Calculus 12 pg.1

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Calculus 12 pg.2

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LG 16 – 17 Worksheet Package Part A: 1. Solve each of the following equations exactly: a) ln x + 3 = 0 b) 2 ln x – 9 = 0 c) (ln x)2 – 4 = 0 d) (ln x)2 + (ln x) – 2 = 0 e) 2 ln x = ln 16 f) 2 ln x = ln(4x + 5)

Calculus 12 pg.3

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Part B: 1. Simplify: a) ln (e-2) b) ln (e-2) + ln (e3) c) e(2 ln 3) d) e(-2 ln 3)

e) e^-ln( 1x

) f) ln( 8e3)

2. Solve each of the following equations exactly: a) e2x + ex – 6 = 0 b) e2x – 2ex = 8 c) ex + 4e-x = 5 d) ex = 6e-x + 1

Calculus 12 pg.4

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3. Differentiate each function: a) y = (ln x)2 + ln (x2) b) y = (x ln x)2

c) y = 2e x2−x( ) d) y = 2xe(2x)

e) y = ln (π + e(2x) ) f) y = ex

ln x

g) y = ln ( sin y) + x2 h) y = e(2y) + xy

Calculus 12 pg.5

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4. Find the equation of the tangent line and of the normal line to the curve y = ln x at the point (e, 1). 5. Find the equation of the tangent line and of the normal line to the curve y = e at the point (2, e2).

6. Find dydx

at the given point:

a) ln y – x = 0 at (1, e) b) x ln y + xy = 2 at (2, 1)

Calculus 12 pg.6

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Part C: 1. Find the equation of the tangent line and of the normal line to the curve y = sin x at the point (0, 0).

2. Find the equation of the tangent line and of the normal line to the curve

y = cos x at the point π3, 12

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3. If tan(xy) = x, find dydx

at the point 1, #$

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Calculus 12 pg.7

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Part D:

1. Find the tangent approximation to the function f (x) = x at x = 100 and use it to approximate the square root of 102. 2. Use Newton’s Method to solve the equation x2 – 2 = 0 using x-initial = 1

Calculus 12 pg.8

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Part E: 1. An open field is bounded by a lake with a straight shoreline. A rectangular enclosure is to be constructed by using 500 m of fencing along three sides and the lake as a natural boundary on the fourth side. What dimensions will maximize the enclosed area and what is the maximum area? 2. Two farmers have 800 m of fencing. They wish to form a rectangular enclosure and then divide it into 3 sections by running two lengths of fence parallel to one side. What should the dimensions of the enclosure be in order to maximize the enclosed area? 3. A piece of wire 24 in long is used to form a square and/or a rectangle whose length is three times its width. Determine their minimum combined area.

Calculus 12 pg.9

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4. Find the dimensions of the rectangle of largest area whose base in on the x-axis and whose upper two vertices lie on the parabola y = 12 – x2. What is the maximum area? 5. An open box by cutting squares of equal size from the corner of a 24 cm by 15 cm piece of sheet metal and folding up the sides. Determine the size of the cutout that maximizes the volume of the box. 6. An open box from a 12 in by 12 in piece of cardboard by cutting away squares of equal size from the other four corners and folding up the sides. Determine the size of the cutout that maximizes the volume of the box.

Calculus 12 pg.10

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7. A rectangular poster which is to contain 50 cm2 of print, must have margins of 2 cm on each side and 4 cm on the top and bottom. What dimensions will minimize the amount of material used? 8. Construct a closed rectangular box with square base which has a surface area of 150 cm2. What is the maximum possible volume of such a box?

9. An open rectangular box with a base twice as long as it is wide. If its volume must be 972 cm3, what dimensions will minimize the amount of material used in its construction?

Calculus 12 pg.11

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Part F: 1. The height, in feet, at any time t, in seconds, of a projectile thrown vertically is given by the formula:

h(t) = -16t2 + 256t a) how fast is the projectile traveling 10 s after it is thrown and how high is it? b) when is the maximum height reached by this projectile and what is this height? c) when does the projectile return to the ground and with what velocity?

Calculus 12 pg.12

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2. The height, in feet, after t seconds of travel of a projectile thrown upward from the top of a 96 ft building is given by the formula:

h(t) = 80t – 16t2 + 96 a) what is the projectile’s initial velocity? b) what is the projectile’s velocity after 5 s and what is its position? c) what is the maximum height reached by the projectile? d) when does the projectile hit the ground and with what velocity? e) what is the acceleration of this projectile at 8 s? f) at what time is the projectile’s velocity 10 ft/sec? g) at what times is the projectile’s speed 10 ft/sec?

Calculus 12 pg.13

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Part G: 1. A ladder 10 ft long is resting against the side of a building. If the foot of the ladder slips away from the wall at the rate of 2 ft/min how is the angle between the ladder and the building changing when the foot of the ladder is 6 ft away from the building? 2. A TV camera is located 5000 m from the base of a rocket launching pad. The camera is designed to follow the vertical path of the rocket. If the rocket’s speed is 500 m/s at the moment when it has risen 2000 m, how fast is the camera’s angle of elevation changing at this instant?

Calculus 12 pg.14

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LG 16 TAKE HOME ASSIGNMENT* All working must be shown, as applicable to obtain full marks. All parts of questions are worth 2 marks each unless otherwise shown. Do not simplify your derivatives! HAND IN BEFORE MOVING ON TO PART H! 1) Solve each of the following equations exactly: a) 4 ln x – 12 = 0 b) 2 ln x = ln(x + 6) 2) Simplify: a) ln(e-4 ) + ln(e 6 ) b) e -2ln4 3) Differentiate each of the following functions: a) y = ln(x3) + (lnx)3 b) y = 4xe4x + 4e4x – 4e4

c) f (x) = ln xex

d) y = xy+ e4y

Calculus 12 pg.15

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4) Find the equation of the tangent line to the curve y = 2 ln x at the point (e, 2).

5) Given x – ln y = 0, find dydx

at the point (1, e).

6) Find the equation of the normal line to the curve y = cos x at the point ( π2

, 0 )

7) If x = tan(xy), find dydx

at the point 1, π4

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Calculus 12 pg.16

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8) Find the tangent approximation to the function f (x) = x at x = 64 and use it to approximate the square root of 66 9) An open field is bounded by a lake with a straight shoreline. A rectangular enclosure is to be constructed using 600 m of fencing along three sides and the lake as a natural boundary on the fourth side. What dimensions will maximize the enclosed area and what is the maximum area? Solve this problem using Calculus. 10) Find the dimensions of the rectangle of largest area whose base is on the x-axis and whose upper two vertices lie on the parabola y = 24 – 2x2. Solve this problem using Calculus.

Calculus 12 pg.17

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11) Pete and Repete create an open box by cutting squares of equal size from the corner of a 48 cm by 30 cm piece of tin and folding up the sides. Determine the size of the cutout that maximizes the volume of the box. Solve this problem using Calculus 12) A ladder 13 ft long is resting against the side of a building. If the foot of the ladder slips away from the wall at the rate of 3 ft/min how is the angle between the ladder and the building changing when the foot of the ladder is 5 ft away from the building? Solve this problem using Calculus. 13) The height, in feet, after t seconds of travel of a projectile thrown upward from the top of a 192 ft building is given by the equation: h(t) = 192 + 160t -32t2 a) what is the projectile’s initial velocity? b) what is the projectile’s position after 4 s? c) what is the maximum height reached by the projectile? d) when does the projectile hit the ground and with what velocity?

Calculus 12 pg.18

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Part H: 1. Evaluate each of the following IMPROPER INTEGRALS if possible:

a)   1x21

∞

∫  dx b)  2ex−∞

0

∫  dx

c)   1x0

1

∫  dx d)  1x2

∞

∫  dx

Part I: 1, Show how the MEAN VALUE THEOREM applies to the given function over the given interval: a) f(x) = x2, 2 < x < 6 b) g(x) = x2 + 3x + 5, 1 < x < 4

Calculus 12 pg.19

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2. Show how the INTERMEDIATE VALUE THEOREM applies to the given function over the given interval: a) f (x) = x2, 2 < x < 8 and f(c) = 36 3. Use the given formula to find the AVERAGE VALUE of the given function over the given interval:

f (c) = 1(b− a)

  f (x)dxa

b

∫

a) f(x) = x2, 2 < x < 4 Part J: 1. Find the exact arc length of the given function between the given values of x:

a) f(x)= 2x, from x = 4 to x = 8 b) g(x) = 23x32 , from x = 1 to x = 3

Calculus 12 pg.20

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Part K: 1. Given f is a function of x and y find the partial derivative of f with respect to x and the partial derivative of f with respect to y of each of the following: a) f(x, y) = 4x3 – 3x2y2 b) f(x, y) = x4 ln y + y Part L: 1. Find the exact value of each of the following multiple integrals:

a) x  dxdy2

4

∫0

1

∫ b) 2x  dxdy−1

5

∫b

a

∫

c) y2  dydx−1

1

∫1

3

∫ d) 2x + 2y dydx−2

2

∫−1

1

∫

Calculus 12 pg.21

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Part M: 1. Find all solutions of each DIFFERENTIAL EQUATION below:

a) dydx

= 2x +3 b) dydt= 4t2 + 5 c)

dvdt= 3cos2t − 4

2. Find the solution of each of the DIFFERENTIAL EQUATION satisfying the given initial condition:

a) dydx

= 3x2 + 6,   y(1) = 9

b) dydt= 3− 2t( )5 ,   y(2) =1

Calculus 12 pg.22

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Part N: 1. Simplify: a)   −9                   b)   −64                   c)   −20                   d)   −80                   e)   −2

f)  i  5                        g) i 6                        h)  i  83                       i)  i  202                        j)  i 484

2. Find the ABSOLUTE VALUE of each of the following complex numbers: a)  4+ 2i  b)  6−3i 

3. Simplify: a) (2 + 6i) + (4 – 8i) b) (3 – 5i) – (6 – 9i) c) (2 + 5i)(3 – 4i)

d) (2 + 5i)(2 – 5i) e) 65i

h) 4 – 6i i) 8 – 2i 2 + i 8 + 2i

Calculus 12 pg.23

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4. Solve each of the following equations over the COMPLEX NUMBERS: a) x2 + 9 = 0 b) 3x2 + 10 = 2x2 + 8 c) x2 + 2x + 2 = 0 d) 2x2 = 2x – 3 5. Find the following integrals over the COMPLEX NUMBERS:

a)  2x  dx−2i

4i

∫ b)  4x3−2i

i

∫  dx

Calculus 12 pg.24

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Part 0: 1. Give the relationship between the POLAR COORDINATES (r, θ ) and the RECTANGULAR COORDINATES (x, y): 2. Give the rectangular coordinates of the point whose polar coordinates are given:

a) 2, π6

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3. Give the polar coordinates of a point with the given rectangular coordinates:

a) (1, 0) b) (1, 1) c) 3, 1( )

Calculus 12 pg.25

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4. Find a rectangular equation equivalent to the given polar equation and describe the graph: a) r = 2 b) r = a

c) θ =π4

d) r = 2sinθ

e) r = 4cosθ f) r = tanθ secθ 5. Change the given rectangular equation into an equivalent polar equation: a) x2 + y2 = 16 b) x = 3

c) 2x2 + 2y2 = 8 d) y = 3x e) x + 2y = 3 f) x2 – y2 = 1

Calculus 12 pg.26

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LG 17 TAKE HOME ASSIGNMENT* Mark _________________ Name _________________ Date _________________ Student # ______________ All working must be shown, as applicable to obtain full marks. All parts of questions are worth 2 marks each unless otherwise shown. Do not simplify your derivatives! 1) Evaluate each of the following improper integrals if possible:

a)   1x22

∞

∫ b)  4ex−∞

2

∫  dx

2) Show how the Mean Value Theorem applies to f(x) = 2x2 over the interval 2 < x < 8

3) Find the exact arc length of f (x) = 23x32 between x = 2 and x = 6

Calculus 12 pg.27

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4) Given f is a function of x and y find the partial derivative of f with respect to x and the partial derivative of f with respect to y:

f (x) = 6x4 − 2x5y+ x ln y

with respect to x with respect to y

5) Find the exact value of each of the following multiple integrals:

a) 4x  dxdy−2

3

∫a

b

∫ b) 2x  dydx−1

3

∫2

π

∫

6) Find all the solutions of the differential equation

7) Find the solution of the differential equation dydx

= 6x3 − 5 satisfying the

initial condition y(2) = 14

Calculus 12 pg.28

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8) Simplify: a)   −24                               b)   −6                             c)  i  65                             d)  i  404 9) Simplify: a) (4 – 6i) – (8 – 10i) b) (2 + 6i)(4 – 4i) c) 4 + 2i 4 – 2i 10) Solve exactly over the complex numbers: 4x2 = 2x −1

11) Integrate over the complex numbers: 4x  dx−2i

4i

∫

Calculus 12 pg.29

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12) If y = 3x2 − 2 and x = 2 and dx = 0.4 find the differential dy

13) Give the rectangular coordinates of the point whose polar coordinates are given: 2, π3

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14) Change the given rectangular equation into an equivalent polar equation: 2x2 + 2y2 = 50