Question

Explain why the slope of the tangent line can be interpreted as an instantaneous rate of change.

Short answer

Answer: The slope of the tangent line to a curve can be interpreted as an instantaneous rate of change because they both are calculated using the derivative of a function, which describes the change in the function value with respect to another variable at a specific point on the curve. The derivative provides the speed or velocity at which the function value changes, hence representing the instantaneous rate of change at a given point.

Step-by-step solution

Define Tangent Line

A tangent line is a straight line that touches a curve at only one point without crossing it. It represents the direction of the curve at that particular point. Mathematically, the slope of the tangent line is given by the derivative of the function defining the curve.

Define Instantaneous Rate of Change

The instantaneous rate of change represents the rate at which a value changes with respect to another, at a specific point in time. In the context of a function, the instantaneous rate of change can be interpreted as the speed or velocity at which the function value is changing. This can be found using the derivative of the function at a particular point.

Relate Tangent Line Slope to Instantaneous Rate of Change

Both the slope of the tangent line and the instantaneous rate of change are calculated using the derivative of a function. Specifically, the derivative at a specific point gives us the slope of the tangent line at that point, which in turn can be interpreted as the instantaneous rate of change. Derivatives measure the change in function values over infinitesimally small intervals, which makes it ideal for representing instantaneous rates of change.

Provide an Example

Consider a function that models the distance (s) travelled by a car over time (t), given by s(t) = t^2. To find the instantaneous rate of change (i.e. the velocity) at a specific time, we need to find the derivative of s(t) with respect to t. The derivative is given by: s'(t) = 2t So, at any given time t, the slope of the tangent line to the curve s(t) at that point is the instantaneous rate of change (velocity) of the car. For example, at t = 3, the slope of the tangent line is: s'(3) = 2(3) = 6 This means that at t = 3, the car is travelling at a velocity of 6 units per unit of time. In conclusion, the slope of the tangent line can be interpreted as an instantaneous rate of change because both concepts are based on the derivative of the function, which describes how the function value changes with respect to another variable at a specific point on the curve.