Question

Estimate the instantaneous rate of change of ( f(t)=3 t^{2}+3) at the point: ( t=-1 ) In other words, choose ( x)-values that are getting closer and closer to -1 and compute the slope of the secant lines at each value. Then, use the trendpattern you see to estimate the slope of the tangent line. Your answer should be accurate to at least 2 decimal places.

Answer 1

To estimate the instantaneous rate of change of f(t) = 3t^2 + 3 at t = -1, we can calculate the slopes of secant lines as t approaches -1 and observe the trend pattern. By using several t-values close to -1 and finding the corresponding f(t) values, we can compute the slopes of the secant lines and estimate the slope of the tangent line at t = -1.

Step-by-step explanation:

The instantaneous rate of change of the function f(t) = 3t^2 + 3 at the point t = -1 can be estimated by finding the slopes of secant lines as (t) values get closer and closer to -1. Let's choose a few (t) values close to -1, calculate the corresponding (f(t)) values, and compute the slopes of the secant lines. This will give us a trend pattern that can help estimate the slope of the tangent line at t = -1.

Step 1: Choose several (t) values close to -1, such as -1.1, -1.01, and -1.001.

Step 2: Calculate the corresponding (f(t)) values: f(-1.1), f(-1.01), and f(-1.001).

Step 3: Compute the slopes of the secant lines using the formula: slope = (f(t) - f(-1)) / (t - (-1)), where t is one of the chosen (t) values.

Step 4: Observe the trend pattern of the slopes of secant lines as (t) gets closer to -1. Estimate the slope of the tangent line by examining this pattern.