Haef Ib - Math Hl

HAEF IB - MATH HL

TEST 5

Derivatives

Date: 27 March 2018

by Christos Nikolaidis

Paper 1: Without GDC

Name:______

Questions

1.[Maximum mark: 5]

Show from first principles that the derivative of is

......

2.[Maximum mark: 6]

Find the derivatives of the following functions

(a) (b) (c)

[2+2+2 marks]

......

3.[Maximum mark: 7]

Consider the differentiable function with

x / 0 / 1
/ / 5

Given that , find

(a) [2 marks]

(b) [3 marks]

......

4.[Maximum mark: 9]

Consider the function.

(a)Write down the domain of . [1 mark]

(b)Find the x-coordinate of the stationary point; determine its nature. [5 marks]

(c)There is a point of inflection at . Find the value of . [3 marks]

......

5.[Maximum mark: 6]

The line is tangent to the curve atx=1.

Find the values of a andb.

......

6.[Maximum mark: 7]

Consider the curve

(a)Find [3 marks]

(b)Find the coordinates of the points on the curve where the tangent lines

are parallel to y-axis [4 marks]

......

Paper 2: With GDC

Name:______

Questions

1.[Maximum mark: 5]

Let . Find

(a) [1 mark]

(b) the equation of the normal line to the curve of at

expressing your answer in the form [4 marks]

......

2.[Maximum mark: 7]

The cubic function has a stationary point of inflection

at (2,3) while the gradient at is 3. Findthe values of a,b,c,d.

......

3.[Maximum mark: 6]

Show by mathematical induction that the n-th derivative of is

given by

for all .

......

4.[Maximum mark: 5]

Find the two points of the curve of which are closest to the point A(2,1).

......

5.[Maximum mark: 7]

Let be the point on the curve which is closest to the point A(2,0).

(a)Show that [5 marks]

(b)Hence find the value of and justify your answer. [2 marks]

......

6.[Maximum mark: 10]

A rectangle is drawn so that its lower vertices are on the x-axis and its upper vertices are on the curve . The perimeter of this rectangle is denoted by P.

(a)Write down an expression for P. [3 marks]

(b)Find the exact value of the minimum perimeter P and justify

that it is a minimum. [5 marks]

(c)Find the value of P if the rectangle is a square. [2 marks]

......

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