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.

Answer 2

a) The slope of the secant line PQ is -16.667.

The slope of the secant line PQ is -19.608.

The slope of the secant line PQ is -20.408.

b) The slope of the tangent line to the curve at $x = 0.5$ is approximately -20.

How to calculate slopes of secant lines and estimate slopes of tangent lines

In this question we must apply slope equation for secant lines and estimate slope of a tangent line based on the information of two adjacent secant lines.

The slope equation for a secant line between points P and Q for the function $(x,y) = \left(x, \frac{5}{x} \right)$ is described below:

$m_{PQ} = \frac{\frac{5}{x_{Q}}-\frac{5}{x_{P}}}{x_{Q}-x_{P}}$

$m_{PQ} = \frac{\frac{5\cdot x_{P}-5\cdot x_{Q}}{x_{Q}\cdot x_{P}} }{x_{Q}-x_{P}}$

$m_{PQ} = -\frac{5}{x_{Q}\cdot x_{P}}$ (1)

a) Now we proceed to find the slope: $x_{P} = 0.5$, $x_{Q} = 0.6$.

$m_{PQ} = -\frac{5}{(0.6)\cdot (0.5)}$

$m_{PQ} = -16.667$

The slope of the secant line PQ is -16.667. $\blacksquare$

Now we proceed to find the slope: $x_{P} = 0.5$, $x_{Q} = 0.51$.

$m_{PQ} = - \frac{5}{(0.51)\cdot (0.5)}$

$m_{PQ} = - 19.608$

The slope of the secant line PQ is -19.608. $\blacksquare$

Now we proceed to find the slope: $x_{P} = 0.5$, $x_{Q} = 0.49$.

$m_{PQ} = -\frac{5}{(0.49)\cdot (0.5)}$

$m_{PQ} = -20.408$

The slope of the secant line PQ is -20.408. $\blacksquare$

b) And the slope of the tangent line to the curve at $x = 0.5$ is estimated by weighted averages:

$m = \frac{(0.51)\cdot (-19.608)+(0.49)\cdot (-20.408)}{0.51 + 0.49}$

$m = -20$

The slope of the tangent line to the curve at $x = 0.5$ is approximately -20. $\blacksquare$

To learn more on secant lines, we kindly invite to check this verified question: https://brainly.com/question/14345030