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Additional info for Calculus: How Calculus Works (Studymates in Focus) (In-Focus)
Sample text
In this chapter we establish: ● the notation for second order differentiation ● the relationships between the gradient function and the second derivative This leads to some very important techniques in Chapter 7. Repeated differentiation We have seen that when we differentiate a function, we get a new function, which represents the gradient of the original function. e. we can obtain the gradient function of the gradient function! 1 y = x 3 dy/dx = 3x 2 The function dy/dx = 3x 2 can be differentiated.
Does the same short cut work this time? 3. 2), (a), (b), (c). Can you take a similar short cut in these examples? Practical assignment (5) (a) Sketch the curve y = 1/x. You should be able to do this without referring back. (b) Describe how the gradient changes: (i) as x increases in the 1st quadrant (x > 0) and (ii) as x decreases in the 3rd quadrant (x < 0) (c) Using your observations from (b), sketch the curve of the gradient function. (d) Sketch the curve y = −1/x2 (e) Comment on your results.
Remember that x0 = 1). 1) Apply the above process to these functions: (a) y = x2 (b) y = 3x (c) y = ax (a is constant) (d) y = 5 (e) y = k (k is a constant) (f) y = 4x2 (g) y = 2x3 (h) y = 0 4 (i) y = x Check your answers before moving on. Now we can extend the range of functions which we can differentiate. The general rule for y = x n The relationship we have discovered between y and dy/dx applies also to higher powers of x. indb 24 1/22/2008 1:00:58 PM How to differentiate algebraic functions This rule is of fundamental importance.