1. The slope of the tangent line to a function at a point is equivalent to __.

a. the slope of the secant at that point
b. the slope of the cosecant at that point
c. the rate of change of the function at that point
d. the slope of the secant near that point

I'm having a hard time understanding this and can't decide between a. and c. Which one is it and why?

The answer is c.

A secant line is a line passing through 2 points of a function; since the tangent line only goes through one point, the slope of the tangent line to a function can't properly be defined as the slope of the secant at that point, because you haven't specified the other point on the graph

To determine the answer to this question, let's review some key concepts.

The slope of a line measures its steepness and can be found by calculating the ratio of the change in y-values (vertical change) to the change in x-values (horizontal change). In calculus, the derivative represents the slope of a function at any given point.

So, when we talk about the slope of the tangent line to a function at a point, we are essentially referring to the derivative of the function at that specific point.

Now, let's consider the given options:

a. The slope of the secant at that point: A secant line is a straight line that intersects a curve at two different points. While the secant line can be used to approximate the slope of the tangent line, it does not give the exact slope at a specific point. Thus, option a is not correct.

b. The slope of the cosecant at that point: The cosecant is a trigonometric function and not directly related to the slope of a curve. Therefore, option b is also not correct.

c. The rate of change of the function at that point: The rate of change refers to how the y-values of a function vary with respect to changes in x-values. Since the derivative of a function represents the rate of change (slope) at any specific point, option c is correct. Mathematically, the derivative of a function f(x) evaluated at a point x=a represents the slope of the tangent line to the curve at that point.

d. The slope of the secant near that point: Similar to option a, the slope of the secant line may approximate the slope of the tangent line, but it does not provide the exact value at a specific point. Therefore, option d is incorrect as well.

In conclusion, the correct option is c. The slope of the tangent line to a function at a point is equivalent to the rate of change of the function at that point.