Answer to Question #301990 in Calculus for Jimjim

Question

1.graph the height of a right triangle as a function of the base ,if the hypotenuse is fixed at 10.

2.graph the diameter of a circle as a function of it's circumstances.

3.determine whether the following function is continuous or discontinuous .

f(x)=3x+2/x^2-6

a.at x =-1

b.at x =-2

Answer 1

1.

$$y=\sqrt{100-x^2},\{0<x<10\}$$

2.

$$y=\dfrac{x}{\pi}, x\ge0$$

3.

$$x^2-6\not=0$$

$$x=-\sqrt{6}, x=\sqrt{6}$$

a.

$$f(-1)=\dfrac{3(-1)+2}{(-1)^2-6}=\dfrac{1}{5}$$

$$\lim\limits_{x\to -1}\dfrac{3x+2}{x^2-6}=\dfrac{3(-1)+2}{(-1)^2-6}=\dfrac{1}{5}=f(-1)$$

The function $f(x)$ is continuous at $x=-1.$

b.

$$f(-2)=\dfrac{3(-2)+2}{(-2)^2-6}=2$$

$$\lim\limits_{x\to -2}\dfrac{3x+2}{x^2-6}=\dfrac{3(-2)+2}{(-2)^2-6}=2=f(-2)$$

The function $f(x)$ is continuous at $x=-2.$