Question

To find the slope of the tangent line at a point, you can use the derivative of the function. The slope of the tangent line at a point (a,f(a)) is equal to the value of the derivative f′(a) at that point.

So, in your case, you want to find the slope of the tangent line at the point (21,20), which corresponds to x=21. To find the limit as x tends to 21 of the expression, you need to set up the following limit:
f′(21)=limx→21[expression involving f(x)]

Answer 1

To find the slope of a curve at a specific point, calculate the derivative at that point or use two points on the tangent line to calculate the difference in y-values divided by the difference in x-values.

Step-by-step explanation:

To determine the slope of a curve at a specific point, you need to find the derivative of the function at that point. This slope is the same as the slope of the tangent line at that point on the curve. When given two points on the tangent line, you can calculate the slope by taking the difference in y-values (final minus initial) and dividing by the difference in x-values (final minus initial).

For example, if the function f(t) represents the position of an object over time and you are given that the endpoints of the tangent line at t = 25 s are (19 s, 1300 m) and (32 s, 3120 m), you would use these points to find the slope as follows:

1. Subtract the y-value of the first point from the y-value of the second point to get the change in position (3120 m - 1300 m).
2. Subtract the x-value of the first point from the x-value of the second point to get the change in time (32 s - 19 s).
3. Divide the change in position by the change in time to find the slope, which represents velocity (v).