NAME: ___________________________
42
KNOW
/ 12 APP
/ 12 INQ
/ 12 COMM
/6
MCV4U1 - UNIT 2 – DERIVATIVES
TEST
ROUND ALL ANSWERS TO THE NEAREST TENTH, UNLESS OTHERWISE STATED.
1) Differentiate each of the following functions. Final answers should not contain negative or
fraction exponents and should be simplified/factored as much as possible.
a) f ( x) 
4x3  9x 2  6x  5
2

c) f ( x)  6 x 2  5
 2 x  1
5
4
(K – 3 marks)
(K – 3 marks)
b) f ( x) 
4 x
3
2
 2x  7
d) f ( x) 
(K – 3 marks)

5
2  7x
5 x
2
3

4
(K – 3 marks)
e) f ( x) 
x
4x  5
2

3
2
(A – 3 marks)
f) f ( x)  3
x2 1
3x  4
(A – 3 marks)
2) Find the derivative of f ( x)  7 x 2  5 x  8 using first principles. (A – 3 marks)
3) The owners of a restaurant are interested analyzing the productivity of their staff. The
600
function N (t )  150 
models the total number of customers, N, served by the
16  3t 2
staff after t hours during an 8-hour workday (0  t  8) . Determine the rate, to the nearest
tenth, at which the customers are being served at 4 hours. (A – 3 marks)
4) Determine the equation of the tangent to f ( x)   x 2  3 x  1 that is perpendicular to the line
y
1
x2.
3
(I – 6 marks)
5) Find numbers a, b, and c such that the graph of y  ax 2  bx  c has x-intercepts of 0 and 8,
and a tangent with a slope of 16 at x = 2. (I – 6 marks)
(more space available on next page)
6) Fill in the boxes below with integers to create the equation of another function that has the
same tangent slopes as y  8 x 3  4 x 2  7 x  5 at all x-values. (C – 2 marks)
y
x3 
x2 
x
7) Prove the Product Rule of differentiation. That is, prove that if p ( x)  f ( x) g ( x) , where
f (x) and g (x) are differentiable, then p ( x)  f ( x) g ( x)  g ( x) f ( x) . (C – 4 marks)