Home assignment (1 to 2 hours work per day)
11A March 2020
Functions and Trigonometry
Exercise 1
Part A
Consider the two functions g and h defined by g ( x) x 6 x 8 and h( x) 7 x 10 x .
2 2
Let (C1 ) and (C 2 ) be their respective representative curves in an orthonornal system ( O; i , j ) .
1) a) Determine the points of intersection of (C1 ) with the x-axis .
b) Prove that (C1 ) lies above the straight line of equation y 2 .
c) Study the relative position of (C1 ) and the straight line (d) of equation y 2x 4 .
2) a) Verify that (C 2 ) passes through the origin and determine an equation of the tangent to (C 2 ) at O.
b) Prove that (C1 ) and (C 2 ) are tangent at a point T to be determined.
3) Study the variations of the function g and draw the curve (C1 ) .
Part B
7 x 2 10 x if x 1
1) Consider the function f defined over by f ( x) 2 .
x 6x 8 if x 1
Let (C ) be its respective representative curve in an orthonornal system.
a) Prove that f is differentiable at 1.
b) Calculate f '( x) for x 1 and for x 1 .
c) Prove that (C ) has two points where the tangent is parallel to the x-axis.
2) Let and be two real numbers such that 1 and 1 .
a) Determine a relation between and so that the tangents to (C ) at the points of abscissas and
are perpendicular.
9
b) Determine when .
4
Exercise 2
Consider the function f defined by f ( x) x 6 x 8 .
2
1) Study the variations of f.
2) Prove that the curve (C) of f has an axis of symmetry (). Draw (C).
3) Determine two points on (C) symmetric with respect to () and such that the
tangents to (C) at these points are perpendicular.
1
Exercise 3
Given the table of variations of a function f.
Designate by (C) the representative curve of f in an orthonormal system.
1) Find the domain of definition of f .
2) Determine the equations of the asymptotes of (C).
3) Let (T) be the tangent to (C) at the point A(3; 3).
a) Verify that the equation of (T) is y 2 x 3 .
b) Prove that (T) passes through the point H(1 cos 2 t ;cos 2t ) for any real number t.
c) Determine the set of values of m so that (T) passes through the point
G( m 2 m 1; 2 m 3 ).
4) Determine the sign of each of the following numbers: a f '( 2) , b f '(0) and
c f '( 5) . Deduce that a b c .
5) Compare f (0.5) and f (0.6) . Justify your answer.
Exercise 4
5 1
Given sin .
10 4
2
Calculate the exact value of : cos and cos .
5 5
Exercise 5
1) Prove the equality: sin (a b ) cos (a b ) sin 2a sin 2b 1 .
2 2
7π π
2) Deduce the exact value of the expression: E sin cos 2 .
2
12 12
Exercise 6
Solve each of the following equations :
5π
1) 3 cos x cos
6
2) sin x 3 cos x
3) 2sin x cos x 1
2
2
Exercise 7
Solve each equation:
1
1) cos x sin x
2
3
2) cos 5 x sin 5 x
2 2
2
3) sin 3 x sin x cos x
Exercise 8
Solve each equation:
1) sin 2 x 2cos 2 x 0
2) tan 2 x 3 3 cot x
3) cos x cos3x cos5x 0
Exercise 9
Let E sin x sin 2 x sin 3 x sin 4 x , F cos x cos 2 x cos 3 x cos 4 x
and G 1 cos 2 x 2 cos x .
x 5x x 5x
1) Verify that E 4cos x cos sin and F 4cos x cos cos .
2 2 2 2
2) Solve each of the following equations:
a) E 0 ; b) F 0 ; c) E F .
Exercise 10
Given a b c π .
cos b cos c a
1) Prove that tan .
sin b sin c 2
3π π
cos cos
2) Deduce the value of 7 14 .
3π π
sin sin
7 14
