Questions on the applications of the derivative are presented below. These problems are designed to help you develop a deep understanding of how derivatives are used in calculus. Complete answers are provided for each question.
True or False. An absolute maximum or minimum must occur at a critical point or at an endpoint.
Answer. True.
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 \( f(x) \) at \( x = a \) is given by
\[ L(x) = f(a) + (x - a) f'(a) \]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. If the derivative \( f'(x) \) is known, Newton’s method can be applied to \( f'(x) = 0 \) to locate extrema of \( f(x) \).
True or False. Newton’s method fails if \( f'(a) = 0 \) and \( a \) is used as an initial guess.
Answer. True.
True or False. L’Hôpital’s rule states that \[ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}. \]
Answer. False.
L’Hôpital’s rule applies only to indeterminate forms such as \[ \frac{0}{0} \quad \text{or} \quad \frac{\infty}{\infty}. \]
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