Power of Maths: Paper 1 – Section 8


REVISION QUESTIONS Concepts and skills


1. Let f (x) = 2x + 1 and g (x) = x 2 + 3x + 2, x ∈R. (a) Find h (x) = f ( g (x)). (b) Find f ′(x), g′ (x) and h′(x). (c) Solve h′(x) > 0.


(d) Find the equation of the tangent to the curve h (x) at x = −2.


(e) Show that h′′(x) is a constant.


2. If u (x) = 3x 2 + 1 and v (x) = x − 2, fi nd: (a) u ′(x) (b) v ′(x) (c) g (x) = u (x) × v (x) (d) g ′(x) (e) u (x) × v ′(x) (f) v (x) × u ′(x) (g) Show that g ′(x) = u (x) × v ′(x) + v (x) × u ′(x).


3. For the curve y = x 3 − x 2 − 2, fi nd: (a)


___ dx ,


dy


(b) the equation k of the tangent at x = 1, (c) the equation l of the tangent at x = −1, (d) the point of intersection of k and l,


(e) the points on the curve where the tangents are parallel to the x-axis.


4. The equation of a curve is y = x 2 − 3x − 4.


(a) Show that P (2, −6) is on the curve. (b) Find the equation of the tangent k to the curve at P (2, −6).


(c) Find the equation of the line l through P perpendicular to k.


(d) Show that l intersects the curve at the point Q, where the curve crosses the y-axis.


(e) Find | PQ |.


5. (a) If a company sells x items per week at S cent each, what is its weekly revenue R, in terms of x, if S = 50x + 20 000.


334 33 34


(b) If the company produces x items per week at a cost of q cents each, fi nd the company’s weekly costs C, in terms of x, if q = 200x.


(c) Write down an expression P for the company’s weekly profi t.


(d) Find dP ___


(e) Find the value of x for which dP ___


dx . to two decimal places.


(f) Find P when dP ___


(g) Show that d 2 P ____


d x 2 < 0.


6. The volume V, in metres cubed of water of height h, in a hemispherical tank of radius 1 m is given by


V = πh2


( 1 − h __


3 ) , where h is the height of the water


from its lowest point A in metres. 1 m


h A


(a) Find V when h = 0. (b) Find V when h = 1, in terms of π. (c) Find dV


(i) h = 0·5 (ii) h = 1


(e) Find d 2 V ____


(f) At what height is dV ___


d h 2 . that


dh = 15π ____


7. (a) If y = 2x 3 − 6x 2 + 6x, fi nd dy


___ dx ≥ 0 for all x ∈R.


(b) For S = 3t 2 + 1, fi nd:


(i) the average rate of change of S with respect to t from t = 3 to t = 5,


(ii) the instantaneous rate of change of S with respect to t at t = −3.


(c) If f ′(3) = 10, fi nd g′(3) if g (x) = 6x 2 − 7f (x).


16 ? dy


Tank dx = 0, correct dx = 0, correct to the nearest euro.


___ dh by multiplying out the brackets.


(d) Find the rate of change of V with respect to h, in terms of π, when:


___ dx and show


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