## 2.11.2 Differentiation (Paper 2), Question 2 & 3

Question 2: Given the equation of a curve is: y = x2 (x – 3) + 1 (a) Find the gradient of the curve at the point where x = –1. (b) Find the coordinates of the turning points. Solution: (a) y= x 2 ( x−3 )+1 y= x 3 −3 x 2 +1 dy … Read more

## 2.11.1 Differentiation Long Questions (Question 1)

Question 1: The curve y = x3 – 6×2 + 9x + 3 passes through the point P (2, 5) and has two turning points, A (3, 3) and B. Find  (a) the gradient of the curve at P. (b) the equation of the normal to the curve at P. (c) the coordinates of B and … Read more

## 2.10.6 Differentiation Short Questions (Question 22 – 25)

Question 22: Given that   y = 3 4 x 2 , find the approximate change in x which will cause y to decrease from 48 to 47.7. Solution: y = 3 4 x 2 d y d x = ( 2 ) 3 4 x = 3 2 x δ y = 47.7 − … Read more

## 2.10.5 Differentiation Short Questions (Question 19 – 21)

Question 19: The volume of water V cm3, in a container is given by   V = 1 5 h 3 + 7 h , where h cm is the height of the water in the container. Water is poured into the container at the rate of 15cm3s-1. Find the rate of change of the … Read more

## 2.10.4 Differentiation Short Questions (Question 15 – 18)

Question 15: Find the coordinates of the point on the curve, y = (4x – 5)2 such that the gradient of the normal to the curve is 1 8 . Solution: y = (4x – 5)2 d y d x = 2(4x – 5).4 = 32x – 40 Given the gradient of the normal is … Read more

## 2.10.3 Differentiation Short Questions (Question 11 – 14)

Question 11: Given that the graph of function f ( x ) = h x 3 + k x 2  has a gradient function f ‘ ( x ) = 12 x 2 − 258 x 3 such that h and k are constants. Find the values of h and k. Solution: f ( x ) = … Read more

## 2.10.2 Differentiation Short Questions (Question 6 – 10)

Question 6: Given that f (x) = 3×2(4×2 – 1)7, find f’(x).  Solution: f (x) = 3×2(4×2 – 1)7 f’(x) = 3×2. 7(4×2 – 1)6. 8x + (4×2 – 1)7. 6x f’(x) = 168×3 (4×2 – 1)6 + 6x (4×2 – 1)7 f’(x) = 6x (4×2 – 1)6 [28×2+ (4×2 – 1)] f’(x) = 6x (4×2 – 1)6 (32×2 – 1) Question 7: Given that y = (1 + 4x)3(3×2 – 1)4, find d y d x … Read more

## 2.10.1 Differentiation Short Questions (Question 1 – 5)

Question 1: Differentiate the expression 2x (4×2 + 2x – 5) with respect to x. Solution: 2x (4×2 + 2x – 5) = 8×3 + 4×2– 10x d d x (8×3 + 4×2 – 10x) = 24x + 8x –10  Question 2: Given that  y = x 3 + 2 x 2 + 1 3 x ,  find  … Read more

## 2.9 Small Changes and Approximations

2.9 Small Changes and Approximations If  δ x  is very small,  δ y δ x  will be a good approximation of  d y d x , , This is very useful information in determining an approximation of the change in one variable given the small change in the second variable.  Example: Given that y = 3×2 + 2x – 4. Use differentiation … Read more

## 2.8 Related Rates of Change

(A) Related Rates of Change 1. If two variables x and y are connected by the equation y = f(x) Notes: If x changes at the rate of 5 cms -1 ⇒  d x d t = 5 Decreases/leaks/reduces Þ  NEGATIVES values!!! Example 1 (Rate of change of y and x) Two variables, x and y are related … Read more