05_2016_-_2017_H2_Maths_Differentiation_and_its_Applications_Notes_(Final)

National Junior College Mathematics Department 2016 Differentiation and its Applications Page 1 of 32 National Junior College 2016 – 2017 H2 Mathematics Topic 5: Differentiation and its Applications Key Questions to Answer: 1. What does the derivative tell you about the behaviour of the graph ? - What does differentiation by first principles mean? - How do you differentiate polynomials by first principles? - What is the geometrical interpretation of f ()x ? 2. What are the meanings of d d y x and d d )f (x x ? - Is d d y x a fraction? - Does d d x y exist? If it does, what does d d x y mean? 3. How do you differentiate the following functions - polynomial functions, - trigonometric functions, - exponential functions, - logarithmic functions, - as well as constant multiples, sums and differences of any combination of the above functions? 4. What are the rules that are useful in differentiation and how do you use them? 5. Under what circumstances would you need to use implicit differentiation, parametric differentiation or logarithmic differentiation? 6. What does 2 2 d d y x mean? - Is this the same as dd dd y xx   or 2 d d y x   ? 7. How do you relate the concavity of the graph with the first and second derivative? - Is the graph of 1y x an example of a graph that is concave upwards? Why or why not? 8. How do you identify a local maximum or minimum on the graph? - What is the difference between a local and a global maximum/minimum? - What is the first derivative test? - What is the second derivative test? Why does it work? Does it work all the time? - When should you use the first derivative test instead of the second derivative test? 9. How do you find the equation of the tangent? - What is the relationship between the gradient of the tangent and that of the normal? - How do you find the equation of the normal? 10. What do you see in common between chain rule, implicit differentiation, parametric differentiation and connected rates of change? Note: This topic is a build -up from the basic ca lculus knowledge acquired under the O -level Additional Mathematics syllabus. Knowledge of O-level calculus is assumed.

National Junior College Mathematics Department 2016 Differentiation and its Applications Page 2 of 32 §1 Differentiation Techniques 1.1 Differentiation by First Principles Recall that the gradient of a straight line is the rate of change of y with respect to x, i.e. change in .change in y x For example, the line 2yx has gradient 2, which means that every unit change in x results in 2 units change in y or equivalently, the rate of change of y with respect to x is 2. For a curve, recall that the gradient is found by differentiation. For example, for the curve 2,yx d 2.d y xx  Hence the gradients of the curve 2yx at x = –