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Lecture 12

Lecture 12 - 2.8.pdf

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Department
Mathematics & Statistics (Sci)
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MATH 140
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Ewa Duma

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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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