Problem 6 - Calculus Early Transcendentals

Question

Explain the relationships among the slope of a tangent line, the instantaneous rate of change, and the value of the derivative at a point.

Step-by-Step Solution

Answer

Answer: The slope of the tangent line, the instantaneous rate of change, and the value of the derivative at a point on a curve are all equal, as they all represent the rate at which the function is changing at that specific point.

Step 1: Define the slope of a tangent line

The slope of a tangent line is the measure of the steepness or inclination of a line that is tangent (touches) to a curve at a specific point. It represents the rate at which the function is changing at that specific point.

Step 2: Define the instantaneous rate of change

The instantaneous rate of change is the rate at which a function is changing at an exact point on the curve. It can be thought of as the speed at which the value of the function is changing at that instant.

Step 3: Define the derivative at a point

The derivative (f'(x) or dy/dx) of a function f(x) represents the rate at which the function is changing at any given point x. The value of the derivative at a specific point gives us the exact rate at which the function is changing at that point.

Step 4: Relationship among the three concepts

The slope of the tangent line, the instantaneous rate of change, and the value of the derivative at a point are all interconnected. At a specific point on a curve: 1. The slope of the tangent line to the curve is equal to the instantaneous rate of change of the function at that point. This is because both the slope and the instantaneous rate of change measure how steep the curve is at that exact point, or how fast the function is changing at that instant. 2. The value of the derivative at a given point is also equal to the slope of the tangent line at that point, as well as the instantaneous rate of change. This is because the derivative represents the rate of change of the function at that point, and as mentioned earlier, the slope and instantaneous rate of change measure the same thing. In summary, the slope of the tangent line, the instantaneous rate of change, and the value of the derivative at a point are all equal, as they all represent the rate at which the function is changing at that specific point on the curve.