Section 2.1 A Preview of Calculus
References.
- OpenStax Calculus Volume 1, Section 2.1 1 . (The last topic The Area Problem and Integral Calculus can be skipped for now; we return to it later in the semester.)
- Calculus, Early Transcendentals by Stewart, Section 2.1.
One of the beauties of mathematics is that often, several problems that seem to be quite different turn out to have very similar mathematical representations and solutions, so that there is a common way to solve them.
Two such problems are:
- Making sense of the slope at a point on a curve.
- Finding the velocity of a moving object from knowing its position as a function of time.
The Tangent Problem.
We know how to compute the slope of a straight line, and how this is related to, say, the slope of an inclined plank when the graph describes height as a function of horizontal position. It is very useful to extend to this idea to calculating the slopes of curves. The slope can vary from point to point along a curve, so what we will calculate is the slope at each point of a curve. The geometrical idea is that near a point on a curve, the curve is very close to a certain straight line: the tangent line to that point.
Find an equation of the tangent line to the parabola \(y=x^2\) at the point \(P(2,4)\text{.}\)
Tangent lines at each point of a curve.
We often want the tangent slope or tangent line at multiple points on the curve, or at all of them, and then it is more efficient to proceed as follows:
Example 2.1.2. Tangent lines to a cubic curve.
Find an equation of the tangent line to the cubic \(y=x^3\) at the point \(P(a,a^3)\) for any value \(a\text{.}\)
Solution.First we approximate the slope by the slope \(m_{PQ}\) of the secant line between this point \(P\) and a nearby point \(Q(x,x^{3})\) for \(x\) near \(a\text{,}\)
\begin{equation*}
m_{PQ} = \frac{x^3-a^3}{x-a}.
\end{equation*}
This should approach the tangent slope \(m\) as \(x\) approaches \(a\) [\(x \to a\)], and to see how \(m_{PQ}\) behaves then, it helps to simplify first.
The numerator vanishes for \(x=a\text{,}\) so has a factor \(x-a\text{,}\) and when we divide out this factor, \(x^3-a^3 = (x-a)(x^2+x \cdot a+a^2).\) This gives
\begin{equation*}
m_{PQ} = \frac{x^3-a^3}{x-a} = \frac{(x-a)(x^2+x \cdot a+a^2)}{x-a} = x^2+x \cdot a+a^2,
\; \mbox{for}\; x \neq a.
\end{equation*}
For \(x\) near \(a\text{,}\) this has values close to what we get by substituting \(a\) for \(x\text{:}\) \(m_{PQ}\) gets close to \(a^2 + a \cdot a + a^{2} = 3a^{2}\text{.}\) Thus, it seems that the tangent slope should be For \(x\) near \(a\text{,}\) this has values close to what we get by substituting \(a\) for \(x\text{:}\) \(m_{PQ}\) gets close to \(a^2 + a \cdot a + a^{2} = 3a^{2}\text{.}\)
Thus the tangent slope should be
\begin{equation*}
m = \lim_{x \to a} m_{PQ} = \lim_{x \to a} \frac{x^3-a^3}{x-a} = \lim_{x \to a}(x^2+xa+a^2) = 3a^{2}.
\end{equation*}
The point-slope formula then gives the tangent line at \(P(a,a^3)\text{:}\)
\begin{equation*}
y=a^{3}+3a^{2}(x-a).
\end{equation*}
Note that \(a\) is some constant, only \(x\) is variable, so this is a line, not a more complicated polynomial. For example, at the point \(Q(2,8)\) given by \(a=2\text{,}\) the tangent line is \(y=8+12(x-2)\text{.}\)
The Velocity Problem.
One exercise is enough to reveal that the solution to this problem comes from the same calculations as seen above for computing tangent slopes:
Example 2.1.3. A falling object.
Suppose that a ball is dropped from the upper observation deck of the CN Tower in Toronto, 450m above the ground.
Find the velocity of the ball after 5 seconds.
Then find the velocity at any given time after the ball is dropped.
Recycling ideas and methods of calculation.
This section on the velocity problem is very short because in fact we have already solved the velocity problem by solving the tangent problem.
The ability to solve a few core problems, like the tangent problem, and then “recycle” the ideas and computational methods discovered for them when solving various other problems, is one key to the efficiency and utility of calculus. The single most central idea discovered so far is finding limits: getting from various approximations to an exact answer, so we study that next.
Exercises Exercises
Study Calculus Volume 1, Section 2.1 2 , Exercises 4, 5, 6, 16, and 17.
openstax.org/books/calculus-volume-1/pages/2-1-a-preview-of-calculusopenstax.org/books/calculus-volume-1/pages/2-1-a-preview-of-calculus
