Grade 12

Mathematics

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Online Classroom Rules

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Understand the link between the slope of a tangent line and a non-tangent line to a graph geometrically.

Learning Objectives

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Write the equation of a tangent line to a graph at a given point using limits.

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Find the average velocity and the instantaneous velocity at a given point.

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Solve mathematical and real-life problems using derivatives.

A traditional slingshot is essentially a rock on the end of a string, which you rotate around in a circular motion and then release. When you release the string, in which direction will the rock travel?

An overhead view of this is illustrated in Figure.

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Many people mistakenly believe that the rock will follow a curved path, but Newton’s first law of motion tells us that the path as viewed from above is straight. In fact, the rock follows a path along the tangent line to the circle at the point of release. Our aim in this section is to extend the notion of tangent line to more general curves.

https://www.geogebra.org/m/zecrjqk4

The point corresponding to x = 1 is (1, 2) and

the line with slope 2 through the point (1, 2) has equation

y = 2(x − 1) + 2 or y = 2x.

Suppose that the function s(t) gives the position at time t of an object moving along a straight line.

That is, s(t) gives the displacement (signed distance) from a fixed reference point, so that s(t) < 0 means that the object is located |s(t)| away from the reference point, but in the negative direction.

Then, for two times, a and b (where a < b), s(b) − s(a) gives the signed distance between positions s(a) and s(b).

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The average velocity vavg is then given by

NOTES

1. Notice that if (for example) t is measured in seconds and s(t) is measured in feet, then velocity (average or instantaneous) is measured in feet per second (ft/s).

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• When used without qualification, the term velocity refers to instantaneous velocity.

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velocity indicates both speed and direction.

s(t) measures the height above the ground.

So, the negative velocity indicates that the object is moving in the negative (or downward) direction.

The speed of the object at the 2-second mark is then 64 ft/s.

The units of the instantaneous rate of change are the units of f divided by the units of x.

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You should recognize this limit as the slope of the tangent line to y = f (x) at x = a.

provided the limit exists.

Solution From the graph in Figure 2.12, notice that, no matter how far we zoom in on (0, 0), the graph continues to look like Figure 2.12.

This indicates that the tangent line does not exist.

Further, if h is any positive number, the slope of the secant line through (0, 0) and (h,|h|) is 1.

However, the secant line through (0, 0) and (h, |h|) for any negative number h has slope −1.

Defining f (x) = |x| and considering one-sided limits, if h > 0, then |h| = h, so that

In exercises 13 and 14, list the points A, B, C and D in order of increasing slope of the tangent line.