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Class Notes
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Canada
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McGill University
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Mathematics & Statistics (Sci)
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MATH 140
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Ewa Duma
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Lecture 12

School

McGill University
Department

Mathematics & Statistics (Sci)

Course Code

MATH 140

Professor

Ewa Duma

Description

C
TEC Visual 2.8 shows an animation of
Figure 2 for several functions. (a)
y
Pª(5,▯1.5)
1 y=fª(x)MATH140 - Lecture 12 Notes
Aª Bª Cª
Derivatives as a Function:
0 1 5 x
We previously considered the derivative of a function f at a ﬁxed number “a”:
f’(a) = lim (h > 0) [f(a+h)-f(a)]/h
We can change the value of “a” and replace it with x, we will obtain:
f’(a) = lim (h > 0) [f(x+h)-f(x)]/h
FIGURE 2 (b) M
V Which is useful to us as we attempt to ﬁnd the derivative of higher functions. The
156 |||| CHAPTER 2 LIMITS AND DERIVATIVES (a)function f’ is called the derivative of f because it has been derived from f by the limiting
(b)operation.e by comparing the graphs of f and f▯.
(b) We use a graphing device to graphfand f▯in Figure 3. Notice thaf▯▯x▯ ▯ 0when
f has horizontal tangents anf▯▯x▯is positive when the tangents have positive slope. So
(a)Example: If f(x) = x^3 - x, ﬁnd a formula for f’(x) using the deﬁnition of a derivative.
these graphs serve as a check on our work in part (a).
is h and that x is temporarily regarded as a constant during the calculation of the limit.
2 23 3
f▯x ▯ h▯ ▯ f▯x▯ ▯▯x ▯ h▯ ▯ ▯x ▯ h▯▯ ▯ ▯x ▯ x▯
f▯▯x▯ ▯h l 0 ▯ lh l 0
f h fª h
3 2 2 3 3
_2 ▯ lim x ▯ 3x h 2 3xh ▯_2 ▯ x ▯ h ▯ x ▯ x 2
h l 0 h
3x h ▯ 3xh ▯ h ▯ h
▯ lim ▯ lim ▯3x ▯ 3xh ▯ h ▯ 1▯ ▯ 3x ▯ 1
FIGURE 3 _2l0 h hl0 _2 M
EXAMPLE 3 If f▯x▯ ▯ sx , ﬁnd the derivative ff . State the domainf▯.
Example #2: If f(x) = sqrt(x), ﬁnd the derivative of f.
SOLUTION
f▯x ▯ h▯ ▯ f▯x▯ s x ▯ h ▯ s x
f▯▯x▯ ▯ lim ▯ lim
hl0 h hl0 h
s x ▯ h ▯ s x s x ▯ h ▯ sx
Here we rationalize the numerator. ▯ lim ▯ ▯ ▯
hl0 h s x ▯ h ▯ sx
y ▯x ▯ h▯ ▯ x 1
▯ lim ▯ lim
hl0 h s x ▯ h ▯ s x ) hl0 s x ▯ h ▯ s x
1 1
1 ▯ s x ▯ s x ▯ 2 s
0 1 x We see that f▯▯x▯xists ix ▯ 0 , so the domain off▯is▯0, ▯▯ This is smaller than the
domain of f whiOther Notations: M
(a) ƒ=œ„ x
Let’s check to see that the result of Example 3 is reasonable by looking at the graphs of
y We do not always use f’ or y’ to represent the derivative. There are other notations
f and f▯ in Figwhich also represent the derivative. These may include dy/dx = df/dx = d/dx⋅f(x)
very large and this corresponds to the steep tangent line▯0, 0▯in Figure 4(a) and the
large values of▯▯x▯just to the right of 0 in Figure 4(b).xis largef▯▯x▯is very small
1
and this corresponds to the ﬂatter tangent lines at the far right of the gfaand thethey indicate the
horiz

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