1.1 Radical Expressions: Rationalizing Denominators

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1 1.1 Radical Expressions: Rationalizing Denominators Recall: 1. A rational number is one that can be expressed in the form a, where b 0. b 2. An equivalent fraction is determined by multiplying or dividing the numerator and denominator by the exact same value. 3. A radical is a square root: 9 = 3 and 3 x 3 = 3. In a rational number (fraction), it can be helpful to visualize the numerator designates the quantity, and the denominator designates the size of the pieces. Because of this, you have been taught that when simplifying fractions, the denominator must be positive. Ex. 4 5 = 4 5 or 4 5 In the same vein, sometimes you will come across a fraction in which the denominator is a radical. It is preferable to divide by integers rather than roots, so we must rationalize the denominator in these cases. 5 Ex. Simplify by rationalizing the denominator. 4 3 Sol. We multiply both the numerator and the denominator by 3 in order to rationalize the denominator. Remember the idea of conjugate radical roots: ( a + b)( a - b) = a b which is a rational number. Ex. Simplify Ex. Simplify Ex. Simplify Ex. Rationalize the numerator: x+4 2 x

2 1.2 The Slope of a Tangent Introduction to Calculus What is Calculus? Calculus is a branch of mathematics which was created to deal with two simple geometric problems: 1. The problem of tangents (finding the slopes of the tangent line to a curve at a point) 2. The problem of areas (finding the area under a curve) 1. Slope =? y = f(x) Area =? (This is what you ll do in university) These questions are thousands of years old mathematicians before Christ were interested in these properties and their applications. Many others worked on developing a consistent and complete way to deal with these issues: but Sir Isaac Newton ( ) and Gottfried Wilhelm von Leibniz ( ) developed Calculus as we know it, independently of each other. Calculus is a very powerful branch of mathematics that has several applications in business, engineering, science, landscaping, economics, driving, etc. The Tangent Recall: A tangent line is a line which meets a curve at exactly one point. Its slope is equal to the slope of the curve at that exact point. This is a very important concept to understand. All lines contain a slope: in the equation y = mx + b, m represents the slope of the line, variously defined as: rise run, Δy Δx, y 2 y 1 x 2 x 1. In order to find the slope of a line, you need any two points on the line to create a secant. Recall: Slopes of parallel lines are equal; slopes of perpendicular lines are negative reciprocals.

3 A general approach is needed for calculating the tangent line to a curve. Consider the function y = f(x) and a point P on the curve. How can we find the slope of the tangent line at point P? Q P y = f(x) Point Q is any other point on the curve. The line joining PQ is called a secant line. We can determine the slope of this point quite easily, but its slope is not close enough to the tangent line to be considered accurate. What happens if we slide Q towards P, so that the secant line PQ moves successively closer to the tangent line at P? The closer Q moves towards P, the closer this line is to approximating the tangent line at P. Q can get so close to P, in fact, that the distance between them is practically 0 we call this distance h. Definition: The slope of the tangent line to the curve at point P is the limiting slope of the secant line PQ as the point Q slides along the curve towards P. The slope of the tangent is said to be the limit of the slope of the secant as Q approaches P along the curve. Think about it: as the two points get closer and closer together, the slope of the tangent line approaches a certain value. This slope is more accurate (and true to the tangent line s slope) the closer the points are. As h gets closer to 0, the limit of the slope becomes apparent. Ex. Find the slope of the tangent line to the parabola y = x 2 at the point (2, 4). Solution: Let P(2, 4) be the point on the parabola. Choose a nearby point Q which is also on the parabola (a distance of h units horizontally from P). Since f(x) = x 2, P can be expressed as (2, 2 2 ), and Q can be expressed as (2+h, (2+h) 2 ). As Q slides closer and closer to P, the distance h decreases it will take on values closer and closer to 0. We say that h approaches 0 and use the notation h 0. We can now calculate the slope of the secant line between P and Q. Δy = (2+h) 2 4 Δx = (2 +h) 2 = h = 4 + 4h + h 2-4 = h 2 + 4h

4 Slope: Δy = 2 +4 Δx This is the slope of the secant line. Keeping in mind the definition above, =+4 the slope of the tangent line is lim + 4 = Therefore, the slope of the tangent line at the point (2,4) is 4. = 4 In general: Consider a function y = f(x) and a fixed point P(a, f(a)) on the function. Let Q be any other point on the curve. The coordinates of Q are (a + h, f(a+h)). The slope of the secant line is: Δy f a+h f(a) = Δx h The slope of the tangent line is: lim 0 f 1.3 a Rates + of f(a) Change Examples 1. Determine the slope of the secant PQ where P(2, 8) and Q(2+h, f(2+h)) where f(x) = x 3 2. Determine the slope of the tangent at y = x + 2 at x = 7 y = 4x 2 at x = -3 y = 1/x at x = 4 3. Determine the equation of the tangent line to y =x 2 +2x +4 at (1, 7)

5 1.3 Rates of Change -- Velocity Velocity refers to the rate of displacement with regards to time. The difference between speed and velocity is that velocity has a direction (can be negative or positive, relative to the origin), whereas speed is the absolute value of velocity. Ex. Determine the average rate of change (the average velocity) for a car that travels east on Highway km in 4h. Sol. It is customary to represent such a rate-of-change model as a straight line with a definite origin. Generally speaking, movement up or right is considered to be in the positive direction, and movement down or left is considered to be in the negative direction. Visualizing our problem, Time = 4h Average Velocity = change in position/change in time = 350km/4h = 87.5 km/h NB: the units reflect the ratio. What would the average velocity be if we had travelled west of the origin? Displacement = 350km east Average velocity thus = s where s represents position and t represents time. We can express t s t+ s(t) average rate of change between (t, s(t)) and [(t+h), s(t+h)] as Ex. A pebble is dropped from a cliff 80m high. After t seconds, the pebble is s metres above the ground, where s(t) = 80 5t 2, 0 t 4. If we want to calculate the instantaneous velocity of the pebble say, at t = 1 we need to find a point very close to 1 sec perhaps at t = (1+h) seconds, where h is a very small amount of time: From the example above, we can see that: INSTANTANEOUS VELOCITY (instantaneous rate of change) The velocity of an object with position function s(t), at time t = a, is v a = lim 0 s a + s a

6 AN ALTERNATE WAY TO USE LIMITS We have been calculating limits by finding the limiting slope between two points P(a, f(a)) and Q(a+h, f(a+h). This provides the slope of the tangent line that intersects the curve at point P. The slope of the curve at this point is the same as the slope of the tangent line. Depending on the situation, it may be better to look at calculating the limit from the perspective of the curve alone. For the example above: s(t) = 80 5t 2, we can calculate the instantaneous velocity (the slope of the curve) at t = 1 by considering the average velocity from t = 1 to a general time t and letting t approach the value of 1. s(1) = 75 Point P is (1, 75) s(t) = 80 5t 2 Point Q is (t, 80 5t 2 ) s t s(1) lim t 1 t is approaching 1 (we re limiting the distance between P and Q) t 1 = lim t t 2 75 t 1 = lim t 1 5(t 2 1) t 1 = lim t 1 5 t 1 (t+1) t 1 = lim t 1 5(t + 1) = -10 m/s

7 1.4 THE LIMIT OF A FUNCTION Limits can also be used to find the value of a function f(x) as x gets closer to a specific number. The expression lim f x = L means that f(x) approaches the value of L as x approaches the value of a. In order for this limit to exist, the function must be approaching the same value from the left as it is from the right. Ex. Determine the limit of y = x 2 1, as x approaches 1. x 1 Sol. We can determine the limit of this function by looking at the value of the function as x gets increasingly close to 1. Note that the function itself is not defined at this point; this is irrelevant for now. X y = x 2 1 x 1 By looking at the behaviour or pattern of numbers as we approach 1 from the left and the right, we can see that the function approaches 2 as x 1. It s important that the function approaches the same value from the left and the right. Therefore, lim x 1 f x = 2. We say that the number L is the limit of a function y = f(x) as x approaches the value a, written as lim f x = L, if and only if lim f(x) = L = lim f x. + Notes: 1. lim f x may exist even if f(a) is not defined. The function just has to be approaching the same value from the left and from the right, but the value itself doesn t have to exist on the curve. 2. For many functions (such as polynomial functions), lim f x will be the same value as the function itself at (a, f(a)), because there are no gaps or asymptotes at this point. The easiest way to evaluate a limit such as this is to substitute. Ex. Sketch each function and evaluate each limit. If it does not exist, explain why. 1 x + 2, if x < 1 a) lim x 1 b) f x = x 2 x + 2, ifx 1, lim x 1 f(x)

8 1.5 PROPERTIES OF LIMITS Recall that lim f x = L means that the values of f(x) become closer and closer to the number L as x gets closer and closer to a, without actually becoming a. The behaviour of f(x) around a is what matters it is not even necessary for f(a) to exist for the limit itself to exist. You do not need to consider x = a, only values nearby. Big Ideas so far: 1) lim f(x) exists if and only if the following three criteria are met: - lim f x exists (the left hand limit exists) - lim + f x exists (the right hand limit exists) There are - some lim properties f x = of limlimits + that f(x) need to (they be considered. both approach the same value) PROPERTIES OF LIMITS For any real number a, suppose that f and g both have limits that exist at x = a. 1. lim k = k for any constant k 2. lim x = a 3. lim f x ± g x = lim f x ± lim g x 4. lim [c f x = c lim f x for any constant c 5. lim f x g x = lim f x [lim g x ] f(x) 6. lim = lim f(x) g(x) lim g(x) as long as lim g(x) 0 7. lim [f x ] n = [lim f x ] n, for any real number n NOTE: IF F(X) IS A POLYNOMIAL FUNCTION, THEN lim f(x)= f(a). Ex. If lim x 4 f x = 3, use the properties of limits to evaluate each limit. a) lim x 4 [f x ] 3 [f x b) lim ]2 x 2 x 4 f x +x c) lim x 4 3f x 2x

9 We may encounter a limit that, on first inspection, seems impossible to evaluate. We have these strategies to evaluate such limits root functions, rational functions when substitution is impossible and sketching the curve is too onerous: Direct substitution a) Factoring b) Rationalizing c) One-sided limits d) Change of variable *after each step, go back to substitution to see if you can evaluate. x+1 1 Ex. Evaluate lim x 0. x 2x Ex. Evaluate the following limit: lim 3 +3x 9 x 3. State the limit laws used. x 2 4x+2 1 (x+8) Ex. Evaluate lim x 0 x 3 2. Ex. Evaluate lim x 2 x 2 x 2. Ex. Evaluate the following limit: lim x 3 x 2 10x+21 x 2 +x 12. Ex. Evaluate the following limit: lim x 2 (x 2) 3 x 2

10 1.6 CONTINUITY Continuity refers to the idea of a graph without any breaks or jumps or gaps. Think of it this way: if you are able to trace or draw a graph without lifting your pencil, the graph is most probably continuous. What functions did you study in 4U that fit this description? Which ones didn t? In order to be continuous at a point, a graph must pass through the point without a break. If not, the graph is discontinuous at this point. Discontinuity can be visualized as an asymptote, a hole, or a jump. The function f(x) is continuous at x = a if: 1. f(a) is defined; 2. lim f x exists (see 1.5); 3. lim f x = f(a) 1. Consider a function y = f(x) such that lim f x = 4, lim + f x = 4, and f(a) = 3. Explain whether each statement is true or false. a) y = f(x) is continuous at x = a. b) The limit of f(x) as x approaches a exists. c) The value of the right-hand limit is 4. d) The value of the left-hand limit is 3. e) When x = a, the y-value of the function is A) Graph the following function: f(x) = x2 3, ifx 1 x 1, if x > 1 b) Determine lim x 1 f(x) c) Determine f(-1) d) Is f continous at x = -1? Explain. 3. Determine whether or not these functions are continuous at x = 2, and give your reasons. a) f x = x 3 x b) f x = x 2 x 2 x 2 c) f x = x 2 x 2 x 2 ifx 2andf 2 = 3 d) f x = 1 (x 2) 2