Back to Section Page Back to Assignments Page

Section Assignment #9 - Done In Class 3/27

  1. Consider a function \(f(x)\) which is continuous and differentiable for all \(x\). Show that if \(f(x)\) has two roots, then \(f'(x)\) must have at least one root.
    Recall, \(r\) is a root of a function \(g\) if \(g(r)=0\).

    2. Let \(a\) and \(b\) be the roots of \(f\). Then \(f(a)=f(b)=0\). Since Since \(f\) is continuous and differentiable for all real numbers, \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\). Thus, by Rolle's Theorem (since we also have \(f(a)=f(b)\)), there is at least one \(c\in (a,b)\) such that \(f'(c)=0\). This means \(c\) is a root of the function \(f'(x)\), so we have verified \(f'\) must have at least one root.

    Second solution using the MVT instead of Rolle's Theorem:

    Let \(a\) and \(b\) be the roots of \(f\). Since \(f\) is continuous and differentiable for all real numbers, \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\). Thus, by the Mean Value Theorem, there exists a point \(c\in (a,b)\) such that
    \[ f'(c)=\frac{f(b)-f(a)}{b-a}=\frac{0-0}{b-a}=0, \]
    since \(a\) and \(b\) are roots of \(f\). This shows that \(c\) is the root the derivative function \(f'\). That is, this shows \(f'\) has at least one root.
  2. Suppose a sailboat travels 184 seamiles in a 24 hour period. Explain why at some point during this window of the the boat's speed must have exceeded 7.5 knots (seamiles per hour).

    3. Let \(s(t)\) be the distance the boat has traveled (in seamiles) from its starting point. Since the boat must travel across the ocean in a smooth and continuous manner, (i.e. the distance can't make any sudden jumps or change sharply since the boat must speed up and slow down gradually) this distance function must be continuous and differentiable. Thus, by the Mean Value Theorem, there is a \(c\in (0,24)\) where the instantaneous rate of change of the boat's position is equal to it's average rate of change over the time period [0,24]. (This is what \(f'(c)=\frac{f(b)-f(a)}{b-a}\) means.) The average rate of change of \(s(t)\), which is the average speed of the boat, over the 24 window is \[ \textrm{Average speed}=\frac{s(24)-s(0)}{24-0}=\frac{184-0}{24}\approx 7.667 \textrm{ knots}. \] So, the MVT says that there is a time \(c\) at which \(s'(c)=v(c)=7.667>7.5\) knots.
    In words, since the average speed is 7.667 knots, the MVT shows that there must be at time where the boat's speed exceeded 7.5 knots.