1 Unit 1 PreCalculus Review & Limits Factoring: Remove common factors first Terms - Difference of Squares a b a b a b - Sum of Cubes ( )( ) a b a b a ab b 3 3 - Difference of Cubes a b a b a ab b 3 3 3 Terms - form of x bx c - form of ax bx c * can use decomposition but we can also (and should use shortcuts) 4 Terms - Try factoring by grouping
Ex) Factor the following fully. 3 6 3 8 a) 1m n 1m n 15m n b) x ( ) 11( x ) 4 c) 3m n 33mn mn 13 9 d) 51 a
3 e) x 3( 4) 13( x 4) 10 3 3 f) 1x y 5x y 7xy 3 3 e) ( x 4) ( x ) g) 4 a 5 Now Try Factoring Worksheet
4 Rationalizing Denominators: Single Term Denominators Ex) 3 10 * multiply the numerator and denominator by the radical found in the denominator Two Term Denominators Ex) 3 5 10 * multiply numerator and denominator by the conjugate of the denominator
5 Ex) Rationalize the following. a) 5 b) 9 6 c) 1 4 11 d) 4x 1 x 1 e) 7 5 3 f) 4 3 5
6 g) 6 13 6 13 h) x x 1 x x i) 3 6 x j) 3 x 7 3 k) 3 3 x 5 1 l) 3 4 3 Now Try Rationalizing Denominators Worksheet
7 Function Notation: In f( x ), f refers to the name of the function (or equation) and x is the variable used within that function. In ga, ( ) g refers to the name of the function (or equation) and a is the variable used within that function. Ex) If f ( x) x 5, g( x) 5 x, and hx ( ) find the following: a) f(4) h(5) b) ( fh)( 7) x, x 3 c) ( f g)( x ) d) 4 hx ( 1) 9
8 g e) ( x) h f) ( h h )(3) g) g( x 1) 5 ( f f )( 3) 3h x Now Try Function Notation Worksheet
9 Equations of Lines: Stuff we need + ve slope - ve slope slope = 0 slope is undefined *line increase *line decrease *horizontal line *vertical line y mx b y y1 m( x x1) m y y y x x x 1 1 Parallel Lines *have the same slope Perpendicular Lines *have slopes that are negative reciprocals of one another Ex) Determine the equation of the line that passes through 35, 10 and has a slope of 7 15.
Ex) Determine the equation of the line that is parallel to 4y 7x 0 0 and has the same x-intercept as 5y 3x 0. 10 Ex) Determine the equation of the line that passes through ( 1, 4) and (9, 5).
11 Ex) Determine the slope of the tangent to the circle ( x 4) ( y 6) 5 x. when 8
Ex) Determine the equation of the tangents to the circle given by ( x3) ( y 6) 100 at the points that the circle intersects the line 4x3y30 0 1 Now Try Equations of Lines Worksheet
13 Solving Trigonometric Equations: Stuff we need S T A C sin30 sin 45 sin 60 1 3 cos30 cos45 cos60 3 1 tan sin cos cot cos sin csc 1 sin sec 1 cos *understand how reference angles can be used y sin x y cos x 360
14 First Degree Equations Ex) Solve the following, express all solutions as a general statement in radians. a) sin x 3 b) sec x 4 0 c) 3cot x 1 0
15 Second Degree Equations Ex) Solve the following, express all solutions as a general statement in radians. a) sin xsin x 0 b) 4cos x 1 0 c) cos x 7cos x 4 0
16 d) x 3sin cos x 0 e) 3sin x cos x 0 Multiple Angle Equations Ex) Solve the following, express all solutions as a general statement in radians. a) sin x
17 b) tan x 3 0 c) 1 cos 1 0 3 x d) sin ( x) sin( x) 1 0 e) sin 1 xcos 1 x sin 1 x 0 Now Try Trigonometric Equations Worksheet
18 Limits: Limits allow us to determine the value that a function gets closer to (approaches) as x gets closer to (approaches) a specified value Notation lim f x a ( x ) b read: as x approaches the value of a, the value of the function f( x ) approaches b think: what value is f( x ) getting closer and closer to as x gets closer and closer to a Ex) x 3x8 limx4 11 x 4 The limit as x approaches 4 for the function x 3x8 f( x) is equal to 11. x 4 This means that the closer x gets to the value of 4, the function f( x ) gets closer to the value of 11.
19 Ex) Consider the following functions. x 5x6 1 f( x) c) f( x) 4 x x as x gets as x gets as x gets closer to 7 closer to - closer to a) f ( x) x 3 b) x f( x ) x f( x ) x f( x ) Thus Thus Thus x 5x6 x7 x limx x x lim 3 1 lim 4 x
0 Continuous and Discontinuous Functions: A continuous function is a function that when graphed has no breaks ( the graph consists of one piece). Ex) A discontinuous function is a function that when graphed consists of or more parts (is broken). Ex) **These points where the graph is broken are called the Points of Discontinuity.
1 One Sided Limits: One sided limits allow to determine what value a function approaches as x gets closer to a specific value from either the right or the left side of the specified value. lim ( ) f x Means x values are approaching x a value of from the right side Ex) x.5,.4,.1,.0001, etc. lim ( ) f x Means x values are approaching x a value of from the left side Ex) x 1.5, 1.8, 1.9, 1.9999, etc. The general Limit of lim x f( x) will only exist if lim f( x) and lim f( x) equal the same value. This x x means that it shouldn t matter what side the value is approached from to determine the value of the general limit lim x f( x). lim f x a ( x ) b if and only if lim f ( x) b and lim f ( x) b xa xa
Ex) Given the graph of y f ( x), determine the following limits. a) lim x 6 f( x) b) lim f( x) c) lim x x 0 f( x) d) lim f( x) x 5 e) x lim f( x) f) lim x 5 f( x) g) lim f( x) h) lim x x 8 f( x) i) lim f( x) x 4 j) lim f( x) lim f( x) lim f( x) x k) x 4 l) x
Ex) Given the function defined below, find the following limits. 3 x x x 4 9, if f ( x) x 3, if x 4 x5, if x4 a) lim f( x) x b) x lim f( x) c) lim x f( x) d) lim f( x) x 4 e) x 4 lim f( x) f) lim x 4 f( x) Graph the function y f ( x) Now Try Page 7 # 1 to 3, 5 to 11
4 Properties of Limits: lim f ( x) g( x) lim f ( x) lim g( x) xa xa xa 3x 3x lim 5 lim lim 5x x x Ex) x x x x lim cf ( x) clim f ( x) xa xa Ex) x x 1 x x 1 limx 35 5limx 3 x3 x3 lim f ( x) g( x) lim f ( x) lim g( x) xa xa xa Ex) 3 4 3 4 limx 4 x 3 x x limx 4 x 3 lim x x x4 x4 x4
5 lim xa f ( x) lim xa f ( x) g( x) lim g( x) xa lim x5 lim x6( x5) x6 Ex) x x1 limx x x 1 6 n f x f x lim ( ) lim ( ) xa xa n Ex) lim 4 4 x 8 x 15 8 15 lim x x x 3x 10 x 3x 10 x5 x5 lim n f ( x) n lim f ( x), if the root exists xa xa Ex) lim x 3 3 4 x 3 x 1 4 3 1 3 3 lim x x 3 x 3 3x 6x 11 3x 6x 11
6 Solving Limits Algebraically: 1) Solve by Direct Substitution Many limits can be solved by substituting the value for x directly into the function. This can be done if the function is continuous at the point in question. Ex) Solve the following limits. a) lim x5 x x 3 b) lim x1 x 5x 1 x 4 c) lim x 3 x x
7 ) Solve by Factoring Many times if direct substitution is attempted a result of 0 0 occurs. This happens when we attempt to find a limit as x approaches a value where the function is discontinuous (we are trying to find a limit at a restriction). Eg) lim x4 x 16 x 4 Factor first, reduce, then solve by direct substitution Eg) Solve the following limits. a) lim x x x7 3x8 10x1 b) lim h 0 ( ) 4 h h
8 c) lim x x 3 x 8 3x 3) Solve by Multiplying by a Conjugate If direct substitutions results in 0, and the function 0 cannot be factored, try multiplying numerator and denominator by a conjugate (especially if a radical appears in the function). Ex) Solve the following limits. a) lim x 0 x 11 x
9 b) limx47 x 47 7 x c) lim a5 x x 3 50 5 x d) limx 1 1 1 x x 1 Now Try Page 18 # 1 to 9
30 Infinite Limits: These are limits where x approaches n lim x x, n 0 1 limx 0, n 0 n x To evaluate these limits, divide all terms of the numerator and denominator by the largest power of the denominator, then use the above identities to evaluate. Ex) Evaluate the following limits. a) lim x 3x 3 4x 1
31 b) lim x 5x 7x x c) lim x 3 4x x 11 3 6x 5 d) lim n n n n 1 e) lim a 3 4 7a 5a a 3 4 3a 1 5a f) lim x 18x1x 4 x g) lim h 3 5h 6h 1 h 14h 7
3 Infinite Sequences: A sequence will either: Converge Diverge Approach a specific value Approach positive or negative infinity or bounce between or more values To determine if a sequence converges or diverges we find the limit as n for the general term Ex) Determine whether the following sequences converge or diverge. If the sequence converges, to what value does it converge to? a) 1 3 n,,,..., 3 4 n 1
33 b) 6 1 0,,,,..., 9 19 33 n 1 n n c) t 5n d) n t n 1 n e) 4, 1, 36,... f) t n 1 n Now Try Page 50 # 1 to 8
34 Infinite Series: Like sequences, if the sum of a series approaches a specific value it is called Convergent, if not then it is Divergent Ex) State whether the following series are convergent or divergent. a)... b) 1 1 8 4 1... 4
35 Infinite Geometric Series: If r 1, then the sum of the series will converge. Proof: S n a 1 r 1 r n sum of a geometric series lim n a 1 r 1 r n, r 1 Ex) Find the sum of each series given below. a) 1 1 1... 4 16
36 b) 1 9 3 1... c) 3 3 3 3... 5 5 Ex) Express.135 as a fraction. Now Try Page 56 # 1 to 5
37 Slopes of Tangent Lines: When dealing with circles, we know that a tangent is a line that touches the circle once and only once. When dealing with more complicated curves, a tangent line cannot be defined so simply. In calculus we are often concerned with finding the slope of a tangent line as it represents the instantaneous slope of the graph or instantaneous rate of change in a situation where the slope is constantly changing.
38 Ex) Determine the equation of the tangent line to the graph of y x, 4. at the point Finding the slope of the tangent: x 4 m y y 4 y x x x 1 x y x 1 Slope of the tangent is? Equation of the tangent x 4 is? x y x m y y 4 y x x x 1 1
39 Finding Slopes of Tangents Using Limits: Method 1: Method :
Ex) Determine the slope of the tangent line to the graph of y x, 4. at the point 40 Method 1: use f ( x) f ( a) m lim x a x a Method : use m 0 f ( a h) f ( a) lim h h
41 Ex) Find the slope of the tangent line to the curve y x 4x 1, 15. Use both at the point methods. f ( x) f ( a) m lim x a x a f ( a h) f ( a) m lim h 0 h
4 Ex) Determine the slope of the tangent to the curve xy 1 at the point 1,. Use both methods. f ( x) f ( a) m lim x a x a f ( a h) f ( a) m lim h 0 h
43 Ex) Determine the slope of the tangent to the curve y x when x 6. Use both methods. f ( x) f ( a) m lim x a x a f ( a h) f ( a) m lim h 0 h
44 Ex) Determine the equation of the tangent line to the 3 curve y x x when x
Ex) Determine the slope of the tangent line to the curve y x 5x at the general point whose x-coordinate is a. Use this to determine the slope of the tangent when x 1, 4, 6. 45 Now Try Page 9 # 1,, 6, 7, Page 35 # 8, 9 10
46 Slope as a Rate of Change: Slope is defined as: m rise run y y y change in means y x x x change in x 1 1 Slope refers to how quickly y changes with respect to each unit increase in x. Ex) Slope of 5 means: Ex) Slope of 3 means: Ex) Slope of the graph below means:
47 Average Rates of Change vs. Instantaneous Rates of Change: Average Rate of Change: the slope is constantly changing or the rate of change is constantly changing use two points on the curve to find the slope, this treats the curve like a straight line and gives an average rate of change Instantaneous Rate of Change: allows one to find the slope or rate of change at an exact moment to find this determine the slope of the tangent at a particular point
Ex) A ball is dropped from the upper observation deck of the CN Tower, 450 m above the ground. The distance the ball has fallen is given by d 4.9t, where d is the distance in metres and t is the time in seconds. a) Determine the average speed of the ball in the first 3 seconds. 48 b) Determine the speed of the ball at exactly the 3 second mark.
Ex) The displacement, in meters of a particle moving in a straight line is given by d t t, where t is measured in seconds and d in metres. a) Determine the average velocity between t 3.5 s and t 4.5 s. 49 b) Determine the instantaneous velocity at 4 s.
Ex) A spherical balloon is being inflated. If the volume 3 of the balloon, V, is given by V r, where r is the radius of the balloon in cm and V is measured in cm, find the following. a) The average change in volume with respect to the radius between r 7 cm and r 9 cm. 4 3 50 b) The instantaneous rate of change in the volume with respect to the radius when the radius is 8 cm. Now Try Page 43 # 1,, 3, 7, 8