Question

How would you determine the slope of the tangent line using a calculator?

a) By finding the derivative at a specific point
b) By evaluating the function at two different points
c) By calculating the average rate of change
d) By utilizing the midpoint rule

Answer 1

To determine the slope of the tangent line using a calculator, the most precise method is by finding the derivative at the specific point where the tangent occurs. This derivative represents the instantaneous rate of change and thus the slope of the tangent line at that point.

Step-by-step explanation:

To calculate the slope of the tangent line using a calculator, the most precise method is by finding the derivative at a specific point on the function. This is because the derivative at a point gives the instantaneous rate of change, which is equivalent to the slope of the tangent line at that point. Here's a step-by-step strategy to find the slope:

1. Identify the function for which you want to find the tangent's slope.
2. Find the derivative of the function—this represents the slope of the tangent line for every point along the curve.
3. Evaluate the derivative at the specific point of interest (e.g., t = 25s) to get the slope at that exact point.

An alternative method, which is less accurate for curves, is by evaluating the function at two different points and calculating the slope using the formula (rise over run), but this gives an average rate of change over the interval between points, not the instantaneous slope.

It should be noted that the midpoint rule is typically used for approximating the area under a curve, not for finding slopes of tangents.