Calculus Some Questions 1

1. (12 points) Evaluate each of the limits.

4x2 + 11x − 3
a. (3 pts) lim Ans:
x→−3 x2 − 9

4x2 + 11x − 3 (4x − 1)(x + 3)

lim 2
= lim
x→−3 x −9 x→−3 (x − 3)(x + 3)
4x − 1
= lim
x→−3 x − 3
−13
=
−6
13
=
6

4 − 21x3
b. (3 pts) lim
x→∞ 7x3 + 6x2 + 3x − 4

Ans:
4 3
4 − 21x3 x3
− 21x
x3
lim = lim 7x3 2
x→∞ 7x3 + 6x2 + 3x − 4 x→∞ + 6x + 3x − x43
x3 x3 x3
4
x3
− 21
= lim 6
x→∞ 7 +
x
+ x32 − x43
−21
=
7
= −3
Calculus Some Questions 2

Evaluate each of the limits.

5x − x2
c. (3 pts) lim
x→∞ 1 + 2x

Ans:
5x 2
5x − x2 x2
− xx2
lim = lim 1
x→∞ 1 + 2x x→∞
x2
+ 2x
x2
5
−1
= lim 1x
x→∞ 2 + 2
x x
−1
= lim
x→∞ + 2
x
−1
= 0
2

+
x

= −∞

√ √
7− x
d. (3 pts) lim
x→7 x−7
Ans:
√ √ √ √ √ √
7− x ( 5 − x)( 7 + x)
lim = lim √ √
x→7 x−7 x→7 (x − 7)( 7 + x)
7−x
= lim √ √
x→7 (x − 7)( 7 + x)
−1
= lim √ √
x→7 7+ x
−1
=√ √
7+ 7
1
=− √
2 7
Calculus Some Questions 3

√
2. (6 points) Use the first principles to find the derivative of y = 2x − 3

Ans:
f (x + h) − f (x)
f 0 (x) == lim
h→0 h
p √
2(x + h) − 3 − 2x − 3
= lim
h→0 h
p √ p √
( 2(x + h) − 3 − 2x − 3)( 2(x + h) − 3 + 2x − 3)
= lim p √
h→0 h( 2(x + h) − 3 + 2x − 3)
(2(x + h) − 3) − (2x − 3)
= lim p √
h→0 h( 2(x + h) − 3 + 2x − 3)
2x + 2h − 3 − 2x + 3
= lim p √
h→0 h( 2(x + h) − 3 + 2x − 3)
2h
= lim p √
h→0 h( 2(x + h) − 3 + 2x − 3)
2
= lim p √
h→0 2(x + h) − 3 + 2x − 3
2
=p √
2(x + 0) − 3 + 2x − 3
2
=√ √
2x − 3 + 2x − 3
1
=√
2x − 3
Calculus Some Questions 4

3. (28 points) Determine the derivative in simplified form

2
a. (3 pts) z = 9x3 + 2x + π 3 − x

Ans:

z = 9x3 + 2x + π 3 − 2x−1

z 0 = 27x2 + 2 + 0 − 2(−1)x−2
2
= 27x2 + 2 + 2
x

√
b. (3 pts) y = 3 3x2 + 1

Ans:
1
y 0 = 3( √ )6x
2 3x2 + 1

1
y 0 = 3( √ )3x
3x2 + 1
9x
y0 = √
3x2 + 1
Calculus Some Questions 5

Determine the derivative in simplified form

5−t3
c. (6 pts) r= 4t2 −3

Ans:

0 −3t2 (4t2 − 3) − 8t(5 − t3 )

r =
(4t2 − 3)2

−12t4 + 9t2 − 40t + 8t4

r0 =
(4t2 − 3)2
0 −4t + 9t2 − 40t
4
r =
(4t2 − 3)2
−t(4t3 − 9t + 40)
r0 =
(4t2 − 3)2

d. (5 pts) v = (3t2 − 4)5 (t + 2)4

Ans:

v 0 = 5(3t2 − 4)4 (6t)(t + 2)4 + (3t2 − 4)5 (4)(t + 2)3 (1)

v 0 = 2(3t2 − 4)4 (t + 2)3 [5(3t)(t + 2) + 2(3t2 − 4)]

v 0 = 2(3t2 − 4)4 (t + 2)3 [15t2 + 30t + 6t2 − 8]
v 0 = 2(3t2 − 4)4 (t + 2)3 (21t2 + 30t − 8)
Calculus Some Questions 6

Determine the derivative in simplified form

√
e. (5 pts) y = (2x3 + 7) 2x − 5

Ans:
√ 1
y 0 = 6x2 2x − 5 + (2x3 + 7) √ (2)
2 2x − 5

√ 1
y 0 = 6x2 2x − 5 + (2x3 + 7) √
2x − 5
2 3
6x (2x − 5) + (2x + 7)
y0 = √
2x − 5
12x − 30x2 + 2x3 + 7
3
y0 = √
2x − 5
14x − 30x2 + 5
3
y0 = √
2x − 5

3x+2 5


f. (6 pts) y= 2x3 −1

Ans:
 4  0
0 3x + 2 3x + 2
y =5
2x3 − 1 2x3 − 1

4 
3(2x3 − 1) − 6x2 (3x + 2)
 
0 3x + 2
y =5
2x3 − 1 (2x3 − 1)2
4  3
6x − 3 − 18x3 − 12x2
 
3x + 2
y0 =5
2x3 − 1 (2x3 − 1)2
5(3x + 2)4 −12x3 − 12x2 − 3
 
y0 =
(2x3 − 1)4 (2x3 − 1)2
−15(3x + 2)4 (4x3 + 4x2 + 1)
y0 =
(2x3 − 1)6
Calculus Some Questions 7

4. (8 points) For the curve defined by 4y 2 + x2 y 3 = −x + 2

dy
a. (6 pts) differentiate implicitly to determine dx

Ans:

4y 2 + x2 y 3 = −x + 2

8yy 0 + 2xy 3 + 3x2 y 2 y 0 = −1

8yy 0 + 3x2 y 2 y 0 = −1 − 2xy 3
y(8 + 3x2 y)y 0 = −(1 + 2xy 3 )
1 + 2xy 3
y0 = −
3y(8 + 3x2 y)

dy
b. (2 pts) determine the value of dx
at the point (2, −1)

Ans:
1 + 2xy 3
y0 = −
3y(8 + 3x2 y)

0 1 + 2(2)(−1)3
y (2, −1) = −
3(−1)(8 + 3(2)2 (−1))
−3
=−
12
1
=
4
Calculus Some Questions 8

5. (8 points) Given the curve y = (2 − 3x)3 .

a. (4 pts) Determine the value of the slope of this curve at the point ( 13 , 1)

Ans:

y = (2 − 3x)3

y 0 = 3(2 − 3x)2 (−3)

y 0 = −9(2 − 3x)2

1 1
y 0 ( ) = −9(2 − 3( ))2
3 3
2
= −9(2 − 1)
= −9

b. (4 pts) At what value(s) of x is this curve parallel to the line y + 9x − 11 = 0

Ans:

y + 9x − 11 = 0
y = −9x + 11

y20 = −9
then...
−9(2 − 3x)2 = −9
(2 − 3x)2 = 1
9x2 − 12x + 3 = 0

1
There are two solutions: x = 3
and x = 1
Calculus Some Questions 9

6. (6 points) For what value(s) of x is the slope of the curve y = x2

x−2
equal to -3

Ans:
x2
y=
x−2

2x(x − 2) − x2
y0 =
(x − 2)2
x2 − 4x
y0 =
(x − 2)2
then...
2
x − 4x
= −3
(x − 2)2
x2 − 4x = −3(x − 2)2
4x2 − 8x + 12 = 0
x2 − 2x + 3 = 0

There are two solutions: x = 1 and x = 3

Calculus Some Questions 10

7. (12 points) Evaluate each of the limits.

7x3 + 6x2 + 3x − 8
a. (3 pts) lim
x→∞ 8 − 21x3
Ans:
7x3 6x2
7x3 + 6x2 + 3x − 8 x3
+ x3
+ 3x
x3
− x83
lim = lim 3
x→∞ 8 − 21x3 x→∞ 8
x3
− 21x
x3
6
7+ x
+ x32 − x83
= lim 8
x→∞
x3
− 21
−7
=
21
1
=−
3

x2 − 9
b. (3 pts) lim
x→−3 4x2 + 11x − 3

Ans:
x2 − 9 (x − 3)(x + 3)
lim 2
= lim
x→−3 4x + 11x − 3 x→−3 (4x − 1)(x + 3)
x−3
= lim
x→−3 4x − 1
−6
=
−13
6
=
13
Calculus Some Questions 11

Evaluate each of the limits.

√ √
5− x
c. (3 pts) lim
x→5 x−5
Ans:
√ √ √ √ √ √
5− x ( 5 − x)( 5 + x)
lim = lim √ √
x→5 x−5 x→5 (x − 5)( 5 + x)
5−x
= lim √ √
x→5 (x − 5)( 5 + x)
−1
= lim √ √
x→5 5+ x
−1
=√ √
5+ 5
1
=− √
2 5

5x − x2
d. (3 pts) lim
x→∞ 1 − 2x

Ans:
5x 2
5x − x2 x2
− xx2
lim = lim 1
x→∞ 1 − 2x x→∞
x2
− 2x
x2
5
−1
= lim 1x
x→∞ 2 − 2
x x
−1
= lim
x→∞ − 2
x
−1
= 0
−
x2

= +∞
Calculus Some Questions 12

√
8. (6 points) Use the first principles to find the derivative of y = 3x − 2

Ans:
f (x + h) − f (x)
f 0 (x) == lim
h→0 h
p √
3(x + h) − 2 − 3x − 2
= lim
h→0 h
p √ p √
( 3(x + h) − 2 − 3x − 2)( 3(x + h) − 2 + 3x − 2)
= lim p √
h→0 h( 3(x + h) − 2 + 3x − 2)
(3(x + h) − 2) − (3x − 2)
= lim p √
h→0 h( 3(x + h) − 2 + 3x − 2)
3x + 3h − 2 − 3x + 2
= lim p √
h→0 h( 3(x + h) − 2 + 3x − 2)
3h
= lim p √
h→0 h( 3(x + h) − 2 + 3x − 2)
3
= lim p √
h→0 3(x + h) − 2 + 3x − 2
3
=p √
3(x + 0) − 2 + 3x − 2
3
=√ √
3x − 2 + 3x − 2
3
= √
2 3x − 2
Calculus Some Questions 13

9. (28 points) Determine the derivative in simplified form

3
a. (3 pts) z = 7x4 + 3x + π 4 − x

Ans:

z = 7x4 + 3x + π 4 − 3x−1

z 0 = 28x3 + 3 + 0 − 3(−1)x−2
3
= 28x3 + 3 + 2
x

√
b. (3 pts) y = 3 5x2 + 1

Ans:
1
y 0 = 3( √ )10x
2 5x2 + 1

1
y 0 = 3( √ )5x
5x2 + 1
15x
y0 = √
5x2 + 1
Calculus Some Questions 14

Determine the derivative in simplified form

5−t3
c. (6 pts) r= 3t2 −4

Ans:

0 −3t2 (3t2 − 4) − 6t(5 − t3 )

r =
(3t2 − 4)2

−9t4 + 12t2 − 30t + 6t4

r0 =
(3t2 − 4)2
0 −3t + 12t2 − 30t
4
r =
(3t2 − 4)2
−3t(t3 − 4t + 10)
r0 =
(3t2 − 4)2

d. (5 pts) v = (4t2 − 3)5 (t + 3)4

Ans:

v 0 = 5(4t2 − 3)4 (8t)(t + 3)4 + (4t2 − 3)5 (4)(t + 3)3 (1)

v 0 = 4(4t2 − 3)4 (t + 3)3 [5(2t)(t + 3) + (4t2 − 3)]

v 0 = 4(4t2 − 3)4 (t + 3)3 [10t2 + 30t + 4t2 − 3]
v 0 = 4(4t2 − 3)4 (t + 3)3 (14t2 + 30t − 3)
Calculus Some Questions 15

Determine the derivative in simplified form

√
e. (5 pts) y = (2x3 + 5) 2x − 7

Ans:
√ 1
y 0 = 6x2 2x − 7 + (2x3 + 5) √ (2)
2 2x − 7

√ 1
y 0 = 6x2 2x − 7 + (2x3 + 5) √
2x − 7
2 3
6x (2x − 7) + (2x + 5)
y0 = √
2x − 7
12x − 42x2 + 2x3 + 5
3
y0 = √
2x − 7
14x − 42x2 + 5
3
y0 = √
2x − 7

2x+3 5


f. (6 pts) y= 2x3 −1

Ans:
 4  0
0 2x + 3 2x + 3
y =5
2x3 − 1 2x3 − 1

4 
2(2x3 − 1) − 6x2 (2x + 3)
 
0 2x + 3
y =5
2x3 − 1 (2x3 − 1)2
4  3
4x − 2 − 12x3 − 18x2
 
2x + 3
y0 =5
2x3 − 1 (2x3 − 1)2
5(2x + 3)4 −8x3 − 18x2 − 2
 
y0 =
(2x3 − 1)4 (2x3 − 1)2
−10(2x + 3)4 (4x3 + 9x2 + 1)
y0 =
(2x3 − 1)6
Calculus Some Questions 16

10. (8 points) For the curve defined by 9y 2 + x2 y 3 = −x + 3

dy
a. (6 pts) differentiate implicitly to determine dx

Ans:

9y 2 + x2 y 3 = −x + 3

18yy 0 + 2xy 3 + 3x2 y 2 y 0 = −1

18yy 0 + 3x2 y 2 y 0 = −1 − 2xy 3
3y(6 + x2 y)y 0 = −(1 + 2xy 3 )
1 + 2xy 3
y0 = −
3y(6 + x2 y)

dy
b. (2 pts) determine the value of dx
at the point (3, −1)

Ans:
1 + 2xy 3
y0 = −
3y(6 + x2 y)

1 + 2(3)(−1)3
y 0 (3, −1) = −
3(−1)(6 + (3)2 (−1))
−5
=−
9
5
=
9
Calculus Some Questions 17

11. (8 points) Given the curve y = (2 − 5x)3 .

a. (4 pts) Determine the value of the slope of this curve at the point ( 51 , 1)

Ans:

y = (2 − 5x)3

y 0 = 3(2 − 5x)2 (−5)

y 0 = −15(2 − 5x)2

1 1
y 0 ( ) = −15(2 − 5( ))2
5 5
2
= −15(2 − 1)
= −15

b. (4 pts) At what value(s) of x is this curve parallel to the line y + 15x − 9 = 0

Ans:

y + 15x − 9 = 0
y = −15x + 9

y20 = −15
then...
−15(2 − 5x)2 = −15
(2 − 5x)2 = 1
25x2 − 20x + 3 = 0

1 3
There are two solutions: x = 5
and x = 5
Calculus Some Questions 18

12. (6 points) For what value(s) of x is the slope of the curve y = x2

x−3
equal to -8

Ans:
x2
y=
x−3

2x(x − 3) − x2
y0 =
(x − 3)2
x2 − 6x
y0 =
(x − 3)2
then...
2
x − 6x
= −8
(x − 3)2
x2 − 6x = −8(x − 3)2
9x2 − 54x + 72 = 0
x2 − 6x + 8 = 0

There are two solutions: x = 2 and x = 4

 FORMULA SHEET

• First principles
f (x + h) − f (x)
f 0 (x) = lim
h→0 h
• Derivative of a constant
dc
=0
dx
• Derivative of a power
dxn
= nxn−1
dx
• Derivative of the product of a constant and a differentiable function

d(cu) du
=c
dx dx

• Derivative of a sum
d(u + v) du dv
= +
dx dx dx
• Derivative of a product
d(uv) du dv
=v +u
dx dx dx
• Derivative of a quotient
d( uv ) v du
dx
dv
− u dx
=
dx v2
• Chain rule for a power of a differentiable function

d(un ) du
= nun−1
dx dx
13. (4 points) Consider the following function: y = 2x3 − 5x2 + 7x − 3.

a. (3 pts) Find its second derivative

b. (1 pt) At what value of x is the second derivative equal zero?

14. (4 points) Determine y 00 by implicit differentiation for 2xy − 3x2 = −7. Your final
result should be simplified and in terms of x and y.
15. (8 points) For the curve: y = x3 + 3x2 + 1.

a. (4 pts) Find the equation in standard form of the tangent line to the curve at
x = −3

b. (4 pts) Find the equation in standard form of the normal line to the curve at
x = −3

16. (7 points) The x- and y-coordinates (in meters) of a moving object are given by
x = −5t and y = 3 + 11t − 5t2 , where t is time in seconds. Find the magnitude and
direction (angle) of the instantaneous velocity and acceleration of the object at t = 2 sec.
Round your answers to one decimal place.
17. (5 points) A circular puddle of water shrinks by evaporation so that its area
decreases at 6000 cm2 /h. How fast is its radius decreasing when its length is 90 cm.
Leave your answer simplified in exact form.

18. (5 points) Oil is leaking out of the bottom of an inverted conical tank. The height
of the cone is 2.30 m and the radius at the top of the cone is 1.15 m. If the level of oil in
the tank is dropping at a rate of 1.5 m/min, at what rate is the volume decreasing when
the oil level is 1.50 m deep?
19. (5 points) A ship starting at 11:00 AM travels east at 30 km/h, while another
ship starting from the same port and at the same time travels south at 40 km/h. How
fast are they separating at 1:00 PM?
20. (5 points) For the function y = x + 12 x2 − 23 x3 find any maximum or min points
by using the appropriate derivative test(s).

x2 − 2x − 3
21. (7 points) Consider the following: y =
3x2 − 48
a. (2 pts) State the x- and y- intercept (if any):

b. (3 pts) State the vertical asymptote(s) (if any):

c. (2 pts) State the horizontal asymptote(s) (if any):

22. (12 points) For the function y = −x3 − 3x2 + 4x − 12 determine.

a. (2 pts) State the x- and y- intercept (if any):

b. (3 pts) intervals of increase/decrease

c. (2 pts) local max and min points

d. (2 pts) intervals of concavity and points of inflection

 e. (3 pts) Sketch the graph on the graphing paper provided and label all critical
points and points of inflection
23. (4 points) Consider the following function: y = 2x3 − 7x2 + 5x − 3.

a. (3 pts) Find its second derivative

b. (1 pt) At what value of x is the second derivative equal zero?

24. (4 points) Determine y 00 by implicit differentiation for 3xy − 2x2 = −7. Your final
result should be simplified and in terms of x and y.
25. (8 points) For the curve: y = x3 + 3x2 + 2.

a. (4 pts) Find the equation in standard form of the tangent line to the curve at
x = −3

b. (4 pts) Find the equation in standard form of the normal line to the curve at
x = −3

26. (7 points) The x- and y-coordinates (in meters) of a moving object are given
by x = −3t and y = 3 + 9t − 5t2 , where t is time in seconds. Find the magnitude and
direction (angle) of the instantaneous velocity and acceleration of the object at t = 2 sec.
Round your answers to one decimal place.
27. (5 points) A circular puddle of water shrinks by evaporation so that its area
decreases at 5000 cm2 /h. How fast is its radius decreasing when its length is 80 cm.
Leave your answer simplified in exact form.

28. (5 points) Oil is leaking out of the bottom of an inverted conical tank. The height
of the cone is 2.10 m and the radius at the top of the cone is 0.70 m. If the level of oil in
the tank is dropping at a rate of 1.4 m/min, at what rate is the volume decreasing when
the oil level is 1.40 m deep?
29. (5 points) A ship starting at 11:00 AM travels east at 40 km/h, while another
ship starting from the same port and at the same time travels south at 30 km/h. How
fast are they separating at 2:00 PM?
30. (5 points) For the function y = x + 12 x2 − 23 x3 find any maximum or min points
by using the appropriate derivative test(s).

x2 − 7x + 12
31. (7 points) Consider the following: y =
5x2 − 45
a. (2 pts) State the x- and y- intercept (if any):

b. (3 pts) State the vertical asymptote(s) (if any):

c. (2 pts) State the horizontal asymptote(s) (if any):

32. (12 points) For the function y = x3 − 3x2 − 4x + 12 determine.

a. (2 pts) State the x- and y- intercept (if any):

b. (3 pts) intervals of increase/decrease

c. (2 pts) local max and min points

d. (2 pts) intervals of concavity and points of inflection

 e. (3 pts) Sketch the graph on the graphing paper provided and label all critical
points and points of inflection
33. (4 points) A spherical satellite of a planet, with radius 800 km, becomes coated
with ice to a depth of 23.4 m. Approximate the total volume of the ice to two significant
digits

d2 y
34. (5 points) Determine 2 for tan 3x.
dx
35. (6 points) If a box with open top and square base is to be made from 6400 cm2
of material. Determine the largest possible volume of the box.
36. (6 points) A zookeeper with 1200 m of fencing wants to enclose a rectangular area
and then divide it into 5 pens with fences parallel to one side of the rectangle. What is
the largest possible total area of the five pens.
37. (6 points) A cylindrical can is to be made to hold 2.5 l of paint. Determine
(correct to two decimal places) the height of the can that will minimize the cost of the
material required
38. (27 points) Find the derivative for each of the following. Write your answer in
simplified fully factored form without negative exponents.

a. (3 pts) y = sin(5x3 − 4x)

2
b. (3 pts) y = ex − e3

c. (3 pts) f (x) = 55x

Find the derivative for each of the following. Write your answer in simplified fully factored
form without negative exponents.

d. (4 pts) y = cos x2 sin x2

e. (3 pts) y = tan(sin 2x)

x
f. (4 pts) ln y =
y
Find the derivative for each of the following. Write your answer in simplified fully factored
form without negative exponents.

g. (3 pts) y = sec(ln 3x)

2
h. (4 pts) y = x(x )

39. (4 points) Find the slope of the tangent for the function y = xex at x = ln 2.
(Leave your answer in exact form).
40. (7 points) A circle has an equation: (x + 5)2 + (y + 1)2 = 18 Determine:

a. (3 pts) the centre and radius of the circle

b. (4 pts) the equation in general form of the tangent line at the point (-8, 2)

41. (4 points) Determine the centre and radius of the circle x2 + y 2 − 8x + 10y − 26 = 0

42. (3 points) For the parabola given by the following equation, determine the
coordinates of the focus and the equation of the directrix. x2 = y
43. (6 points) Evaluate each of the limits.

3x2 + 5x − 12
a. (3 pts) lim
x→−3 2x2 − 18

4 − 21x2
b. (3 pts) lim
x→∞ 7x3 + 6x2 + 3x − 4
44. (22 points) Determine the derivative in simplified form

a. (3 pts) y = −2x4 + 2x3 + x2 − x + 6

√
b. (3 pts) y = 3 x2 − 5
Determine the derivative in simplified form

5 − t2
 
c. (4 pts) y = ln
4t2 − 3

d. (4 pts) y = 4 tan 2x sec2 2x

Determine the derivative in simplified form
√
e. (4 pts) y = etan x

2
f. (4 pts) y = 24x sin x
45. (11 points) differentiate implicitly to find the indicated derivative

dy
a. (3 pts) 3y = 2e3yt find:
dt

dr
b. (3 pts) r2 + 2t = 3t2 + sin(2r + t2 ) find:
dt

d2 w
c. (5 pts) u − cos w = w find:
du2
46. (8 points) The motion of an object (in meters) is given by the parametric equations

x = t3 − t and y = −2t3

a. (4 pts) Find the magnitude and direction of the instantaneous velocity at

t=2s

b. (4 pts) Find the magnitude and direction of the instantaneous acceleration at

t=2s

47. (4 points) Find the equation of the normal line for the curve y = 2 sin x at x = π
6
48. (5 points) A cylindrical water tank is being filled at a rate of 42.0 m3 /min. If the
radius of the tank is 4.2 m how fast is the depth of the water increasing?
49. (5 points) A box-shaped wire frame consists of two identical wire squares whose
corners are connected by four straight wires of equal length. If the frame is to be made
from a wire of 600 cm, what should the dimensions be to obtain a box of greatest volume?
50. (4 points) Identify each of the following equations below as representing a conic:
a circle, an ellipse, a parabola or a hyperbola.

a. (1 pt) 2x2 + 7x − 2 = 3y − 2x2

b. (1 pt) 2x2 + 7x − 2 = 3y

c. (1 pt) 2x2 + 7x = 2 + 3y − 5x2

d. (1 pt) 2x2 + 7x + xy = 2 + 3y + 5x2

51. (4 points) Determine the vertex, the focus and the equation of the directrix of the
parabola: (x + 2)2 = 6y − 12
 (x − 3)2 (y − 3)2
52. (7 points) For the ellipse
9
+
25
= 1, determine:

a. (1 pt) coordinates of the centre

b. (4 pts) coordinates of the foci

c. (2 pts) Sketch the graph

 (x − 3)2 (y − 3)2
53. (8 points) For the hyperbola
16
−
25
= 1, determine:

a. (1 pt) coordinates of the centre

b. (2 pts) coordinates of the vertices

c. (2 pts) equations of the asymptotes

d. (3 pts) Sketch the graph of the hyperbola and its asymptotes