Haef Ib - Math Hl

HAEF IB - MATH HL

A TEST ON DERIVATIVES

(and a taste of integrals!)

by Christos Nikolaidis

Name:______

Date:______

Questions

1. [Maximum mark: 7]

Let . Find

(a) . (b)

......

2. [Maximum mark: 9]

Find the derivatives of the following functions (do not simplify the answers)

(a) (b)

(c)

......

3. [Maximum mark: 5]

Show from first principles that the derivative of is .

......

4. [Maximum mark: 7]

Consider the differentiable function with some values given by

x / 1 / 2 / 3 / 4
/ 2 / 3 / 5 / 10
/ 5 / 4 / 0 / 6

(a) Find the derivative of at

(b) Find the derivative of at

......

5. [Maximum mark: 7]

The point P(1,1) lies on the curve

(a) Find the value of a.

(b) Find the equation of the tangent line to the curve at point P.

......

6. [Maximum mark: 6]

The diagram below shows the graph of which has three stationary points.

(a) Sketch the graph of the function .

(b) Sketch the graph of the function .

7. [Maximum mark: 7]

The total surface area of two cubes is 300 cm2. Given that the lengths of the edges of the two cubes are x and y respectively,

(a) find an expression of y in terms of x.

(b) find the minimum total volume and the corresponding dimensions of the two cubes; justify your answer.

......

8. [Maximum mark: 6]

Find the tangent lines to the curve which pass through the point (0,–48).

......

9. [Maximum mark: 10]

Let

(a) Find the first three derivatives of f.

(b) Guess a formula for , the n-th derivative of f.

......

10. [Maximum mark: 9]

A ladder of length 10 m on horizontal ground rests against a vertical wall. The bottom of the ladder is moved away from the wall at a constant speed of 0.5 ms-1.

(a) Calculate the speed of descent of the top of the ladder when the bottom of the ladder is 6 m away from the wall.

(b) Calculate the rate of change of the area of the triangle formed by the ladder, the wall and the ground at the moment that this triangle is an isosceles.

......

11. [Maximum mark: 15]

Consider the function

(a) Find the domain and the root of the function.

(b) Find the coordinates of the stationary point of the curve and determine

its nature.

(d) Sketch the graph of f (x); indicate the information found above.

(e) Write down the range of the function.

......

12. [Maximum mark: 12]

Consider the graph G of the function and its reflection G΄ about the vertical line .

(a) Sketch the graphs of G and G΄; indicate the coordinates of the intersection point of the two graphs.

(b) Find the area enclosed by the two curves and the x-axis.

A rectangle is drawn so that its lower vertices are on the x-axis and its upper vertices on the curves G and G΄ respectively. The area of this rectangle is denoted by S.

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ROUGH PAPER

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