AP Calculus AB Exam Question

Question: Define the derivative of a function at a given point and explain its geometrical interpretation.

Answer:

The derivative of a function at a specific point represents the rate at which the function is changing at that point. It measures how fast the function is changing as the input (x-value) increases or decreases near that specific point.

Geometrically, the derivative can be thought of as the slope of the tangent line to the graph of the function at the given point. It describes the steepness or slope of the function's curve at that point.

Mathematically, the derivative of a function f(x) at a point x=a is denoted by f'(a) or dy/dx |x=a and is defined as:

where h is a small real number approaching 0.

Example Explanation:

Let's consider the function f(x) = 2x^2 - 3x + 1.

To find the derivative of f(x) at the point x = 2, we can use the definition of the derivative mentioned above.

First, we substitute the function into the definition:

f'(2) = lim(h → 0) [(f(2 + h) - f(2)) / h]

Now, let's evaluate this limit step-by-step:

f(2 + h) = 2(2 + h)^2 - 3(2 + h) + 1 = 2(4 + 4h + h^2) - 6 - 3h + 1 = 8 + 8h + 2h^2 - 6 - 3h + 1 = 2h^2 + 5h + 3

f(2) = 2(2)^2 - 3(2) + 1 = 8 - 6 + 1 = 3

Substituting these values back into the derivative definition:

f'(2) = lim(h → 0) [(2h^2 + 5h + 3 - 3) / h] = lim(h → 0) [(2h^2 + 5h) / h] = lim(h → 0) [2h + 5] = 2(0) + 5 = 5

Therefore, the derivative of the function f(x) = 2x^2 - 3x + 1 at x = 2 is 5. This means that at x = 2, the function has a slope of 5, and its tangent line at this point is rising at a rate of 5 units for each unit increase in x.