Answer to Question #110180 in Calculus for MANNU KUMAR GUPTA

Question

Find an expression for the function whose graph consists of the line segment from the point (-2,2) to the point (-1,0) together with the top half of the circle with centere at the origin and radius 1.

Answer 1

Equation of line joining the two points $(-2,2)$ and $(-1,0)$ is,

$$y-2=\frac{0-2}{-1+2}(x+2)$$

$$y-2=-2(x+2)$$

$$y=-2x-2$$

Here, the line segment lies in the interval $[-2,-1]$.

Now, consider the circle of radius $1$ centered at origin as $x^2+y^2=1$

So, the top half of the circle is given by,

$$y=\sqrt{1-x^2}$$

Here, the circular arc lies in the interval $[-1,1]$.

Therefore, the function is defined piece-wise and the expression for the function is given as piece-wise function,

$$f(x) = \begin{cases} -2x-2 &\text{if } -2\le x\le -1\\ \sqrt{1-x^2} &\text{if } -1\le x\le 1 \end{cases}$$

The sketch of the curve is as shown in the figure below: