Table of Contents

Chapter 5 - Untitled

Monday, 1 January 0001
3-minute read
467 words

Optimization

  1. Assign symbols to all given quantities and quantities to be determined. Make a sketch (if possible).
  2. Write a PRIMARY EQUATION for the quantity being maximized or minimized. (Call this M.)
  3. Reduce the primary equation to one having a single independent variable.
  4. Determine the maximum/minimum using critical values.
  5. Use the first (or second) derivative test and choose the feasible answer.
  6. Be sure to answer the question asked. Include units (if applicable).

Example 1

Let the sum of two numbers equal to 16. What are those values that yield the greatest product (maximum)?

$a + b = 16$

$\frac{d}{db}(ab) = 0$

$a = 16 - b$

$(16 - b)(b) = 16b - b^2$

$16 - 2b = 0$

$b = 8$

$a = 8$

Example 3

Example 4

Closest point to $y = 4 - x^2$

$m = \sqrt{x^2 - (2 - x^2)^2}$

$m' = 4x^3 - 6x = 0$

$x = \{ 0, \pm \frac{\sqrt{3}}{2}\}$

Example 5

Find the largest area you can make with 800 meters of wire, open on the top.

$800 = 2x + y$

$m = xy$

$m = x(800 - 2x)$

$m = 800x - 2x^2$

$m' = 800 - 4x$

$m'' = -4$ (this is a minimum)

$m' = -4(x - 200)$

$x = 200$

$800 = 2(200) + y$

$y = 400$

Linearization

If $f$ is differentiable at $x = a$, then the equation of the tangent line, sometimes called $L(x)$, defines the linearization of $f$ at $a$. The approximation is called the linear approximation of $f$ at $a$. The point $x = a$ is called the center of the approximation. Values of $f(x)$ will be "close" to $L(x)$ near the center.

Differentials

Example 1

Let $y = 1 - 2x^2$. Find $dy$ when $x = 1$ and $dx = -0.1$. Compare this value with $\Delta y$ for $x = 1$ and $\Delta x = -0.1$

$\frac{dy}{dx} = -4x$

$dy = -4x \cdot dx$

Related Rates

  1. Make a sketch (if possible). Name the variables and label the constants on your sketch
  2. Write down the known information and the variable we are to find
  3. Write an equation that relates the variables
  4. Differentiate implicitly the equation with respect to $t$ using the chain rule
  5. Answer the question that was asked with correct units

Sample AP Question 1

The radius of a circle is increasing at a constant rate of 0.2 meters per second. What is the rate of increase in the area of the circle at the instant when the circumference of the circle is $20\pi$ meters?

\[ C = 20\pi \]

\[ \frac{dr}{dt} = 0.2 \]

\[ C = 20\pi = 2\pi r \]

\[ 20\pi = 2\pi r, r = 10 \]

\[ A = \pi r^2 \]

\[ \frac{dA}{dt} = 2\pi r \frac{dr}{dt} = 2\pi (10)(0.2) = 4\pi \]