Tangent lines

Optimization: Differentiation

Tangent lines

As we saw earlier, the derivative function gives the slope of a tangent line at each point. We will now discuss the tangent in more detail, because it can be useful in all kinds of situations, such as approximating answers when functions are difficult to calculate.

If #f# is a graph of a function on the plane and #P# is a point on that graph, then a line #l# through #P# is called a tangent to #f# if

#P# is the only point that #f# and #l# have in common in a small neighbourhood of #P#, and
the points of #l# near #P# are all on the same side of #f#.

In plain English, the line #l# and the graph of #f# both pass through #P# but do not intersect near #P#.

A tangent line concerns the vicinity of the point #P#. It is possible that the line #l# and the graph #f# intersect at a point further away.

Below we see that the tangent at point #P# within a small neighbourhood of that point is the best linear approximation of that function.

Best linear approximation Let #f# be a differentiable function on an open interval containing the number #p#. The tangent line to the graph of #f# in #P = \rv{p,f(p)}# is the line that best approximates #f# to #P#. Instead of the tangent to the graph of #f# in #P#, we also speak of the tangent to #f# in #p#.

The information that the tangent line #l# passes through #\rv{p,f(p)}# and has slope #f'(p)# is sufficient to formulate an equation for the line:

\[ y= f(p)+f'(p)\cdot(x-p)\]

The derivative function gives the slope of the tangent line. In the equation of the line, this slope #f'(p)# is used in #p#. Furthermore, #\rv{x,y} = \rv{p,f(p)}# satisfies this equation, so the point #\rv{p,f(p)}# lies on this line. This determines the tangent uniquely. After all, if we know a point of a line and its slope, then we know the line.

At the point #\rv{p, f(p)}# the tangent line #l# and the function #f# have the same slope; and therefore they run at the same angle. Because they run at the same angle at the point #\rv{p, f(p)}#, they will not intersect in the vicinity of the point #\rv{p, f(p)}#. This means that line #l# is indeed a tangent.

The fact that the line #l# is the best approximation for the graph of #f#, means that also #f(x)#, for #x# close to #p# can be easily and effectively approximated by the linear function of the line:

\[f(x)\approx f(p)+ f'(p)\cdot (x-p)\qquad \text{for }x\text{ sufficiently close to }p\]

Application In economics, the marginal cost of a production process indicates the amount by which total costs increase when the level of production increases by one additional unit. If we write #K(Q)# for the cost as a function of the production level #Q# measured in numbers, and apply the formula #f(x)\approx f(p)+ f'(p)\cdot (x-p)# for the best linear approximation of a function #f(x)# for #x# sufficiently close to #p# with #f=K#, #p=Q# and #x=Q+1#, then we get \(K(Q+1)\approx K(Q)+ K'(Q)\), or

\[K'(Q) \approx K(Q+1)-K(Q)\]

In other words: for sufficiently large production units, the marginal costs can be seen as the derivative of the cost function.

Graph Below is a graph of the quadratic function #f# and the tangent to #f# at a point #\rv{p,f(p)}#. You can move the points #\rv{a,f(a)}# and #\rv{b,f(b)}# to see how the tangent moves.

Determine the equation for the tangent at the point #[2,-21]# to the graph of the function #f(x)=-x^3-9x+5#.

#y=# #-21x+21#

We write the equation for the tangent to #P# as #y=ax+b#, in which #a# and #b# are yet to be determined real numbers. Because #a# is the slope of the tangent #f'(p) = a# applies. According to the derivative of a polynomial the derivative of #f# is given by #f'(x) =-3 x^2 -9#. Consequently: #a=f'(2)=-3\cdot(2)^2-9=-21#.

Now we determine the value of #b# in the equation #y=-21x+b# of the tangent. Since the point #[2,-21]# is on the tangent, #-21=-21\cdot (2) + b# must apply. This is a linear equation with one unknown (namely #b#) which we can solve easily by reduction. This gives #b=21#.

The following figure shows the graph of the function #f(x)=-x^3-9x+5# and the tangent line to #f# at the point #[2,-21]#.

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