Question

Given f(x)=2x³+3, use a table to estimate the slope of the tangent line to f at the point P(−2,−13). 1. Find the slope of the secant line PQ for each point Q(x,f(x)) with the x values given in the table. (Round each answer to 6 decimal places if necessary.)

Answer 1

To estimate the slope of the tangent line at P(-2,-13) for the function f(x)=2x³+3, the slopes of the secant lines through P and nearby points Q are calculated and used to approximate the slope of the tangent line as Q approaches P.

Step-by-step explanation:

To estimate the slope of the tangent line to the function f(x)=2x³+3 at the point P(-2,-13), we need to calculate the slope of the secant lines through P and various nearby points Q(x, f(x)). This process will allow us to approximate the slope of the tangent line at P.

First, we calculate f(x) for the x values around -2, for example, -2.1, -1.9, -2.01, -1.99, etc. Then, for each point Q(x, f(x)), we find the slope of the secant line PQ using the formula slope = (f(x) - f(-2)) / (x - (-2)). By choosing points closer and closer to -2, the slope of the secant line will approach the slope of the tangent line at P.

Once the secant line slopes are calculated, we can estimate the slope of the tangent line by observing the trend as the x values approach -2. The more refined the x values (i.e., the closer they are to -2), the better the estimation of the tangent slope we will have.