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Applications of Derivatives

24 min Calculus

Derivatives are used to analyze increasing/decreasing behavior, extrema, and real-world optimization.

Critical Points

Critical points occur where $f'(x)=0$ or $f'(x)$ is undefined. These are candidates for local maxima and minima.

Example 1

Find the critical points of $f(x)=x^2-6x+5$.

$$f'(x)=2x-6$$ so $2x-6=0$ gives $x=3$.

Example 2

Find the minimum value of $f(x)=x^2-6x+5$.

The parabola opens upward, so the critical point is a minimum. $$f(3)=9-18+5=-4$$

Motion

If $s(t)$ is position, then $v(t)=s'(t)$ is velocity and $a(t)=v'(t)=s''(t)$ is acceleration.

Example 3

Given $s(t)=t^3-6t^2+9t$, find velocity.

$$v(t)=3t^2-12t+9$$

Example 4

A rectangle has perimeter 20. What dimensions maximize area?

If sides are $x$ and $10-x$, then $$A(x)=x(10-x)=10x-x^2.$$ Since $A'(x)=10-2x$, the maximum occurs at $x=5$. The rectangle is a square.

Practice Problems

1. Find the critical point of $f(x)=x^2-8x+1$.

2. Is that critical point a max or min?

3. If $s(t)=2t^2+3t-1$, find $v(t)$.

4. If $v(t)=6t-4$, find $a(t)$.

5. What shape maximizes area for a fixed rectangle perimeter?

Show Answer Key

1. $x=4$

2. Minimum

3. $v(t)=4t+3$

4. $a(t)=6$

5. A square