Question

In Exercises \(15-18\) , determine whether the curve has a tangent at the indicated point, If it does, give its slope, If not, explain why not. $$f(x)=\left\\{\begin{array}{ll}{\sin x,} & {0 \leq x<3 \pi / 4} \\ {\cos x,} & {3 \pi / 4 \leq x \leq 2 \pi}\end{array}\right.\( at \)x=3 \pi / 4$$

Short answer

The curve \( f(x) \) does have a tangent line at \( x = 3 \pi / 4 \) and its slope is \(-\sqrt{2}/2\).

Step-by-step solution

Find the derivative of \( \sin x \) and \( \cos x \)

Derivate separately the two parts of the function:The derivative \(\sin x\) is \(\cos x\), and the derivative of \(\cos x\) is \(-\sin x\)

Evaluate these derivatives at \( x = 3 \pi / 4 \)

To do this, we will substitute \( 3 \pi / 4 \) into \(\cos x\), which gives \(\cos(3 \pi / 4)\). This yields \(-\sqrt{2}/2\).Substituting \( 3 \pi / 4 \) into \(-\sin x\), gives \(-\sin(3 \pi / 4)\). This yields \(-\sqrt{2}/2\).

Compare the two results

Seeing that both limits are equal, we can conclude that the limit as \( x \) approaches \( 3 \pi / 4 \) of the derivative exists and therefore there is a tangent at \( x = 3 \pi / 4 \)