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MATH 140
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Lecture 6

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Mathematics & Statistics (Sci)

MATH 140

Ewa Duma

Summer

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THE TANGENT AND VELOCITY PROBLEMS
2.1
In this section we see how limits arise when we attempt to ﬁnd the tangent to a curve or
the velocity of an object.
MATH140 - Lecture 6 Notes
THE TANGENT PROBLEM
Tangent Problem:
The word tangent is derived from the Latin word tangens, which means “touching.” Thus
t
A tangent to a curve is a line that touches the curve. A tangent line should have thee should
have the same direction as the curve at the point of contact. How can this idea be made
same direction as the curve at the point of contact.
precise?
For a circle we could simply follow Euclid and say that a tangent is a line that intersects
Ithe circle once and only once as in Figure 1(a). For more complicated curves this deﬁni-
tangent. Typically, this is not enough information to solve the problem because you needcurve
(a) C. The line l intersects C only once, but it certainly does not look like what we think of as
two points to ﬁnd the slope and y-intercept.
a tangent. The line t, on the other hand, looks like a tangent but it intersects C twice.
P To be speciﬁc, let’s look at the problem of trying to ﬁnd a tangent line t to the parabola
t C Fy ▯ x in the following example.e equation of the tangent line, you need to get solve for
the equation y = mx+b where m is the slope and b is the y-intercept.
2
l V EXAMPLE 1 Find an equation of the tangent line to the parabola y ▯ x at the
Tpoint P▯1, 1▯.e, you can use the following equation: m = (y2-y1)/(x2-x1)
(b) SOLUTION We will be able to ﬁnd an equation of the tangent line t as soon as we know its
Tslope m. The difﬁculty is that we know only one point, P, on t, whereas we need two
FIGURE 1
points to compute the slope. But observe that we can compute an approximation to m by
Example: Find an equation of the tangent line to the parabola y = x^2 at the point P(1,1)
y
*NOTE* Prof used example from textbook.
Q{x,▯▯} t We choose x ▯ 1 so that Q ▯ P. Then
2
y=▯ P(1,▯1) m ▯ x ▯ 1
PQ x ▯ 1
0 x
For instance, for the point Q▯1.5, 2.25▯ we have
FIGURE 2 2.25 ▯ 1 1.25
m PQ▯ 1.5 ▯ 1 ▯ 0.5 ▯ 2.5
x mPQ The tables in the margin show the values of PQfor several values of x close to 1. The
closer Q is to P, the closer x is to 1 and, it appears from the tables, the clis to 2.
2 3 PQ
1.5 2.5 This suggests that the slope of the tangent line t should be m ▯ 2.
1.1 2.1 We say that the slope of the tangent line is the limit of the slopes of the secant lines,
1.01 2.01 and we express this symbolically by writing
1.001 2.001
2
lim m PQ▯ m and lim x ▯ 1 ▯ 2
x mPQ Q lP xl1 x ▯ 1
0 1
Assuming that the slope of the tangent line is indeed 2, we use the point-slope form
0.5 1.5
0.9 1.9 of the equation of a line (see Appendix B) to write the equation of the tangent line
0.99 1.99 through ▯1, 1▯ as
0.999 1.999
y ▯ 1 ▯ 2▯x ▯ 1▯ or y ▯ 2x ▯ 1
Velocity Problems:
83
Similar to tangent problems, using the same equation, but to ﬁnd the instantaneous
velocity. EXAMPLE 1
A24 ||||APPENDIX D TRIGONOMETRY
(a) Find the radian measure of 60.▯ (b) Express 5▯▯rad in degrees.
SOLUTION
(a) From Equation 1 or 2 we see that to convert from degrees to radians we multiply by
D TRIGONOMETRY ▯ ▯180. Therefore
ANGLES 60▯ ▯ 60 ▯ ▯ ▯ rad
Trigonometry Review: ▯180 3
Angles can be measured in degrees or in radians (abbreviated as rad). The angle given by
a complete revolutioAngles can be measured in degrees or in radian

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