MHR • Calculus and Vectors 12 Solutions 485 Chapter 5 Section 1 ...

Chapter 5 Section 1
Rates of Change and the Number e
Chapter 5 Section 1 Question 1 Page 256
a)
b)
c)
d)
MHR • Calculus and Vectors 12 Solutions 485

Chapter 5 Section 1 Question 2 Page 256
a)
b)
c)
d)
Chapter 5 Section 1 Question 3 Page 256
a) f ( x ) = 2 x : {x !!}
b) No
c) No
x
f ( x)
= e : {x !!}
MHR • Calculus and Vectors 12 Solutions 486

Chapter 5 Section 1 Question 4 Page 257
a) B. The graph of the derivative of a quadratic function is a straight line.
b) C. The graph of the derivative of a line is of the form y = a, where a is a constant.
c) D. The graph of the derivative of an exponential function is also an exponential function.
d) A. The graph of the derivative of a cubic function is a quadratic function.
Chapter 5 Section 1 Question 5 Page 257
a) b > e
b) 0 ≤ b < e
Chapter 5 Section 1 Question 6 Page 258
a)
b) Answers may vary. For example:
The graph of the rate of change of
y in the x-axis.
c)
! 1 "
y = # $
% 2 &
x
will be a compression and a reflection of the graph of
MHR • Calculus and Vectors 12 Solutions 487

Chapter 5 Section 1 Question 7 Page 258
Answers may vary. For example:
x
If 0 < b < 1, the graph of y = b will be above the x-axis and the graph of the rate of change of this
x
function will be below the x-axis. If b > 1, the graph of y = b and the graph of the rate of change of
this function will both be above the x-axis.
Chapter 5 Section 1 Question 8 Page 258
a)
b)
c) Answers may vary. For example:
The graph of the combined function g( x ) will be a horizontal straight line.
MHR • Calculus and Vectors 12 Solutions 488

d)
Answers may vary. For example:
The graph of g( x ) = ln 4 is a constant function. Therefore the graph is a horizontal straight line.
Chapter 5 Section 1 Question 9 Page 258
a) Answers may vary. For example:
No. The shape of the graph of g will not change. The shape of the graph of g will be a horizontal
straight line. If the base is other than 4, the graph will be parallel to the graph of g and shifted up or
down depending on the numerical value of the base.
If the value of the base is greater than 4, the graph will be shifted up. If the value of the base is
greater than 1 and less than 4, the graph will be shifted down, but will still be above the x-axis.
If the value of the base is greater than 0 and less than 1, the graph will be shifted down and will be
below the x-axis.
b) The graph is the line g( x) = ln e , which is the horizontal straight line
f !( x)
g( x)
= where g( x ) = 1.
f ( x)
Chapter 5 Section 1 Question 10 Page 258
Solutions to the Achievement Checks are shown in the Teacher’s Resource.
Chapter 5 Section 1 Question 11 Page 258
a) Answers may vary. For example:
The graph of the function g! ( x)
will be the horizontal straight line, y = 0.
b) Answers may vary. For example:
x
If f ( x ) = 2 x
x
2 ln 2
, then f !( x) = 2 ln 2 , and g( x ) = .
x
2
g( x ) = ln 2 , which is just a constant so g! ( x)
= 0. This is applicable for any base. Therefore, the
graph of g! ( x)
will be the graph of y = 0.
The graph of g( x ) is a constant function. The derivative of a constant function is 0.
MHR • Calculus and Vectors 12 Solutions 489