Applications

To determine the equation of the tangent line on a given \(f(x)\), note that a tangent line is a LINE. Consequently, the equation of the tangent can be found by updating the point-slope equation \(y-y_0=m(x-x_0)\). Recall that the point \((x_0,y_0)\) must reside on the tangent line, whereas the slope of the line can be found by evaluating the derivative of \(f(x)\) at the given point.

The linear approximation (tangent line approxiaiton) for a function \(f(x)\) at the center of \(x=a\) is the tagent line 

\(L(x)=f(a)+f'(a)(x-a)\)

Linear approximations are sometimes formulated in terms of differentials such as \(dt\) where \(dy=f'(x)dx\). Applications that require an investigation of incremental changes may require anaylizing the difference \(\Delta y=f(x+\Delta x)-f(x)\)

Optimization problems naturally arise when there is a need to determine the most efficient usage of resources. Common applications included determining the maximizing area that can be enclosed by using a given amount of fencing material, as well as minimizing costs for manufacturing objects from raw material.

A critical number of a function \(f(x)\) is a number \(c\) that is contained within the function’s domain such that either \(f'(c)=0\) or \(f'(c)\) does not exist (DNE). 

Relative Extrema

To determine the relative extrema for a function \(f(x)\) on an open interval \((a,b)\)

1. Determine the critical values of \(f(x)\) on the interval \((a,b)\)
2. Apply either First Derivative Test or the Second Derivative Test.
3. Compute the value of the function at these critical points. The largest of these values is the absolute maximum, whereas the smallest of these values is the absolute minimum.

Absolute Extrema

To find the absolute extrema for a function \(f(x)\) on a given closed interval \([a,b]\) requires determining the relative extrema and insepcting the endpoints.

1. Determine the critical values of \(f(x)\) on the interval \([a,b]\) and locate any relative extrema.
2. Discard any critical values that are NOT contained within the interval \([a,b]\).
3. Inspect the endpoints, i.e. compute \(f(a)\) and \(f(b)\)
4. Compute the value of the function at the critical points. The largest of these values is the absolute maximum, whereas the smallest of these values is the absolute minimum.

CAUTION: The \(x\)-value for the critical point is NEITHER the relative extrema NOR the absolute extrema. This \(x\)-value however tell us where to find these extrema values, just as compass points us in the direction of travel. The extrema values for a function corresponds to the \(y\)-value in the critical point, i.e. value for \(f(c)=y\).

Another useful application of the derivative involves analyzing the graph of a given function in terms of monocity, concavity, as well as optimization.

A function \(f(x)\) is said to be monotonic if it is either nondecreasing or nonincreasing. A function \(f(x)\) is said to be nondecreasing provided that \(f(a)\leq f(b)\) where \(a<b\). If the inequality is strict i.e. \(f(a)\leq f(b)\) then the function is said to be increasing, or strictly increasing. A function \(f(x)\) is said to be nonincreasing provided that \(f(a)\leq f(b)\) where \(a<b\). If the inequality is strict i.e. \(f(a)\leq f(b)\) then the function is said to be decreasing, or strictly decreasing.

An inflection point is a point on a graph \(f(x)\) in which the concavity of the graph changes. An inflection point for \(f(x)\) can be found by finding the critical values for \(f'(x)\). In other words, if \(c\in(a,b)\) such that \((c,f(c))\) is an inflection point then one of the following must hold true. 

1. \(f”(c)=0\)
2. \(f”(c)\) DNE

Related rates an application of implicit differential whose variable of differentiation is typically in terms of time \(t\). These problems, as the name suggests, relate various rates by analyzing differentials. 

1. Construct a diagram that will capture the meaning of the problem at hand.
2. Use the diagram to better understand the problem.
3. Develop a system of equations
4. Compute the necessary differentials
5. Plug-in the given values and solve for the indicated variable

A particle changes direction whenever \(v(t)=0\)

When the object is moving in the

• right direction or moving upward then the velocity is positive.
• left direction or moving downward then the velocity is negative.

The Total Distance for an object from \(x=a\) to \(x=b\) whose velocity is given by \(v(t)\) can be found by evaluating the integral \(\displaystyle \int_{a}^b|v(t)|dt\).

Displacement measures the net change of the object’s position, which can be found by computing \(\displaystyle \int_{a}^b v(t)dt\). Notice the subtle difference between these integral. When is it guaranteed that the total distance travel is the same as the displacement?

To determine whether a function \(f(x)\) is invertible over a given region it must pass the horizontal line test. Functions that pass the horizontal line do not change in their monocity, in other words, either \(f'(x)>0\) always or \(f'(x)<0\) always. 

What happens if the monocity  of a function changes?