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.

Questions with Solutions

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
\( g'(x) = \dfrac{1}{f'(g(x))} \).

Question 4

True or False. The derivative of \( \ln(ax) \), where \( a \) is a constant, is equal to \( \dfrac{1}{x} \).
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) \( \dfrac{1}{10} \)
(C) 1
(D) None of the above
Answer :
(B). Use \( g'(x) = \dfrac{1}{f'(g(x))} \) given as the answer to question 3 above to write \( g'(3) = \dfrac{1}{f'(g(3))} \).
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 \( f'(x) = 3x^2 - 6x + 1 \).
\( f'(g(3)) = 3 \cdot 3^2 - 6 \cdot 3 + 1 = 10 \); and then substitute in the formula that gives \( g'(3) = \dfrac{1}{10} \).

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
\( \left(\dfrac{1}{y}\right) \dfrac{dy}{dx} = \ln a \).
Solve for \( \dfrac{dy}{dx} \)
\( \dfrac{dy}{dx} = y \ln a = a^x \ln a \).

References and Links

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