Applications

Tangent Line

$$y-f(a)=f^{\prime }(a)(x-a)$$

Straight Line Motion

• position: $x(t)=s(t)=p(t)$
• velocity: $v(t)$
• acceleration: $a(t)$
• $x^{\prime }(t)=v(t)$
• $x^{\prime \prime }(t)=v^{\prime }(t)=a(t)$
• speed = $|v(t)|$

Local Linearization

• A function has the point $(a,f(x))$
• For a value near a, the tangent line can be a good approximation for the y value
• To approximate, plug in the known value for a and the target value for x

Related Rates

1. Recognize when some unit is changing over time (look for “inc” or “how fast”)
2. Make a sketch including all important sides and angles
3. Make a list of known variables (from the sketch)and variables you want to find. Pay attention to what is constant.
4. Choose a formula that ties your variables together (may need to combine and manipulate formulas)
5. Plug in any constant
6. Implicitly differentiate with respect to time
7. Plug in all known values and solve for the unknown

Optimization

• Determining the maximum and minimum values of functions
• Steps
  1. Identify all quantities (make a sketch if needed)
  2. Write an equation needed to be maximized/minimized
  3. Use a secondary equation to get the first in terms of one variable
  4. Use a secondary equation to get the first in terms of one variable
  5. Determine the max/min value
  6. Ensure your answer is within the feasible domain of the problem (ex: time and size can’t be negative)