Chapter 2.5, Problem 46E

To determine

To calculate: The equations of tangent to the given curve.

Answer to Problem 46E

The equation of tangent line is $y=-1-x-\pi$ .

Explanation of Solution

Given information:

The equation of the curve is $y=\sin x+\cos x$ , $x=\pi$

Concept used:

The formulae used are $\frac{d}{dx}\left(\sin x\right)=\cos x$ , $\frac{d}{dx}\left(\cos x\right)=-\sin x$ .

Calculation:

Slope of tangent line.

  $\begin{array}{c}m=\frac{dy}{dx}\\ =\frac{d}{dx}\left(\sin x+\cos x\right)\\ =\frac{d}{dx}\left(\sin x\right)+\text{}\frac{d}{dx}\left(\cos x\right)\\ =\cos x-\sin x\end{array}$

Solve further by substituting $\pi$ for $x$ in $m=\cos x-\sin x$

  $\begin{array}{c}m=\cos \pi -\sin \pi \\ =-1-0\\ =-1\end{array}$

The slope of tangent line is $-1$

Substitute $\pi$ for $x$ in $y=\sin x+\cos x$ for the value of $y_1$ .

  $\begin{array}{c}y=\sin x+\cos x\\ =0+\left(-1\right)\\ =-1\end{array}$

So, the value of $\left(x_1,y_1\right)=\left(\pi ,-1\right)$

Substitute the values in the standard equation of line $y-y_1=m\left(x-x_1\right)$

  $\begin{array}{l}y-\left(-1\right)=-1\left(x-\pi \right)\\ y=-1-x-\pi \end{array}$

The equation of tangent line is $y=-1-x-\pi$.

Conclusion: The equation of tangent line is $y=3-\left(x-\pi \right)$ , option (d) is correct.