Applications of Derivatives

Apply derivatives to solve optimization and rate-related problems.

Rates of Change

The derivative of a function measures its instantaneous rate of change. This is one of its most fundamental and useful applications.

If a function represents position over time, its derivative is velocity.
If a function represents velocity over time, its derivative is acceleration.
In economics, the derivative of a cost function is the marginal cost.

Increasing and Decreasing Functions

The sign of the first derivative tells us whether a function is increasing or decreasing on an interval.

If f'(x) > 0 on an interval, then f(x) is increasing on that interval.
If f'(x) < 0 on an interval, then f(x) is decreasing on that interval.

Tangents and Normals

The derivative is a powerful tool for analyzing the geometry of curves.

A tangent line to a curve at a given point is a straight line that "just touches" the curve at that point and has the same instantaneous slope as the curve. The slope of the tangent at point `(x₀, y₀)` is given by `f'(x₀)`.
A normal line is a line that is perpendicular to the tangent line at the point of tangency. Its slope is the negative reciprocal of the tangent's slope, which is `-1 / f'(x₀)`.

Use the slider in the visualizer below to move the point of tangency and observe how the tangent and normal lines change.

Point of Tangency (x₀)

Maxima and Minima

Derivatives are crucial for finding the maximum and minimum values of a function (optimization). These occur at critical points, where the derivative is either zero or undefined.

First Derivative Test

This test helps identify local maxima and minima by looking at the sign change of the derivative at a critical point `c`:

If f'(x) changes from positive to negative at `c`, then there is a local maximum at `c`.
If f'(x) changes from negative to positive at `c`, then there is a local minimum at `c`.

Second Derivative Test

This test uses the second derivative to classify critical points where f'(c) = 0:

If f''(c) > 0, then there is a local minimum at `c` (concave up).
If f''(c) < 0, then there is a local maximum at `c` (concave down).

Rolle's Theorem

Rolle's Theorem is a special case of the Mean Value Theorem. It states that if a function `f` is:

Continuous on the closed interval [a, b],
Differentiable on the open interval (a, b), and
f(a) = f(b)

Then there exists at least one number `c` in (a, b) such that f'(c) = 0.

Geometrically, this means that if a smooth curve has the same height at two different points, there must be at least one point between them where the tangent line is horizontal (i.e., its slope is zero).

Lagrange's Mean Value Theorem

The Mean Value Theorem (MVT) is a more general version of Rolle's Theorem. It states that if a function `f` is:

Continuous on the closed interval [a, b], and
Differentiable on the open interval (a, b)

Then there exists at least one number `c` in (a, b) such that:

f'(c) = [f(b) - f(a)] / [b - a]

This means that there is at least one point on the curve where the slope of the tangent line (`f'(c)`) is equal to the slope of the secant line connecting the endpoints (a, f(a)) and (b, f(b)).

Applications of Derivatives

Rates of Change

Increasing and Decreasing Functions

Tangents and Normals

Maxima and Minima

First Derivative Test

Second Derivative Test

Rolle's Theorem

Lagrange's Mean Value Theorem

Example Problems

1. Find the local maxima and minima of f(x) = x³ - 6x² + 5

2. A particle moves along a straight line with its position given by s(t) = t³ - 6t² + 9t. Find its velocity, acceleration, and when it is at rest.

3. Find the absolute maximum and minimum values of f(x) = 2x³ - 15x² + 36x + 1 on the interval [1, 5].

4. Show that of all the rectangles with a given area, the square has the smallest perimeter.

5. An open-top box is to be made by cutting small congruent squares from the corners of a 12-inch-by-12-inch sheet of tin and bending up the sides. How large should the squares cut from the corners be to make the box hold as much as possible?

6. Find the dimensions of the rectangle of largest area that can be inscribed in a semicircle of radius r.

7. A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall?

8. Water is leaking out of an inverted conical tank at a rate of 10,000 cm³/min. The tank has a height of 6 m and a diameter at the top of 4 m. Find the rate at which the water level is dropping when the water is 3 m deep.

10. A man walks along a straight path at a speed of 4 ft/s. A searchlight is located on the ground 20 ft from the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 15 ft from the point on the path closest to the searchlight?

11. Find the equation of the tangent and normal to the parametric curve x = a cos³(θ), y = a sin³(θ) at θ = π/4.

12. Find the condition that the line x cos(α) + y sin(α) = p should be a tangent to the curve x²/a² + y²/b² = 1.

13. Find the angle of intersection of the curves y² = 4ax and x² = 4ay.

14. If the normal to the curve x^(2/3) + y^(2/3) = a^(2/3) at any point 'P' makes an angle θ with the x-axis, show that its equation is y cos(θ) - x sin(θ) = a cos(2θ).

15. If the normal to the curve y = f(x) at the point (3,4) makes an angle 3π/4 with the positive x-axis, then f'(3) is equal to:

16. Can Rolle's Theorem be applied to f(x) = tan(x) on [0, π]?

17. Verify Rolle's Theorem for f(x) = sin(x) on [0, π].

18. Find the value of c for f(x) = x(x-2) on [1, 2] that satisfies the Mean Value Theorem.

19. Find c for f(x) = x³ - 5x² - 3x on [1, 3].

20. Using MVT, show that |cos(a) - cos(b)| ≤ |a - b| for all a,b.