Cambridge International AS and A Level Mathematics Pure Mathematics 1

Differentiation P1^ 5 KEY POINTS 1   y = kxn ⇒ d d y x =knxn–1 y = c ⇒ d d y x=^0 2  y = f(x) + g(x) ⇒ d d y x = f′(x) + g′(x). ...

P1^ 6 Reversing differentiation Integration Many small make a great. Chaucer ●?^ In^ what^ way^ can^ you^ say^ that these four c ...

Integration P1^ 6 The equation d d y x = 2 x is an example of a differential equation and the process of solving this equation ...

P1^ 6 Reversing (^) differentiation The rule for integrating xn Recall the rule for differentiation: y = xn ⇒ d d y x = nxn^ −^1 ...

Integration P1^ 6 EXAMPLE 6.2 A curve is such that d d y x x x =+ (^382). Given that the point (4, 20) lies on the curve, find t ...

P1^ 6 Exercise (^) 6A EXERCISE 6A  1  Given that d d y x (^) = 6 x^2 + 5 (i) find the general solution of the differential ...

Integration 178 P1^ 6 7  A curve passes through the point (4, 1) and its gradient at any point is given by d d y x (^) = 2 x − ...

P1^ 6 Finding the area under a curve 15  The equation of a curve is such that d d y x x =−^3 x. Given that the curve passes ...

Integration P1^ 6 If T is close to P it is appropriate to use the notation δx (a small change in x) for the ...

P1^ 6 Finding (^) the (^) area (^) under (^) a (^) curve Integrating, A = x^6 + 6 x + c. When x = −1, the line PQ coincides wit ...

Integration P1^ 6 EXAMPLE 6.5 Find the area between the curve y = 20 − 3 x^2 , the x axis and the lines x = 1 and x ...

P1^ 6 Area (^) as (^) the (^) limit (^) of (^) a (^) sum The estimated value of A is 2 + 5 + 10 + 17 = 34 square ...

Integration P1^ 6 Notation This process can be expressed more formally. Suppose you have n rectangles, each of width δx. Notice ...

P1^ 6 Area as the limit of a sum Notice that in the limit: ●●● is replaced by = ●●●δx is replaced by dx ●●● ...

Integration P1^ 6 EXAMPLE 6.7  Evaluate the definite integral 49 (^32) ∫ xxd^ SOLUTION 4 9 4 9 4 9 (^32) (^52) (^52) 52 ...

P1^ 6 Area as the limit of a sum ACTIVITY 6.2  Figure 6.13 shows the region bounded by the graph of y = x + 3, the ...

Integration P1^ 6 EXAMPLE 6.8 Evaluate (^) ∫ 12 ()^3142 −+ 4 xx d.x SOLUTION 1 2 (^421) (^242) 31 (^31434) 3 31 ∫ ()−+ =−∫ ( + ) ...

P1^ 6 Exercise (^) 6B EXERCISE 6B  1  Find the following indefinite integrals. (i) (^) ∫ 3 x^2 dx (ii) (^) ∫(5x^4 + 7 x^6 ) dx ( ...

Integration P1^ 6 5  The graph of y = 2 x is shown here. The shaded region is bounded by y = 2 x, the x axis and the lines ...

P1^ 6 Exercise 6B 10  (i)  Sketch the graph of y = (x + 1)^2 for values of x between x = −1 and x = 4. (ii) Shade ...