Question

The point P(3,20) lies on the curve y=x²+x+8. If Q is the point (x,x²+x+8), find the slope of the secant line PQ for the following values of x.

x=3.1, x=3.01, x=2.9, x=2.99.
Based on the above results, guess the slope of the tangent line to the curve at P(3,20).

Answer 1

To find the slope of the secant line PQ, substitute the x-values into the equation to find the corresponding y-values, and calculate the slope of the secant line using the coordinates of P and Q. Make an educated guess about the slope of the tangent line at point P (3, 20) based on the results.

Step-by-step explanation:

To find the slope of the secant line PQ, we need to first find the coordinates of point Q. The coordinates of point Q are given by (x, x²+x+8). For each value of x (3.1, 3.01, 2.9, 2.99), substitute the x-value into the equation to find the corresponding y-value. Then, calculate the slope of the secant line PQ using the formula: (change in y) / (change in x).

For example, when x = 3.1, the coordinates of Q are (3.1, 3.1²+3.1+8). Substitute the value of x into the equation to find the y-value, and then calculate the slope of the secant line PQ using the coordinates of P and Q.

Repeat this process for each value of x to find the slopes of the secant lines for x=3.1, x=3.01, x=2.9, and x=2.99. Based on the results, make an educated guess about the slope of the tangent line at point P (3, 20).

To find the slope of the tangent line at P, take the limit of the secant line slopes as x approaches 3. This limit will give you the slope of the tangent line at P.