17.
Periodic Functions Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that f(x) = f(x+b) for all x.
18.
Periodic Functions Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that f(x) = f(x+b) for all x. The graph of a periodic function: Frank Ma 2006 p
19.
Periodic Functions Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that f(x) = f(x+b) for all x. The smallest number p>0 such that f(x) = f(x+p) is called the period of f(x). The graph of a periodic function: Frank Ma 2006 p
20.
Periodic Functions Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that f(x) = f(x+b) for all x. The smallest number p>0 such that f(x) = f(x+p) is called the period of f(x). The graph of a periodic function: Frank Ma 2006 p one period
21.
Periodic Functions Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that f(x) = f(x+b) for all x. The smallest number p>0 such that f(x) = f(x+p) is called the period of f(x). Over every interval of length p, the graph of a periodic function repeats itself. The graph of a periodic function: Frank Ma 2006 p one period
22.
Periodic Functions Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that f(x) = f(x+b) for all x. The smallest number p>0 such that f(x) = f(x+p) is called the period of f(x). Over every interval of length p, the graph of a periodic function repeats itself. The graph of a periodic function: Frank Ma 2006 p x x+p p one period
23.
Periodic Functions Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that f(x) = f(x+b) for all x. The smallest number p>0 such that f(x) = f(x+p) is called the period of f(x). Over every interval of length p, the graph of a periodic function repeats itself. The graph of a periodic function: Frank Ma 2006 p x x+p p one period f(x) = f(x+p) for all x
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Periodic Functions sin(x) and cos(x) are periodic with period p=2π.
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Periodic Functions sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period. For y=cos(x):
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Periodic Functions sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period. For y=cos(x):
27.
Periodic Functions sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period. For y=cos(x):
28.
Periodic Functions sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period. For y=cos(x):
29.
Periodic Functions sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period. For y=cos(x):
30.
Periodic Functions sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period. For y=sin(x): 0
31.
Periodic Functions sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period. For y=sin(x): 0
32.
Periodic Functions sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period. For y=sin(x): 0
33.
Periodic Functions sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period. For y=sin(x): 0
34.
Periodic Functions The basic period for: y=sin(x)
35.
Periodic Functions The basic period for: y=sin(x) y=cos(x) 1 -1
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Periodic Functions The basic period for: y=sin(x) y=cos(x) 1 -1 The Graph of Tangent
37.
Periodic Functions The basic period for: y=sin(x) y=cos(x) 1 -1 The Graph of Tangent The function tan(x) is not defined when cos(x) is 0, i.e. when x = ±π/2, ±3π/2, ±5π/2, ..
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Periodic Functions The basic period for: y=sin(x) y=cos(x) 1 -1 The Graph of Tangent The function tan(x) is not defined when cos(x) is 0, i.e. when x = ±π/2, ±3π/2, ±5π/2, .. Frank Ma 2006 As the values of x goes from 0 to π/2 the values of sin(x) goes from 0 to 1,
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Periodic Functions The basic period for: y=sin(x) y=cos(x) 1 -1 The Graph of Tangent The function tan(x) is not defined when cos(x) is 0, i.e. when x = ±π/2, ±3π/2, ±5π/2, .. Frank Ma 2006 As the values of x goes from 0 to π/2 the values of sin(x) goes from 0 to 1, but the values of cos(x) goes from 1 to 0. So tan(x) goes from 0 to +∞.
40.
The Graph of Tangent Since tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x).
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The Graph of Tangent Since tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x). Specifically, x π/6 0 π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 tan(x)
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The Graph of Tangent Since tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x). Specifically, x π/6 0 π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 ∞ 0 1/3 1 3 tan(x)
43.
The Graph of Tangent Since tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x). Specifically, x π/6 0 π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 ∞ 0 1/3 1 3 -1/3 -1 -3 -∞ tan(x)
44.
The Graph of Tangent Since tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x). Specifically, x π/6 0 π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 ∞ 0 1/3 1 3 -1/3 -1 -3 -∞ tan(x) 0 π/2 -π/2
45.
The Graph of Tangent Since tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x). Specifically, x π/6 0 π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 ∞ 0 1/3 1 3 -1/3 -1 -3 -∞ tan(x) The same pattern repeats itself every πinterval. 0 π/2 -π/2
46.
The Graph of Tangent Since tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x). Specifically, x π/6 0 π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 ∞ 0 1/3 1 3 -1/3 -1 -3 -∞ tan(x) The same pattern repeats itself every πinterval. In other words, y = tan(x) is a periodic function with period πas shown in the graph. 0 π/2 -π/2
47.
The Graph of Tangent Since tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x). Specifically, x π/6 0 π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 ∞ 0 1/3 1 3 -1/3 -1 -3 -∞ tan(x) The same pattern repeats itself every πinterval. In other words, y = tan(x) is a periodic function with period πas shown in the graph. π -π 0 π/2 -π/2 3π/2 -3π/2 y = tan(x)
48.
The Graph of Inverse Trig-Functions Recalll that for y = cos-1(x), then 0 < y < π.
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The Graph of Inverse Trig-Functions Recalll that for y = cos-1(x), then 0 < y < π. It's graph may be plotted in the similar manner.
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The Graph of Inverse Trig-Functions Recalll that for y = cos-1(x), then 0 < y < π. It's graph may be plotted in the similar manner. (-1, π) (0, π/2) (1, 0) -1 1 The graph of y = cos-1(x)
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The Graph of Inverse Trig-Functions Recalll that for y = cos-1(x), then 0 < y < π. It's graph may be plotted in the similar manner. (-1, π) (0, π/2) (1, 0) -1 1 The graph of y = cos-1(x) Remark: The above graphs of y = sin-1(x) and y = cos-1(x) are the complete graphs (i.e. that's all there is).
52.
The Graph of Inverse Trig-Functions The domain of y = tan-1(x) is all real numbers and the output y is restricted to -π/2 < y < π.
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The Graph of Inverse Trig-Functions The domain of y = tan-1(x) is all real numbers and the output y is restricted to -π/2 < y < π. x ∞ 0 1/3 1 3 -1/3 -1 -3 -∞ tan-1(x)
54.
The Graph of Inverse Trig-Functions The domain of y = tan-1(x) is all real numbers and the output y is restricted to -π/2 < y < π. x ∞ 0 1/3 1 3 -1/3 -1 -3 -∞ tan-1(x) π/6 0 π/4 π/3 π/2
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The Graph of Inverse Trig-Functions The domain of y = tan-1(x) is all real numbers and the output y is restricted to -π/2 < y < π. x ∞ 0 1/3 1 3 -1/3 -1 -3 -∞ tan-1(x) π/6 0 π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2
56.
The Graph of Inverse Trig-Functions The domain of y = tan-1(x) is all real numbers and the output y is restricted to -π/2 < y < π. x ∞ 0 1/3 1 3 -1/3 -1 -3 -∞ tan-1(x) π/6 0 π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 Here is the graph of y = tan-1(x) y = π/2 (1,π/4) (0,0) (-1,-π/4) y = -π/2
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The Graph of Inverse Trig-Functions The domain of y = tan-1(x) is all real numbers and the output y is restricted to -π/2 < y < π. x ∞ 0 1/3 1 3 -1/3 -1 -3 -∞ tan-1(x) π/6 0 π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 Here is the graph of y = tan-1(x) y = π/2 (1,π/4) (0,0) (-1,-π/4) y = -π/2 Remark: y =tan-1(x) has two horizontal asymptoes.
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