Download Calculus with Trigonometry (Day 2) - Notes File

IB Math HL
Calculus with Trigonometry – Day 2
Name: ____________________________________ Block: ______
Derivatives of Inverse Trig Functions
If y  arcsin x , then
dy
1

dx
1  x2
If y  arccos x , then
Ex1) Differentiate the following with respect to x.
a) y  arccos 3x 
dy
1

dx
1  x2
If y  arctan x , then
dy
1

dx 1  x 2
b) f ( x)  arcsin(cos x)
Optimization using Calculus with Trigonometry
Ex2) Two corridors meet at right angles and are 2 m and 3 m wide respectively. θ is the angle marked on the figure. [AB] is a thin
metal tube which must be kept horizontal and cannot be bent as it moves around the corner from one corridor to the other.
a) Show that the length AB is given by L = 3secθ + 2cscθ.
b) Show that
dL
 0 when   tan 1
d
   41.1 . Thus, find the maximum length the tube can be to fit around the corner.
3 2
3
Related Rates with Trigonometry
Ex3) Triangle ABC is right angled at A, and AB = 20 cm. ABˆ C increases at a rate of
changing at the instant when ABˆ C measures

6

180
radians per minute. At what rate is BC
?
Ex4) A farmer has a water trough of length 8 m which has a semi-circular cross-section of diameter 1m. Water is pumped into the
trough at a constant rate of 0.1 m3 per minute.
a) Show that the volume of water in the trough is given by V    sin  , where θ is the angel illustrated (in radians).
b) Find the rate at which the water level is rising at the instant when the water is 25 cm deep.
(Hint: First find ddt and then find dh
at the given instant.)
dt
Homework:
Exercise 18I: #3bde, 4c, 5a
Exercise 20C: #27, 29
Exercise 20D: #10, 14