Use the method of Example 1 to calculate the slope of the line tangent to the curve of each of the given functions. Let Q₁  have the indicated x-values. Sketch the curve and tangent lines.

EXAMPLE 1 Limit of slopes of secant lines

Find the slope of a line tangent to the curve y = x² + 3x at the point P (2,10) by finding the limit of the slopes of the secant lines PQ as Q approaches P. Let point Q have the x-values of 3.0, 2.5, 2.1, 2.01, and 2.001. Then, using a calculator, we tabulate the necessary values. Because P is the point (2,10) x₁ = 2 and y₁ = 10. Thus, using the values of x₂we tabulate the values of  and thereby the values of the slope m:

We see that the slope of PQ approaches the value of 7 as Q approaches P. Therefore, the slope of the tangent line at (2, 10) is 7. See Fig. 23.15.