MEI C2 Study Resources Core2 Further Differentiation And by Catherine Berry

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I) Find the gradient of the graph y = x − 5. Find the coordinates of the stationary points of the graph y = passes through the point (1, 2). [3] [3] 1 1 1 and + − x x 2 x3 determine the nature of each. 6. Find the equation of the graph with gradient function [2] [5] dy 1 = − x x which dx x 2 [3] 7. A closed rectangular tank is to have dimensions x m, 2x m and y m, and is to have a capacity of 9 cubic metres. To keep costs down the surface area of the tank is to be as small as possible. (i) Write down expressions for the volume and the surface area of the tank in terms of x and y, and hence find an expression for A in terms of x only.

Find the equation of the graph with gradient function [2] [5] dy 1 = − x x which dx x 2 [3] 7. A closed rectangular tank is to have dimensions x m, 2x m and y m, and is to have a capacity of 9 cubic metres. To keep costs down the surface area of the tank is to be as small as possible. (i) Write down expressions for the volume and the surface area of the tank in terms of x and y, and hence find an expression for A in terms of x only. [3] (ii) Find the value of x for which A has a stationary point, and show that this is a minimum point.

Find the following indefinite integrals 3 ⎞ ⎛ (i) ∫ ⎜ 2 x − ⎟ dx x⎠ ⎝ 1 ⎞ ⎛ 2 (ii) ∫ ⎜ 2 − 3 ⎟ dx ⎝ x 2x ⎠ [3] [3] 3. Find the following definite integrals (i) ∫ 9 (ii) ∫ 2 1 1 (2 − x) x dx [4] ⎛ 6 8⎞ ⎜ 2 − 3 ⎟ dx x ⎠ ⎝x [4] 2 at the point where x = -1. x2 (ii) Find the equation of the tangent to the graph at this point. (iii) Find the equation of the normal to the graph at this point. 4. (i) Find the gradient of the graph y = x − 5. Find the coordinates of the stationary points of the graph y = passes through the point (1, 2).

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