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JingleBells

  • 2 years ago

Problem Set 1: 1D-10 Show that

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  1. JingleBells
    • 2 years ago
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    \[g(h)=\left[ f(a+h)-f(a) \right]/h\]has a removable discontinuity at h=0 \[f \prime(a) exist\]

  2. JingleBells
    • 2 years ago
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    I gave it a try, is anyone convinced? \[g(0^{-})=\lim_{h \rightarrow 0}\left[ f(a+h)-f(a) \right]/h=f \prime (a)\] \[g(0^{+})=\lim_{h \rightarrow 0}\left[ f(a+h)-f(a) \right]/h=f \prime (a)\] Since the left-hand limit=right-hand limit at h=0, i.e. \[f \prime(a)=f \prime(a) \] Therefore f'(a) exists and g(h) has a removable discontinuity at h=0 Sounds about right? seems too simple an answer...

  3. JXP
    • 2 years ago
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    Yes, it is correct. The first problem set goes over some basics of high school calculus since some of the students have not taken calculus yet. So keep on going if you're fairly confident with your answer!

  4. Stacey
    • 2 years ago
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    What you have is correct. It is removable because by defining g(0) = f'(a) the function becomes continuous at h=0.

  5. JingleBells
    • 2 years ago
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    Thanks a million! I'm using these videos to prepare for A-levels so I'm bound to be out of my depth most of the time.

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