B). Use g '(x) = 1 / f ' (g(x)) given as the answer to question 3 above to write g '(3) = 1 / f ' (g(3)).

First find g(3) which is the solution to the equation f(x) = 3 by definition of the inverse function.

x 3 -3x 2 + x = 3

The above equation has one real solution x = 3. So g(3) = 3, the solution of the above equation.

Then compute f '(x) = 3 x 2 -6 x + 1.

f ' (g(3)) = 3 (3) 2 -6 (3) + 1 = 10; and then substitute in the formula that gives g '(3) = 1 / 10.

Question 7:

True or False. The derivative of f(x) = a x, where a is a constant, is x a x-1.

Answer:

False. Let y = a x so that ln y = x ln a

Differentiate both sides of ln y = x ln a with respect to x to obtain

(1 / y) dy / dx = ln a

Solve for dy / dx

dy / dx = y ln a = a x ln a

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