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Differentiation Applications

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.

  1. Start with the point-slope form. \(y-y_0=m(x-x_0)\)
  2. If a point has been provided, determine the point of tangency and update the equation for the point-slope form for the tangent line.
  3. Compute \(f'(x)\) and evaluate \(m=f'(x_0)\)
  4. Update the point-slope form with the newly computed slope value \(m\)
  5. Simplify the equation in either the general form for a line or in slope-intercept form.
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 \(\Delta y=f(x+\Delta x)-f(x)\)
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.  To solve these problems,
  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

Another useful application of the derivative involves analyzing the graph of a given function in terms of monocity and concavity.

First Derivative Test Let \(f(x)\) be a differentiable function on a given interval \((a,b)\)
  • If \(f'(x)<0\) on the interval \((a,b)\) then \(f(x)\) is a decreasing on the given interval.
  • If \(f'(x)>0\) on the interval \((a,b)\) then \(f(x)\) is a increasing on the given interval.
Second Derivative Test Let \(f(x)\) be a twice differentiable function on a given interval \((a,b)\)
  • If \(f”(x)<0\) on the interval \((a,b)\) then \(f(x)\) is a concave downwards on the given interval.
  • If \(f'(x)>0\) on the interval \((a,b)\) then \(f(x)\) is a concave upwards on the given interval.
Function Interpretation Polarity of \(v(x)\) Polarity of \(a(x)\) Object’s Motion
\(f(x)\) Position of a particle \(v(x)<0\) \(a(x)<0\) Speeding Up
\(v(x)=f'(x)\) Velocity of a particle \(v(x)<0\) \(a(x)>0\) Slowing Down
\(a(x)=v'(x)=f”(x)\) Acceleration of a particle \(v(x)>0\) \(a(x)<0\) Slowing Down
\(v(x)>0\) \(a(x)>0\) Speeding Up
  • When the object is moving in the right direction or moving upward then the velocity is positive.
  • When the object is moving in the left direction or moving downward then the velocity is negative
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. 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 \([a,b]\)
  1. Determine the critical values of \(f(x)\) on the interval \([a,b]\)
  2. Inspect the endpoints, i.e. compute \(f(a)\) and \(f(b)\)
  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.

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 \(f(c)=y\).

Let \(f(x)\) be a function that satisifes the two conditions
  1. Continuous on the closed interval \([a,b]\)
  2. Differentiable on the open interval \((a,b)\)
Then, there exists a number \(c\) within the interval, i.e. \(a<b<c\), such that the following holds true

\(\displaystyle f'(c)=\frac{f(b)-f(a)}{b-a}\)