Question

What is the slope of the line through (2,4) and (x,y) for y = x2 + x - 2 and x=1.99? x=2.004? x=2+h.

What happens to this last slope when h is very small?

Answer 1

Answer:

Part 1) $m=4.99$

Part 2) $m=5.004$

Part 3) $m=h+5$

Part 4) The slope tends to 5.

Step-by-step explanation:

we know that

The formula to calculate the slope between two points is equal to

$m=\frac{y2-y1}{x2-x1}$

we have

the points (2,4) and (x,y)

$y=x^{2} +x-2$

Part 1)

For x=1.99

substitute the value of x in the quadratic equation and solve for y

$y=(1.99)^{2} +(1.99)-2$

$y=3.9501$

we have the points

(1.99,3.9501) and (2,4)

substitute the values in the formula of slope

$m=\frac{4-3.9501}{2-1.99}$

$m=\frac{0.0499}{0.01}$

$m=4.99$

Part 2)

For x=2.004

substitute the value of x in the quadratic equation and solve for y

$y=(2.004)^{2} +(2.004)-2$

$y=4.020016$

we have the points

 (2,4) and (2.004,4.020016)

substitute the values in the formula of slope

$m=\frac{4.020016-4}{2.004-2}$

$m=\frac{0.020016}{0.004}$

$m=5.004$

Part 3)

For x=2+h

substitute the value of x in the quadratic equation and solve for y

$y=(2+h)^{2} +(2+h)-2\\\\y=4+4h+h^{2} +2+h-2\\\\y=h^{2}+5h+4$

we have the points

(2,4) and (2+h,h^{2}+5h+4)

substitute the values in the formula of slope

$m=\frac{h^{2}+5h+4-4}{2+h-2}$

$m=\frac{h^{2}+5h}{h}$

Simplify

$m=h+5$

Part 4) What happens to this last slope when h is very small

$m=h+5$

If the value of h is very small, then the value of h tends to zero and the value of m tends to 5.

therefore

$m=5$