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Worksheet Derivatives of Other Trigonometric Functions - Activity 2.4.5

Activity 28.

The purpose of this activity is to confidently produce the derivative of the secant function by hypothesizing its properties, deriving its formula algebraically, and then looking back to see if our answer is reasonable.

(a)

Using GeoGebra, graph \(f(x) = \sec x\text{.}\) GeoGebra can be found by β€œgoogling” the phrase β€œGeoGebra classic”.
Answer.
Graph of f(x) = sec(x), shown as repeating U-shaped and inverted U-shaped branches with vertical asymptotes at odd multiples of Ο€/2.
Figure 159. GeoGebra gives the graph of \(\sec(x)\text{.}\)

(c)

  1. \(f(x) = \sec x\) is periodic. What is its period?
  2. True or False: We expect \(f'(x)\) to be periodic. Defend your response.
  3. At various \(x\)-values, we expect \(f'(x) = 0\text{.}\) In fact, we expect this to happen periodically. How often should we expect \(f'(x)=0\text{?}\)
Hint.
This of period as the length of the motif.
Answer.
  1. \(\displaystyle 2\pi\)
  2. True. The slope is periodic if the function is periodic.
  3. Every \(\pi\) units.

(e)

In GeoGebra, use the second input box to estimate \(f'(x)\) by typing the expression \(\displaystyle \frac{f(x+0.01)-f(x)}{0.01}\text{.}\) GeoGebra will label this as \(g(x)\) and plot it for you. Does the estimated graph of \(f'(x)\) found in this manner appear to have all of the properties guessed in parts (b)-(d)? If not, which ones does it fail to have?
Answer.
Yes, it has all of the expected properties.
A GeoGebra screen showing the graph of f(x) = sec(x) in solid blue and the difference quotient g(x) = (sec(x + 0.01) βˆ’ sec(x)) / 0.01 in dashed red. The graphs display repeated vertical asymptotes at multiples of Ο€/2.
Figure 160. GeoGebra gives the graph of \(\sec(x)\) and an estimate of its derivative by using a difference quotient.

(f)

Using the quotient rule, compute \(f'(x)\) if \(f(x) = \sec x\text{.}\) Use trigonometric identities to simplify your result.
Hint.
\(\displaystyle \sec(x) = \frac{1}{\cos(x)}\)
Answer.
\begin{align*} \displaystyle \frac{d}{dx}\left( \frac{1}{\cos(x)} \right) = \amp \mathstrut \frac{ (\cos(x))(0)-(1)(-\sin(x))}{\cos^2(x)} \\ = \amp \mathstrut \frac{\sin(x)}{\cos^2(x)} \\ = \amp \mathstrut \tan(x)\sec(x) \end{align*}

(g)

In the third box in GeoGebra, enter the formula for \(f'(x)\) you found above. Does the graph GeoGebra produces appear to be similar to the estimate you produced in part (e)? Should it?
Answer.
Yes and yes.
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