Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals and radical epressions, limits of functions and limits of geometric sequences. General Outcomes: Perform operations on polynomials. Use eact values, arithmetic operations and algebraic operations on real numbers to solve problems. Solve coordinate geometry problems involving lines and line segments. Give eamples of the differences between intuitive and rigorous proofs in the contet of limits. Epress final algebraic and answers in a variety of equivalent forms, with the form chosen to be the most suitable form for the task at hand. Specific Outcomes Perform operations on irrational numbers of monomial and binomial form, using eact values. Factor polynomial epressions of the form a + b + c, a b y, difference and sum of squares. Determine the equation of a line, given information that uniquely determines the line Solve problems using slopes of: o parallel lines o perpendicular lines. Eplain the concept of a limit Solve of functions with limits, with left-hand or right-hand limits, or with no limit Solve bounded and unbounded functions, and of bounded functions with no limit Eplain, and give eamples of, continuous and discontinuous functions Define the limit of an infinite sequence and an infinite series Eplain and illustrate the limit theorems for sum, difference, multiple, product, quotient and power Solve limits using intuitive and rigorous proofs in the contet of limits Find limits of functions and sequences, both at finite and infinite values of the independent variable Compute limits of functions, using definitions, limit theorems and calculator/ computer methods. Determine the limit of any algebraic function as the independent variable approaches finite or infinite values for continuous and discontinuous functions Calculate the sum of an infinite convergent geometric series Use definitions and limit theorems to determine the limit of any algebraic function as the independent variable approaches a fied value Using definitions and limit theorems to determine the limit of any algebraic function as the independent variable approaches ±. This unit is worth 10% of your mark in Mathematics 31.
The time line is: Lesson 1: Factoring and Rationalizing Review Lesson : Factoring More Comple Epressions Lesson 3: Linear Functions Lesson 4: The Tangent Problem Lesson 5: Solving Limits Using Intuitive Reasoning Lesson 6: The Limit theorems Lesson 7: Finding the Limits of Functions by Factoring Lesson 8: Finding the Limits of Rational and Radical Functions Lesson 9: One Sided Limits Lesson 10: Discontinuities Lesson 11: Using Limits to Find Tangents Lesson 1: Velocity as a Rate of Change Lesson 13: Infinite Sequences Pre-calculus and Limits Page of 37
Outcomes: Lesson 1 Factoring and Rationalizing Review Review skills factoring. Review: 1. Factor a) + 7 + 10 b) 5y 40y + 60 3 4 c) m 18m 0m d) a + 18a + 36. Factor a) + 3 b) 15a + 7a c) 1 15 18 d) 3y + 13y 10 e) 6a + 17ab + 1b 4 3. Factor a) 5 16 b) 4 5m c) 7y 50 d) 16n 4 e) 16 1 f ) 1 p 4 6 Pre-calculus and Limits Page 3 of 37
4. Rationalize the following denominator. 7 a) b) 7 5 50 3 c) d) 5 7 3 + + 3 4. Rationalize the following numerator. 8 7 a) b) c) 81 + Homework: Page 3 #1,a,g,h, Page 4 # 1ab,ab Pre-calculus and Limits Page 4 of 37
Lesson Factoring More Comple Epressions Outcomes: Factor epressions with fractional eponents and a sum and difference of cubes. Warm up: Multiply the following epressions a) ( a b)( a + ab + b ) b) ( a + b)( a ab + b ) From the above one can deduce the following formulas. Factoring a Difference and Sum of Cubes. 3 3 a b = ( a b)( a + ab + b ) 3 3 a + b = ( a + b)( a ab + b ) 1. Use the formula for sum and difference of cubes to factor the following. 3 a) 8 15 b) 7 6 8y 3 3 c) 7 + 64 d) 15 + 64y 9 6 Pre-calculus and Limits Page 5 of 37
In pure mathematics 0 you worked with the factor theorem. Recall The Factor Theorem If polynomial P() is divided by b, the remainder P( b ) = 0. Then b is a factor of the polynomial. Factor the following epressions completely. 3 a) 5 + 6 b) 3 3 4 5 + Factoring Epressions With Fractional Eponents When factoring epressions with fractional eponents factor out the smallest eponent. You will have to recall the eponent laws. 3. Factor the following epressions. a) 3 1 4 1 + 6 3 3 3 b) 15a 8 + 7a 5 a Homework: Page 3 #1, b,c,d,e,f, 3,4 Pre-calculus and Limits Page 6 of 37
Outcomes: Lesson 3 Linear Functions Solve problems involving linear equations. Warm up: In mathematics 10 you found the slope of an equation through the use of the formula. y = y y 1 1 Find the slope between the following points. m = a) A( 1, 3), B( 8, 5 ) b) M( 6, 4), N(, 1 ) y y 1 1 1. Use the formula to find the equation for the following situations. a) Find the equation of a line with a slope of 3 4 which passes through the point (,3) b) Determine the equation of the line that passes through the points Q(3,6) and R(-1,). If we manipulate the formula we can arrive at y y = m( ) 1 1 a) Find the equation of a line with a slope of 4 which travels through the point C(5,7). 5 b) Find the equation of a line, which passes through the point (-4,5) and (,7) Pre-calculus and Limits Page 7 of 37
Investigate: Since slope is defined as y the slope is the ratio of the change in y to the change in, it can be interpreted at the rate of change of y with respect to. Eamples: 1. A linear function is given by y = 3 + 4. If increases by 3 how does y change?. A linear function is given by y = 6 5. If increases by 3 how does y change? 3. The deeper one enters the earth the warmer the temperature becomes. For every 5 m of depth the temperature increases about 1 0 C, up to 10km. If the ground temperature is 5 0 C, find an epression for the temperature T, as a function of the depth d. Homework: Page 9 # 1 6, 11, 1 Pre-calculus and Limits Page 8 of 37
Outcomes: Warm up: Eplain the concept of a limit Lesson 4 The Tangent Problem In pure mathematics 0 we found the tangent of a line to a circle. The tangent line was a line that touched the circle in only one place. In math 31 we will find tangent lines to curves at various positions. Investigate: At which points are the lines tangent to the given curve. Eamples: 1. Find the equation of a tangent line to the parabola y = at the point (1,1). Pre-calculus and Limits Page 9 of 37
. Find the equation of a tangent line to the cubic function y = 3 at the point (,8). The slopes in the above two eamples illustrate the concept of a limit. The rest of this unit will deal with the concept of limits. We will use limits to solve various problems in this unit. Homework: Page 9 # 8,9 Pre-calculus and Limits Page 10 of 37
Lesson 5 Solving Limits Using Intuitive Reasoning Outcomes: Give eamples of functions with limits, with left-hand or right-hand limits, or with no limit Warm up: 1. Consider the function f ( ) = 1 a) Find the values of f( -1 ), f( -10 ), f(-100 ), f( -1000 ) What is happening to the value of f( ), as is becoming smaller. From above we can intuitively conclude lim 1 = 0 This is stated: the limit of 1 as approaches negative infinity is 0. b) Find the values of f( 1 ), f( 10 ), f( 100 ), f( 1000 ) What is happening to the value of f( ), as is becoming larger. 1 From above we can intuitively conclude lim = 0 This is stated the limit of 1 as approaches infinity is 0.. Consider the function f ( ) = a) Find the values of f( -1 ), f( -10 ), f(-100 ), f( -1000 ) What is happening to the value of f( ), as is becoming smaller. In the function f ( ) =, lim f ( ) = This is stated the limit of f () as approaches negative infinity is infinite. b) Find the values of f( 1 ), f( 10 ), f( 100 ), f( 1000 ) What is happening to the value of f( ), as is becoming larger. In the function f ( ) =, lim = This is stated the limit of f () as approaches infinity is infinite. Pre-calculus and Limits Page 11 of 37
Limits of continuous functions at a in f (a). Eamples: 1. Given f ( ) = + 4, Sketch the function. Find lim f ( ). 4 f ( ) f ( ) 3 5 3.5 4.5 3.9 4.01 3.99. Find the lim 3 4 + 3. Then draw a sketch of the function. 3. Find the lim 4 3 + 7 4. Find the lim 9 Functions with the property lim f ( ) = f ( a ) are called continuous at a. a P Likewise in a rational function lim ( ) P( a) = is continuous at a. The function a Q( ) Q( a) is therefore defined at a. Pre-calculus and Limits Page 1 of 37
Limits of discontinuous functions at a in f (a). 16 1. Investigate the function f ( ) = 4 a) State the non-permissible values for f ( ) = 16 4 b) Determine the value of f ( 4 ) for the function 16 4 16 c) Determine the lim 4 4. f ( ) f ( ) 3 5 3.5 4.5 3.9 4.01 3.99 d) Graph the function f ( ) = 16 on your calculator. 4 e) Graph the function f ( ) = 16 4 manually. 16 f) What is the difference between the lim 4 4 and f ( 4 )? Eplain why this difference eists. Pre-calculus and Limits Page 13 of 37
. Investigate the function f ( ) = 3 4 + 3 a) State the non-permissible values for f ( ) = 3 4 + 3 b) Determine the value of f ( 3 ) for the function f ( ) = 3 4 + 3 3 c) Determine the lim 3 4 + 3. f ( ) f ( ) 4.5 3.5.9 3.01.99 d) Graph the function f ( ) = 3 on your calculator. 4 + 3 e) Graph the function f ( ) = 3 manually. 4 + 3 f) What is the difference between the lim 3 eists. 3 4 + 3 and f ( 3 )? Eplain why this difference Homework: Page 18 # 1,,4a,,7,8 Pre-calculus and Limits Page 14 of 37
Outcomes: Lesson 6 Limit Theorems Solve limit problems using the properties of limits. Investigate: Suppose that the limits lim f ( ) a 1. Find lim 5 4 and lim g ( ) a both eist and let c be a constant. The limit of a constant is the constant itself lim c = c a. Find lim( + ) and lim + lim 5 5 5 The limits of a sum is the sum of the limits lim f ( ) + g( ) = lim f ( ) + lim g( ) a a a 3. Find lim( ) and lim lim 5 5 5 The limits of a difference is the difference of the limits lim f ( ) g( ) = lim f ( ) lim g( ) a a a 4. Find lim( 5 ) and 5lim 5 a The limit of a product of a constant and a function is the product of the constant and the limit of the function lim cf ( ) = c lim f ( ) a a 5. Find lim( ) then lim lim 5 5 5 The limit of a product is the product of the limits lim f ( ) g( ) = lim f ( ) lim g( ) a a a Pre-calculus and Limits Page 15 of 37
6. Find lim 5 3 and lim 5 lim3 5 The limit of a quotient is the quotient of the limits f ( ) lim f ( ) z lim = a g( ) lim g( ) a 7. Find lim( ) 5 3 and lim( ) 5 3 The limit of a power is the power of the limit lim f ( ) a n = lim f ( ) a n 8. Find lim ( ) 5 4 and lim( ) 5 4 The limit of a power is the power of the limit lim n f ( ) = n lim f ( ) a a Pre-calculus and Limits Page 16 of 37
Eamples: 1. Use the limit theorems to find lim( 3 ). Find lim( + 3 ) using the properties of limits 5 3. Find lim 1 5 + 1 + 4 using the properties of limits 4. Find lim + 3 using the properties of limits Since all of the above functions were continuous at the value approaches. lim f ( ) = f ( a ) a The limit Theorems may seem rather useless at this time. That is because we chose equations in which the theorems were evident. Net class you will consider the eamples in which you will need to use the limit theorems to arrive at the correct solution. Practice the limit theorems you will need them. Homework: Page 19 # 3 Pre-calculus and Limits Page 17 of 37
Lesson 7 Finding the Limits of Functions by Factoring Outcomes: Solve limit problems which involve 0 0 Warm up: Find f ( ) of 4 0 Investigate: Since f ( ) = is meaningless you must simplify to obtain a function which is 0 continuous at, before you can find the limit of the function. Eamples: Find the following limits by factoring. 4 1. Find lim 16. Evaluate lim 4 4 4 3. Find lim 8 4. Evaluate lim 3 + 3 Pre-calculus and Limits Page 18 of 37
5. Find lim h 0 + h 4 h 6. Find lim h 0 1 h 1 h If a function cannot be written as another function which is continuous for the approached value of then the function does not have a limit, for the approached value of. 1 7. Show lim does not eist. Draw a graph to help with your understanding 0 + 8. Find lim 4 4 Homework: Page 19 # 4(a f), 5 (a,b,d),6 (a,b,c,d,e,f,i) Pre-calculus and Limits Page 19 of 37
Lesson 8 Finding the Limits of Rational and Radical Functions Outcomes: Solve limit problems which involve 0 0 Warm up: 1. Rationalize the following denominators 3 a) b) 5 7 3+ + 3 5. Simplify the following epressions. 1 3 3 a) + b) ( + 1) 3 9 Eamples: 1. Find lim 0 + 1 1. Find lim 0 + 4 Pre-calculus and Limits Page 0 of 37
1 1 3 3. Find lim 3 3 4 + 4. Find lim 0 5. Find lim 4 4 4 Homework: Page 19 # 4(g,h), 5 (c,e,f),6 (g,h,j) Pre-calculus and Limits Page 1 of 37
Outcomes: Warm up: Lesson 9 One Sided - Limits Solve functions with limits, left-hand or right-hand limits, or with no limit 5 Consider the function f ( ) = 3. Complete the table for the function. f ( ) f ( ) 4.5 3.5.9 3.1.99 3.01.9999 3.0001 Sketch the above function. Investigate: approach. The limit as you approach 3 from the left is different than the limit when you The left-hand limit The right-hand limit 5 lim = 5 lim 3 3 + = 3 3 the limit as approaches from the left is from the right is infinite negative infinite the limit as approaches 5 The lim does not eist. Since the left and right limits are not equal. 3 3 Pre-calculus and Limits Page of 37
Eamples: 1. Find the following using a table of values. a) lim b) lim 4 4 4 + 4 c) lim 4 4. Find the following using a table of values. a) lim b) lim 4 ( ) 4 4 + ( 4 ) lim 4 ( ) 4 c 3. Find the following from the above graph a) lim f ( ) b) lim f ( ) 0 0 + c) lim f ( ) 0 d) lim f ( ) 5 e) lim f ( ) 5 + f) lim f ( ) 5 Pre-calculus and Limits Page 3 of 37
Some functions are not described by a single equation but rather multiple equations. Consider the follow. = if 1 4. f 3 if > 1 a) Find the values of f ( -1 ) f ( 0 ) f ( 1 ) f ( ) b) Sketch the function c) Find lim f ( ) 1 lim f ( ) 1 + lim f ( ) 1 5. Find a) lim 0 + b) lim 0 c) lim 0 6. Show that lim = 0 0 Pre-calculus and Limits Page 4 of 37
7. The Heaviside function H is defined by (details are in your book page 3) 0 if t < 0 H( t) = 1 if t 0 Find lim H( t) 0 lim H( t) 0 + lim H( t) 0 8. a) If if 1 f ( ) = if 1< < 1determine whether or not lim f ( ) and lim f ( ) eist. if 1 1 1 Homework: Page 7 # 1,,4,5,6,7 Pre-calculus and Limits Page 5 of 37
Lesson 10 Discontinuous Functions Outcomes: Warm up: Distinguish between continuous and discontinuous functions. In lesson 5 we discussed continuous functions. Recall the following. Functions with the property lim f ( ) = f ( a ) are called continuous at a. a P Likewise in a rational function lim ( ) P( a) = is continuous at a. The function a Q( ) Q( a) is therefore defined at a. If a functions is not continuous at a, we say the function is discontinuous or the function has a discontinuity at a. If lim f ( ) does not eist then the function is discontinuous at a. a Eamples: Where are the following functions discontinuous? 3 if < 0 a) f ( ) = if 0 1 b) 1+ if > 1 + 1 f ( ) = 0 1 < 0 = 0 > 0 + 1 c) f ( ) = 1 = Homework: Page 7 # 3,8,9,10(a,b,c) Pre-calculus and Limits Page 6 of 37
Outcomes: Warm up: Lesson 11 Using Limits to Find Tangents Find Tangents to Curves In pure mathematics 0 we found the tangent of a line to a circle. The tangent line was a line that touched the circle in only one place. In math 31 we will find tangent lines to curves at various positions. Investigate: We can use the concept of limits to find the equation of a line tangent to a given curve. Consider the following eamples. Develop a formula for finding slope of a line tangent to a curve (, f()) (a,f(a)) Eamples: 1. Find the equation of a tangent line to the parabola y = at the point (1,1).. Find the equation of a tangent line to the cubic function y = 3 at the point (,8). Pre-calculus and Limits Page 7 of 37
Develop a second algebraic method to find the slope of a line tangent to a given curve by moving the point ( a + h) successively closer to a. This is often referred to as first principle of calculus. (a,f(a)) (a+h, f(a+h)) 3. Find the slope and the equation of the tangent line to the curve y = + 4 1 at the point (, 15). 4. Find the tangent line to the hyperbola y = 1 at the point ( -, - ½ ). Pre-calculus and Limits Page 8 of 37
5. Find the tangent line to the curve y = at the point where = 6. Graph the curve and the tangent line. 6. Find an equation to find the slope of a tangent line at any point tangent to the curve y = + 3 7. If f ( ) = 7, what is the value of where the slope is? Homework: Page 35 #1,,6,7(a,b),8,9 Pre-calculus and Limits Page 9 of 37
Lesson 1 Velocity as a Rate of Change Outcomes: Solve rate of change problems using limits Warm up: Find the slope of a line tangent to y = 3 5 through the point when = -1. distance travelled Average velocity = If you drive to Edmonton your average time elapsed velocity may be 100km/h. We all know that you would go slightly faster and slightly slower than that speed at any given moment. This speed at any moment is known as instantaneous velocity. Consider the speed at an instantaneous instance how much time is elapsed at that moment, how much distance is traveled. Consider the following investigation carefully. Investigate: Suppose that a ball is dropped from the upper observation deck of the CN tower, 450 m above the ground. How fast is the ball falling after 3 s? Approimate the speed using a table of values distance travelled Remember Average velocity = time elapsed Use time intervals closer and closer to 3 seconds to predict the speed. Time Distance Traveled Average Velocity 3.5 3.1 3.05 3.01 3.001 s Average velocity = t f (3 + h) f (3) = h Pre-calculus and Limits Page 30 of 37
So the instantaneous velocity can be found using f lim ( 3 + h ) f ( 3 ) h 0 h Find the instantaneous velocity at 3 seconds. Eamples: 1. The displacement in metres, of a particle moving in a straight line is given by s = t + t, where t is measured in seconds. Find the velocity of the particle after 3 s. Homework: Page 43 # 1,,3 Pre-calculus and Limits Page 31 of 37
Lesson 13 Infinite Sequences Outcomes: Warm up: Investigate: Solve Geometric Sequence Problems Infinite sequences and their limits are basic to the understanding of Calculus. Even though sequences are more fully developed in pure mathematics 30, there are a number of key concepts you will need to know for mathematics 31. In a geometric sequence each term is multiplied by a constant to obtain the following term. In an arithmetic sequence each term is added to a constant to obtain the following term. Many other sequences can be generated using various methods. 1. The first two terms of a two geometric sequences are as follows: Sequence 1 Sequence t1 = 64 and t = 3 t = 1 1 1 and t 8 = 4 a) Write the first 5 terms for each sequence. b) The first sequence is called a convergent sequence. What value does the sequence appear to be approaching. An infinite sequence is the range of a function which has the set of natural number as its domain. If the terms of an infinite sequence approach a unique finite value, that sequence is called a convergent sequence. A sequence which does not converge is called divergent. Eamples: 1. a) Determine the first five terms of the sequence defined by the function f ( n) = n 1, n N. b) Plot the points of sequence. c) What do you think lim f ( n) is? n Pre-calculus and Limits Page 3 of 37
. a) Determine the first five terms of the sequence defined by the function n t( n) = n N n + 1 b) Plot the points of sequence. c) What do you think lim f ( n) is? n 3. a) Determine the first five terms of the sequence defined by the function 1 t( n) = n N n b) Plot the points of sequence. 1 c) What do you think lim is? n n The following statements will prove very useful in your study of sequences 1 lim n n r = 0 if r > 0 n lim r = 0 if r < 1 n Pre-calculus and Limits Page 33 of 37
n 3 4. Find lim n n n n 5. Find lim n n + 1 3n 5n + 8 6. Find lim n n + 3n 7 7. Find the following limits if they eist. Which sequence is a convergent sequence. a) lim 1 n n 1 b) lim Homework: Page 50 #1 6, 8 Pre-calculus and Limits Page 34 of 37
Outcomes: Warm up: Investigate: Find S 1 Lesson 14 Infinite Series Define the limit of an infinite sequence. A series is the sum of a sequence. Consider the geometric series following geometric series 3 + 3 3 3 4 + 3 16 + 64 + 56 S S 3 S 4 S 5 Plot above partial sums on the grid below. The value of n partial sums can be found using the formula. n a( 1 r ) Sn = r 1 1 r Find the value of the 6 th partial sum using the formula. Predict the value of S Pre-calculus and Limits Page 35 of 37
To find the sum of an infinite geometric series we would have to find lim ( n a 1 r ) 1 r To find a sum of a infinite geometric series S a =. 1 r Eamples: 4 1. Find the sum of the infinite geometric series 4 4 5 + 5,.... Which infinite geometric series is convergent? What is the sum? a) 4 6 + 9 13.5 + b) 6 + + 3 + 9 +... 3. Write each of the following as reduced fractions. a) 0. b) 0. 3 c) 3. 0517 Pre-calculus and Limits Page 36 of 37
Note: r < 1 then the denominator would be less than 1 and the resulting series would be convergent. E: 3 + 3 3 3 4 + 3 16 + 64 + 56 If r = 1 the common ratio is 1. So the series is a + a + a + a. therefore the series does not have a sum. E: + + + +. If r = - 1 the common ratio is -1. So the series is a - a + a - a. therefore the series does not have a sum. E: - + - +. If r > 1, then the series grows indefinitely. Therefore lim ( n a 1 r ) 1 r 1 lim = 0 only if r < 1 n r n E: + 4 + 8 + 16 + 3. does not eist. Remember 4. For what values of are the following series convergent? In each case find the sum of the series for those values of. 3 6 1 4 a) ( + 1), ( + 1), ( + 1) 3,... b,,,,... c) 3,,,,... 3 4 Homework: Page 56#1,3,4,5 Pre-calculus and Limits Page 37 of 37