# On The Shoulders Of Giants A Course In Single Variable by Geoff Smith;Gordon McClelland

By Geoff Smith;Gordon McClelland

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Extra resources for On The Shoulders Of Giants A Course In Single Variable Calculus

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Function of . Particular values of are denoted by the function value notation. Thus The derivative is written and this again denotes a function with particular values written with the function value notation, as in . There is also an alternative notation for derivatives which originated with Leibniz and which is notation. Recall that the derivative of a function at a point is still used today—the ✥ ④ ✥ ④ ✡ ✥ ④ ✑ ✥ ✑ ④ ④ ✰ ✛ ✲ ✡ ✛ ✡ ✒ ✑ ④ ð ð ④ ✟ ④ ⑦ ✟ ✰ ❈ ✲ ✥ ❨ ❨ ✰ ✥ ↕ ✲ ❇ ❨ ✰ ✥ ❺ ✥ ❺ ✲ ð ❨ ✰ ✥ ❺ ✲ ✡ ❮ ↕ ❰ Ð ✪ Ï Ñ ✥ where ➔ ✥ ↕ → ➯ ✡ ✥ ↕ ❇ ❇ ✥ ❺ is any special sequence that converges to .

3 . This is a function whose graph we would certainly Let us deﬁne the function by the rule expect to satisfy our intuitive notion of a smooth function. 1 above, we estimated the gradient at a point on the graph of by drawing up a table, but such a table cannot give an exact answer for the gradient at a particular point. In this example, we will use what we know about convergence of sequences to get an exact answer for the gradient of the tangent at a particular point on the graph. be any real number (so that is in the domain of ).

Can be any sequence converging to 2, we conclude that is continuous at 2. Since ✼ ✓ ✒ ➜ ❈ ✑ ➔ ✥ ↕ → ❮ ❰ ↕ ✩ ✥ ✡ Ï ✥ Ð ↕ ✡ ❈ ❮ Ñ ↕ ❰ Ð Ï ✩ ✰ ✥ ↕ ✲ ✡ ❨ ✰ ⑦ ❿ ➔ → ✥ ↕ ✲ ❈ ➔ ✖ ❈ Ñ → ➔ ❈ ❞ ✥ → ✑↕ ❈ ➔ ✼ ✓ ✒ ✥ → ✑ ➔ ✥ ↕ ✼ ✓ ✒ ➜ ❈ ✑↕ ➔ ✩ ✰ ✥ ↕ ✲ → ✩ ✰ ❈ ✲ ✑ → ✩ ❊ In this last example we have made assumptions about the convergence of combinations of sequences. For example, we assumed converges to if converges to . There is a theorem which deals with these matters. We state it without proof. 1 Let be a sequence converging to ➔ ✥ (a) ↕ ➧ (b) ➔ ✥ ✥ ↕ ➔ ✥ ↕ ➔ ✥ ↕ ➔ ✥ ↕ ↕ ➔ (c) (d) (e) → converges to → ❞ ❇ ④ ⑦ ↕ ↕ ↕ → ④ ↕ → ✥ converges to ✥ ❺ and let ❺ for any constant ❺ converges to converges to → ✥ converges to ④ → ④ ➧ ✥ ④ ✥ ❺ ✥ ❺ ❞ ❇ ④ ❺ ④ ❺ ➔ ④ ↕ be a sequence converging to → ④ ❺ .