Question

The point P( 0.5 , 10 ) lies on the curve y = 5 / x . Let Q be the point (x, 5 / x ) . a.) Find the slope of the secant line PQ for the following values of x. If x= 0.6, the slope of PQ is: If x= 0.51, the slope of PQ is: If x= 0.4, the slope of PQ is: If x= 0.49, the slope of PQ is: b.) Based on the above results, guess the slope of the tangent line to the curve at P(0.5 , 10 ).

Answer 1

Answer:

Based on the results  it is expected that the slope of the tangent line at the point (0.5,10) is -20.

Step-by-step explanation:

To see that compute, the slopes for the given points as follows:

1. For $x=0.6$ we have:

$m=\dfrac{(5/0.6)-10}{0.6-0.5}=-16.66$

2. For $x=0.51$ we have:

$m=\dfrac{(5/0.51)-10}{0.51-0.5}=-19.60$

3. For $x=0.4$ we have:

$m=\dfrac{(5/0.4)-10}{0.4-0.5}=-25.00$

4. For $x=0.49$ we have:

$m=\dfrac{(5/0.49)-10}{0.49-0.5}=-20.40$

So, we observe that the slope of the tangent line at $(0.5, 10)$ is close to the value -20.

The graph shown below is useful to see the secant lines that approximates the tangent line.