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Yr 12 HL Maths Calculus revision Assignment

Part A Questions (Non calculator)

1. Consider the function f : x x x2 for 1 x k, where 1 < k 3. (a) Sketch the graph of the function f.

(3)

(b) Find the total finite area enclosed by the graph of f, the x-axis and the line x = k. (4)

(Total 7 marks) 2. The diagram shows the graph of y = f (x).

y

x

y f x= ( )

Indicate, and label clearly, on the graph

(a) the points where y = f(x) has minimum points;

(b) the points where y = f(x) has maximum points;

(c) the points where y = f(x) has points of inflexion.

Working:

(Total 3 marks)

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3. For what values of m is the line y = mx + 5 a tangent to the parabola y = 4 x2?

Working:

Answers:

..............

(Total 3 marks)

4. The diagram shows a sketch of the graph of y = f(x) for a x b.

a b

y f x= ( )

y

x

3

On the grid below, which has the same scale on the x-axis, draw a sketch of the graph of

y = f(x) for a x b, given that f(0) = 0 and f(x) 0 for all x. On your graph you should clearly indicate any minimum or maximum points, or points of inflexion.

a b

y

x

Working:

(Total 3 marks) 5. The line 16y = 113 x is a normal to the curve y = 2x3 + ax2 + bx 9 at the point (1,7). Find the

values of a and b.

Working:

Answers:

..

(Total 3 marks)

3 marks)

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6. The diagram below shows the graph of y1 = f (x), 0 x 4.

0

y

x1 2 3 4

On the axes below, sketch the graph of y2 = ,d)(0x ttf marking clearly the points of inflexion.

0

y

x1 2 3 4

(Total 6 marks)

Part B (calculator Questions)

7. Consider the tangent to the curve y = x3 + 4x2 + x 6.

(a) Using calculus methods, find the equation of this tangent at the point where x = 1.

(b) Find the coordinates of the point where this tangent meets the curve again.

Working:

Answers:

(a) ..

(b) ..................................................................

(Total 3 marks)

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8. Using calculus methods, find the area between the curves y = 2 + x x2 and y = 2 3x + x2. ..............................................................................................................................................

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9. Let f (x) = 3x2 x + 4. Find the values of m for which the line y = mx + 1 is a tangent to the graph of f.

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.............................................................................................................................................. (Total 6 marks)

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10. In the diagram, PTQ is an arc of the parabola y = a2 x2, where a is a positive constant, and PQRS is a rectangle. The area of the rectangle PQRS is equal to the area between the arc PTQ of the parabola and the x-axis.

y

x

S R

QPO

y=a x2 2

T

Find, in terms of a, the dimensions of the rectangle.

Working:

Answer:

..............

(Total 4 marks)

11. A family of cubic functions is defined as fk (x) = k2x3 kx2 + x, k +. (a) Express in terms of k

(i) f k (x) and f k (x); (ii) the coordinates of the points of inflexion Pk on the graphs of fk.

(6)

(b) Show that all Pk lie on a straight line and state its equation. (2)

(c) Show that for all values of k, the tangents to the graphs of fk at Pk are parallel, and find the equation of the tangent lines.

(5) (Total 13 marks)

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12. The function f is given by f (x) = x

x 25 , x 0. There is a point of inflexion on the graph of f at the point P. Find the coordinates of P.

Working:

Answer:

(Total 6 marks)

13. Find the real number k > 1 for which k

x12

11 dx = 23 .

Working:

Answer:

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(Total 4 marks)