Question

Describe the parallels between finding the instantaneous velocity of an object at a point in time and finding the slope of the line tangent to the graph of a function at a point on the graph.

Short answer

Answer: The main parallel between finding the instantaneous velocity and finding the slope of the tangent line is that they both involve taking the derivative of a function, which provides the rate of change at a specific point.

Step-by-step solution

Define Instantaneous Velocity

Instantaneous velocity is the velocity of an object at a specific point in time. It can be thought of as the limit of the average velocity over smaller and smaller intervals of time, as the time interval approaches zero.

Relate Instantaneous Velocity to Derivatives

Instantaneous velocity can be found using the derivative of the position function with respect to time. The derivative represents the rate of change of a function at a given point, so it gives us the instantaneous rate of change when applied to the position function, which is the instantaneous velocity.

Define Tangent Lines and Slope

A tangent line is a straight line that touches the graph of a function at exactly one point (unless the point is an inflection point) and follows the same direction as the graph at that point. The slope of the tangent line represents the rate of change of the function at that specific point.

Relate Tangent Line Slope to Derivatives

The slope of the tangent line at a point on a function can be found using the first derivative of the function with respect to the independent variable (usually x). This is because the derivative represents the function's rate of change, and the slope of the tangent line is a representation of that rate of change at the specific point.

Establish the Parallels

Both finding the instantaneous velocity and finding the slope of the tangent line involve taking the derivative of a function. In the case of instantaneous velocity, we take the derivative of the position function with respect to time, and in the case of the tangent line slope, we take the derivative of the function with respect to the independent variable (usually x). The derivative provides us with the rate of change at a specific point, which is the main parallel between these two concepts.