CALCULUS REVIEW Limits 1. Evaluate each limit, if it exists.

2. Use the graphs to evaluate the following limits.

iv) f(2) 3. Find the derivative using only first principles.

Answers: 1a) 5 b) 1/5 c) 4 d) DNE e) -1/25 f) -1/2 g) DNE 2a)i) ii) ∞ iii) DNE iv) DNE b)i) 2 ii)-1 iii)DNE iv)-1

3a) 4x+1 b) c)

Derivatives 1. 2. Differentiate. Simplify.

3. Find at the indicated x value.

4. Find .

5. Find the point(s) at which the tangent to the curve is horizontal.

6. Find the equation of the tangent to the curve at the point (1,3).

7. What is the equation of the normal to the graph of when x= -2.

8. Find the point(s) at which the tangent to the curve equals 1.

9. At what point is the tangent to the curve parallel to 2x+2y-5=0 ? Answers:

3a) 3/25 b) -124 c) -2

5. 6. 26x-y-23=0 7. 8. (-1,5), (3,-11)

9. All points where x=y

Applications of Derivatives 1. The position at t seconds of a particle moving along a straight line is given by

, where s is measured in metres. a) Determine the acceleration after 6 seconds. b) When does the particle change directions? c) When is the velocity increasing and when is it decreasing?

2. A ball is thrown up and its motion can be described by where the height is measured in metres and the time in seconds.

a) Find the initial velocity. b) When does the ball reach its maximum height? c) When does the ball hit the ground? d) What is its velocity when it hits the ground? e) What is the acceleration of the ball at 4s? 3. Two cars approach an intersection. At a certain time, each car is 5km from the intersection. One car travels west at 90 km/h and the other travels south at 80 km/h. a) What is the distance between the cars at this time? b) How fast is the distance between the cars decreasing after 2 min?

4. A train leaves at 4:00 and travels due west at 120 km/h. Another train travelling due north at 90 km/h arrives at the same train station at 5:00. When are the trains the closest together?

5. You have 1000m of fencing for a garden. You plan on having one side against the house so you

need to fence only 3 sides. Find the dimensions to maximize the area of the garden. 6. A cone of height 20cm and radius 8cm is constructed. Find the dimensions of the largest cylinder

that can fit inside the cone.

7. The dealership buys cars for $15 000. When the dealer sells each car for $25000, she sells 24

cars per month. For each reduction of $600 in selling price, the dealer sells 2 more cars per

month. Determine the number of cars sold in one month to maximize profit.

8. A box with no lid is to be made from a 40cm by 60cm sheet of metal. What are the dimensions to maximize volume?

9. A pipeline is to be constructed from an island that is 4km from the shore to a town that is 10km

north of the island, along the shoreline. Find the minimum cost if it cost $6000/km to lay pipe in the water and $4000/km to lay pipe on land.

Answers: 1.a) 27m/s2 b) 3, 6 c) incr t>4.5, decr t<4.5 2. a) 6m/s b) 0.6s c) 1.5s d)-8.7m/s e) -9.8m/s2 3. a) 7.1 km b) approx 120 km/h 4. 0.36 hours 5. 250m by 500m 6. r=16/3 cm, h=20/3 cm 7. 29 8. 7.8cm by 24.4cm by 44.4cm 9. 3.58 km from perpendicular to shoreline or 6.42km from town

Curve Sketching 1. Find the maximum and minimum values and classify them.

2. Find the intervals on which each curve is concave up.

3. Find the vertical, horizontal, and oblique asymptotes, if possible.

4. Sketch each function using asymptotes, crossover points, critical points, points of inflection and intercepts.

5. Given the graph of f’(x) below. 1. What are the x-coordinates of the local extrema for f(x)? 2. For what values of x is f(x) increasing?

3. What are the x-coordinates of the point of inflection of f(x)? 4. For what values of x is f(x) concave down? Answers:

1. f(0)=5 max, f(2)=f(-2)=-11 min 2.a) x>-4/5 b)

3a) v.a. x=3,x=- ½ h.a. y=3 b) v.a. x= -1 o.a. y=6x-6 4a) b) c) d) d) 5.a) x= -2,3,5 b) -2<x<3, x>5 c) x=0,4 d) 0<x<4

Derivatives of Exponential and Logarithmic Functions

1. Find the derivative and simplify. 2. Use implicit differentiation to find the derivative of a) x+exy =y b) lny+2x=3y 3. Find the equation of the tangent to the curve y=2-xex at the point where x=0. 4. Find the equation of the tangent to y= xlnx and parallel to 6x-2y+7=0

5. Find the possible maximum and/or minimum point of .

6. The number of bacteria, N, in a culture at time, t, hours, is modelled by the equation

. Determine the maximum number of bacteria in the culture between t=0 and t=40 h.

Answers:

3. y= -x+2 4. y=3x-e2 5. (e, 1/e) 6. approx 536 790 000

Final Exam Review on Vectors Page 1 1. An airplane flies 400 km/h [NW] and is blown off course by a wind blowing

120 km/h [N E]. Find the resultant velocity of the plane relative to the ground.

2. is a parallelogram where and . The diagonals intersect at and is the midpoint of . Express the following in terms of and :

(a) (b) 3. From the given diagram, with and , determine the following without the use of

a calculator.

(a) (b) (c) 4. Given the diagram below, determine the following: (a) (b)

5. An object of mass 25 kg is suspended from the ceiling by

two ropes that make angles of and with the ceiling. Find the tension in each of the two ropes.

6. A lawn roller with a mass of 150 kg is pushed with a force of 320 N along its handle. The handle makes an angle of with the ground.

(a) Find the vertical and horizontal components of the force. (b) Find the magnitude of the force exerted by the moving roller on the ground. Answers 1. 366 km/h .

2. (a) (b)

3. (a) (b) (c)

4. (a) (b) 5. The tension in the rope from the ceiling is 232 N and the tension in the rope from the

ceiling is 256 N. 6. (a) The vertical component of the force is 160 N and the horizontal component

of the force is 277 N. (b) The force exerted by the roller on the ground is 1630 N.

Final Exam Review on Vectors Page 2

1. If is a unit vector directed into the page, evaluate and without the use of a calculator when =10 and =8.

(a) (b) 2. Find the exact area of the following shape.

3. Use resolution of vectors to find the resultant of the following forces (round the magnitude to

one decimal place and the angle to the nearest degree). is 130 N [ N E ]

is 80 N [ N W ]

is 50 N [ N W ] 4. A uniform metre stick is balanced at the 50 cm mark. The stick is in equilibrium under the

actions of the forces in the following diagram. Find .

5. A mathlete in training (mass 75 kg) carries a 1 kg mathketball up a 6 m ladder inclined to the vertical. Find the work done (rounded to the nearest Joule) by the mathlete in training.

Answers 1. (a) , (b) ,

2. 220 square units 3. 52.8 N [N E] 4. = 66 cm 5. 3 661 J

Final Exam Review on Vectors Page 3 1. A vector goes from to . Determine

(a) the position vector equal to ,

(b)

(c) the direction of (correct to the nearest degree).

2. Given and , find a vector of magnitude 3 opposite in direction to .

3. Given and , find the coordinates of when .

4. Given the standard basis vectors in , , , and , and given and

, determine

(a)

5. Evaluate .

6. Determine the coordinates of so that is a parallelogram, where ,

, and .

7. Given , , and , determine the measurement of to the nearest degree.

8. Given and , find .

9. Determine the direction angles of correct to the nearest degree.

10. Calculate the area of (rounded to one decimal place) with vertices ,

, and . 11. Given the parallelepiped below, find the coordinates of . Answers 1. (a)

(b) (c)

2.

3.

4. (a)

5.

6. 7. 8.

9. , , 10. 11.

Final Exam Review on Vectors Page 4

1. Determine the vector, parametric, and Cartesian equations of the line passing through the points and .

2. Calculate the distance (to one decimal place) between the lines and

.

3. Find the Cartesian equation of the line through that is parallel to

. 4. 5. 6. Determine the distance between the following: and .

7. Determine the distance between the following: and .

8. Determine the symmetric equations of the line passing through and parallel to

.

9. For the line , determine a point on the line and a direction vector for

the line. 10. Determine the point of intersection of the following lines:

and .

11. Determine the point of intersection of with

(a) (b) Answers 1. A vector equation is .

Parametric equations are .

The Cartesian equation is . 2. 3. 6. 4 7. 0

8.

9. ,

10.

11. (a) (b) None

Extra Vectors Exam Review

1. Evaluate .

2. Express as a linear combination of and where =(-1,3), =(2,-4) and =(3,-3).

3. A mass of 9kg is suspended from a ceiling by two cords that make angles of 40º and 50º with the ceiling. Find the tension in each cord.

4. An airplane has a velocity, relative to the air, of 150 km/h in a direction N30ºW. If the wind is blowing at

20 km/h from a direction S50ºE, determine the velocity of the plane relative to the ground to the nearest km/h and to the nearest degree.

5. a) Determine the distance from the point (2,-1,4) to the plane .

b) Find the distance between the planes defined by and .

c) Find the distance from the point (-3,0,1) to the line .

6. Given

a) Find

b) Find

7. Find parametric equations of the line through the points (-3, -2, 1) and (5, 4, -6).

8. Find the point of intersection of the line and the plane .

9. Find the scalar equation of the plane through the points (6, 5, -4), (8, 9, -3), (7, 10, -2). 10. If , then determine the value of a:

11. Find parametric equations of the plane through the points A(2,-1,1), B(4,1,5), C(1,2,2). Find the value of k if point (0, k, -3) is in the plane.

12. Solve the following using matrices . x - 5y + 2z = 27 3x + 2y - z = -5 4x - 3y + 5z = 42 Solutions

1. -8 2. 3. 67.6N 56.7N 4. 169km/h N32°W 5. a) b) c)

6. a) b)21 7. x=-3+8t, y=-2+6t, z=1-7t 8. (12, 1, 7) 9. x-y+2z+7=0 10. a=2 11. Л=(1,2,2)+s(1,1,2)+t(-1,3,1) k=-3 12. (2,-3,5)