6.01 what type of polynomial is 3x2 6x 1

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Update you log book Make sure you complete and correct ALL of the Differentiation 1Differentiation 1 questions in the past paper booklet. Outcome 3. Numbers Treasure Hunt Following each question, click on the answer.

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Cancel Download. Presentation is loading. Please wait. Copy to clipboard. Is the Math Wizard correct? What can you deduce about the zeros, the leading coefficient, and the least degree of the polynomial functions represented by the following graphs? The free end of a diving board dips C centimetres when a diver of x kilograms stands on it. The relation is C 0. Calculate the amount of dip when a 95 kg diver stands on the board. Give your answer to the nearest tenth of a centimetre.

Calculate the mass of a diver, correct to one decimal place, if the diving board dips 40 cm. As you got older, you studied the rules of grammar, and you learned that there was a correct way to form these sentences, with rules to follow. As a tool in the service of science, calculus served its purpose very well, but it took over two centuries for mathematicians to identify and agree on its underlying principles—its grammar.

In this chapter, you will see some of the ideas that were brought together to form the underlying principles of calculus. Determine the slope of the line passing through each of the following pairs of points: a. Find the equation of a line determined by the given information. The domain of a function f is the set of all numbers x, and its values are given x. Find each of the following values: by f x 2 x 4 a. Compute each of the following: a. A function s is defined for t 3 by s t Find each of the following: a.

Rationalize each denominator. Rationalize each numerator. What is the domain of each of the following? Obviously, sporting performance is enhanced when an athlete is able to increase his or her heart rate at a slower pace i. Heart rate is the rate of change of the number of heartbeats with respect to time.

A heart rate is given for an instant in time. In calculus terminology, heart rate at an instant in time is known as the instantaneous rate of change of the number of heartbeats with respect to time. When a nurse or doctor counts our heartbeats then divides by the time elapsed, they are not determining the instantaneous rate of change but instead are calculating the average heart rate over a period of time usually ten seconds.

In this chapter, the idea of the derivative will be developed, progressing from the average rate of change being calculated over a smaller and smaller interval until a limiting value is reached at the instantaneous rate of change. Case Study—Assessing Elite Athlete Performance The table below shows the number of heartbeats of an athlete who is undergoing a cardio-vascular fitness test. Complete the discussion questions to determine if this athlete is under his or her maximum desired heart rate of 65 beats per minute at precisely 30 seconds.

Graph the number of heartbeats versus time in minutes on graph paper, joining the points to make a smooth curve. Draw a second relationship on the same set of axes showing the resting heart rate of 50 beats per minute. Use the slopes of the two relationships graphed to explain why the test results indicate that the person must be exercising. Discuss how the average heart rate between two points in time could be calculated on this graph.

Explain your reasoning. Show the progression of these average speed calculations on the graph as a series of secants. Use the progression of your average heart rate secants to make a graphical prediction of the instantaneous heart rate at t 30 s. Use this method to determine the heart rate after exactly 60 s. Two simple geometric problems originally led to the development of what is now called calculus. Both problems can be stated in terms of the graph of a function y f x.

Professor Barrow recognized that there was a close connection between the problem of tangents and the problem of areas. However, it took the genius of both Sir Isaac Newton — and Gottfried Wilhelm von Leibniz — to show the way to handle both problems. Their discovery is considered to be one of the major advances in the history of mathematics. Further research by mathematicians from many countries using these operations has created a problemsolving tool of immense power and versatility, which is known as calculus.

It is a powerful branch of mathematics, used in applied mathematics, science, engineering, and economics. We begin our study of calculus by discussing the meaning of a tangent and the related idea of rate of change.

This leads us to the study of limits and, at the end of the chapter, to the concept of the derivative of a function. What geometric interpretation can be given to a tangent to the graph of a function at a point P? A tangent is the straight line that most resembles the graph near that point. Its slope tells how steep the graph is at the point of tangency.

In the figure, four tangents have been drawn. We begin with a brief review of slopes and lines. To find the equation of a tangent to a curve at a given point, we first need to know the slope of the tangent.

What can we do when we only have one point? We proceed as follows: Consider a curve y f x and a point P that lies on the curve. Now consider another point Q on the curve. The line joining P and Q is called a secant. Think of Q as a moving point that slides along the curve towards P, so that the slope of the secant PQ becomes a progressively better estimate of the slope of the tangent at P. In other words, the slope of the tangent is said to be the limit of the slope of the secant as Q approaches P along the curve.

We will illustrate this idea by finding the slope of the tangent to the parabola y x2 at P 3, 9. Find the y-coordinates of the following points that lie on the graph of the parabola y x2.

Find the y-coordinates of each point on the parabola and then repeat step 2 using the points. Use your results from steps 2 and 3 to estimate the slope of the tangent at point P 3, 9. Graph y x2 and the tangent to the graph at point P 3, 9. In this investigation, you found the slope of the tangent by finding the limiting value of the slopes of a sequence of secants. Since we are interested in points Q that are close to P 3, 9 on the parabola y x2, it is convenient to write Q as 3 h, 3 h 2 , where h is a very small non-zero number.

The variable h determines the position of Q on the parabola. As Q slides along the parabola towards P, h will take on values successively smaller and closer to zero. Using technology or graph paper, draw the parabola f x x2. Let P be the point 1, 1. Graph these secants on the same utility you used in step 1.

Use your results to estimate the slope of the tangent to the graph of f at point P. Draw the tangent at point P 1, 1. Find the slope of the secant PQ through points P 3, 9 and Q 3 h, 3 h 2 , h 0. Explain how you could predict the slope of the tangent at point P 3, 9 to the parabola f x x2. Use your calculator to graph the parabola y 18 x 1 x 7 and plot the points on the parabola from x 1 to x 6, where x is an integer.

Determine the slope of the secants using each point from part a and point P 5, 1. Use the result of part b to estimate the slope of the tangent at P 5, 1. Using the x-intercepts of 1 and 7, the equation of the axis of symmetry is 1 7 3, so the x-coordinate of the vertex is 3. Therefore, the vertex is 3, 2. The y-intercept of the parabola is The points on the parabola are 1, 0 , 0, 0.

Using graphing software, the parabola and the secants through each point and point P 5, 1. Using points 1, 0 and P 5, 1. The slope of the tangent at P 5, 1. It can be determined to be 0. The Slope of a Tangent at an Arbitrary Point We can now generalize the method used above to derive a formula for the slope of the tangent to the graph of any function y f x.

Let P a, f a be a fixed point on the graph of y f x and let Q x, y Q x, f x represent any other point on the graph. If Q is a horizontal distance of h units from P, then x a h and y f a h. Point Q then has coordinates Q a h, f a h. Using the definition of a derivative, determine the slope of the tangent to the curve y x2 4x 1 at the point determined by x 3.

Determine the equation of the tangent. Sketch the graph of y x2 4x 1 and the tangent at x 3. The slope of the tangent can be determined using the expression above. The slope of the tangent at x 3 is 2. Using graphing software, we obtain y 10 5 x —20 —15 —10 —5 5 10 15 20 —5 EXAMPLE 4 3x 6 at Determine the slope of the tangent to the rational function f x x point 2, 6.

Solution At x 9, f 9 9 3. Using the limit of the difference quotient, the slope of the tangent at x 9 is 3. When the numerator has two terms, such as 9 h 3, we multiply both the numerator and denominator by the conjugate radical; that is, by 9 h 3.

Therefore, the slope of the tangent to the function f x x at x 9 is For example, suppose we wish to find the slope of the tangent to y f x x3 at x 1.

Graph Y1 0. Remember that this approximates the slope of the tangent and not the graph of f x x3. This means the slope of the secant passing through the points where x 1 and x 1 0.

Can you improve this approximation? Explain how you could improve your estimate. Try once again by setting Xmin 9, Xmax 10, and note the different appearance of the graph. What is your guess for the slope of the tangent at x 1 now? Explain why only estimation is possible. Another way of using a graphing calculator to approximate the slope of the tangent is to consider h as the variable in the difference quotient.

You can use the table or graph functions of your 1 h 1 in the neighbourcalculator. Graphically, we say we are looking at h 3 1 x 3 1 hood of h 0. Exercise 3. Find the slope of the line through each pair of points. What is the slope of a line perpendicular to the following?

State the equation and sketch the graph of the following straight lines: a. Simplify each of the following: 2 h 2 4 a. Rationalize each of the following numerators to obtain an equivalent expression. Find the slope m, in simplified form, of each pair of points. Consider the function f x x3. Copy and complete the following table of values; P and Q are points on the graph of f x. Use the results of part a to approximate the slope of the tangent to the graph of f x at point P.

Calculate the slope of the secant PR, where the x-coordinate of R is 2 h. Use the result of part c to calculate the slope of the tangent to the graph of f x at point P. Compare your answers for parts b and d.

Sketch the graph of f x and the tangent to the graph at point P. Find the slope of the tangent to each curve at the point whose x-value is given. Find the slope of the tangent to each curve at the given point. Sketch the graph of Question 11, part f.

Show that the slope of the tangent can be found using the properties of circles. Communication Communication Explain how you would approximate the slope of the tangent at a point without using first principles.

Sketch the graph of y 34 16 x2. Explain how the slope of the tangent at P 0, 3 can be found without using first principles. Copy the following figures. Draw an approximate tangent for each curve at point P. Find the slope of the demand curve D p , p 1, at point 5, Application It is projected that t years from now, the circulation of a local newspaper will be C t t2 t Find how fast the circulation is increasing after 6 months.

Hint: Find the slope of the tangent when t is equal to 6 months. Find the coordinates of the point on the curve f x 3x2 4x, where the tangent is parallel to the line y 8x. Find the points on the graph of y 13x3 5x 4x at which the slope of the tangent is horizontal. Show that at the points of intersection of the quadratic functions y x2 and y 12 x2, the tangents to each parabola are perpendicular.

For example, the volume of a balloon varies with its height above the ground, air temperature varies with elevation, and the surface area of a sphere varies with the length of the radius. These and other relationships can be described by means of a function, often of the form y f x. The dependent variable, y, can represent price, air temperature, area, and so forth. The independent variable, x, can represent time, elevation, length, and so on.

We are often interested in how rapidly the dependent variable changes when there is a change in the independent variable. This concept is called rate of change. In this section, we show that a rate of change can be calculated by finding the limit of a difference quotient, in the same way we find the slope of a tangent.

An object moving in a straight line is an example of a rate of change model. It is customary to use either a horizontal or a vertical line with a specified origin to represent the line of motion.

On such lines, movement to the right or upward is considered to be in the positive direction, and movement to the left or down is considered to be in the negative direction. An example of an object moving along a line would be a vehicle entering a highway and travelling north km in 4 h. At any instant you do not know how fast the car is going. Your odometer readings are the following: t in hours 0 1 2 2.

Determine the average velocity of the car over each interval. Do any of the results suggest that you were speeding at any time? If so, when? Explain why there may be other times when you were travelling above the posted speed limit. Compute your average velocity over the interval 4 and s 7 km. After 3 h of driving, you decide to continue driving from Goderich to Huntsville, a distance of km. Using the average velocity from Question 4, how long would it take you to make this trip?

After t seconds, it is s metres above the ground, where s t 80 5t2, 0 t 4. Find the average velocity of the pebble between the times t 1 s and t 3 s. Find the average velocity of the pebble between the times t 1 s and t 1. Explain why the answers to parts a and b are different. Since gravity causes the velocity to increase with time, the smaller interval of 0. Time Interval Average Velocity in metres per second 1 t 1. The average velocity over the time interval 1 t 1 h is s 1 h s 1 average velocity h 5 1 h 5 h 2 5 10h 5h 5 h 2 10 5h, h 0.

The instantaneous velocity when t 1 is defined to be the limiting value of these average values as h approaches 0. The speed of an object is the absolute value of its velocity. It indicates how fast an object is moving, whereas velocity indicates both speed and direction relative to a given coordinate system.

What is the velocity of the ball at t 4? Velocity is only one example of the concept of rate of change. In general, suppose that a quantity y depends on x according to the equation y f x. From the diagram, it follows that the average rate of change equals the slope of the secant PQ of the graph of f x. What is the total cost of manufacturing items of the product? What is the rate of change of the total cost with respect to the number of units, x, being produced when x ?

An Alternative Form for Finding Rates of Change In Example 1, we determined the velocity of the pebble at t 1 by taking the limit of the average velocity over the interval 1 t 1 h as h approaches 0. We can also determine the velocity at t 1 by considering the average velocity over the interval from 1 to a general time t and letting t approach the value 1. The velocity of an object is given by v t t t 4 2. At what times, in seconds, is the object at rest? Give a geometrical interpretation of the following, where s is a position function.

Give a geometrical interpretation of lim. Use the graph to answer each question. Between which two consecutive points is the average rate of change of the function greatest? Is the average rate of change of the function between A and B greater than or less than the instantaneous rate of change at B? Sketch a tangent to the graph between points D and E such that the slope of the tangent is the same as the average rate of change of the function between B and C.

Part B 7. A construction worker drops a bolt while working on a high-rise building m above the ground. After t seconds, the bolt has fallen a distance of s metres, where s t 5t2, 0 t 8. Find the average velocity during the first, third, and eighth seconds. Find the average velocity for the interval 3 t 8.

Find the velocity at t 2. The function s t 8t t 2 describes the distance s, in kilometres, that a car has travelled after a time t, in hours, for 0 t 5. Find the average velocity of the car during the following intervals: i from t 3 to t 4 ii from t 3 to t 3. Use the results of part a to approximate the instantaneous velocity of the car when t 3.

Find the velocity at t 3. Application 9. Suppose that a foreign-language student has learned N t 20t t2 vocabulary terms after t hours of uninterrupted study. How many terms are learned between time t 2 h and t 3 h? What is the rate in terms per hour at which the student is learning at time t 2 h?

A medicine is administered to a patient. Find the rate of change of the amount, M, 2 h after the injection. What is the significance of the fact that your answer is negative? The time, t, in seconds, taken for an object dropped from a height of s metres to reach the ground is given by the formula t 5. Determine the rate of s change of the time with respect to the height when the height of an object is m above the ground. Find the rate of change of temperature with respect to height at T h h2 a height of 3 km.

A spaceship approaching touchdown on a distant planet has height h, in metres, at time t, in seconds, given by h 25t2 t When and with what speed does it land on the surface? A manufacturer of soccer balls finds that the profit from the sale of x balls per week is given by P x x — x2 dollars. Find the profit on the sale of 40 soccer balls. Find the rate of change of the profit at the production level of 40 balls per week.

Using a graphing calculator, graph the profit function and from the graph determine for what sales levels of x the rate of change of profit is positive. Let a, b be any point on the graph of y 1x, x 0. Prove that the area of the triangle formed by the tangent through a, b and the coordinate axes is 2. Show that the rate of change of the cost is independent of fixed costs. A circular oil slick on the ocean spreads outward. Find the approximate rate of change of the area of the oil slick with respect to its radius when the radius is m.

Show that the rate of change of the volume of a cube with respect to its edge length is equal to half the surface area of the cube. We shall use this definition to evaluate some limits. Note: This is an intuitive explanation of the limit of a function.

A more precise definition using inequalities is important for advanced work but is not necessary for our purposes. Copy and complete the table of values. As x approaches 2 from the left, starting at x 1, what is the approximate value of y? As x approaches 2 from the right, starting at x 3, what is the approximate value of y? Graph y x2 1 using graphing software or graph paper.

Using arrows, illustrate that as we choose a value of x that is closer and closer to x 2, the value of y gets closer and closer to a value of 3. Explain why the limit of y x2 1 exists as x approaches 2, and give its approximate value. The graph shown on your calculator is a straight line f x x 1 whereas it should be a line with the point 1, 2 deleted f x x 1, x 1.

We can estimate lim f x. We say that the limit at x 1 exists only if the value approached from the left is the same as the value approached from the right. From this investigation, we x 1 2. We write this as lim f x lim x 1 0. These two limits are referred to as one-sided limits, because in each case only values of x on one side of x 1 are considered.

We can now summarize the ideas introduced in these examples. 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 lim f x L lim f x.

What do you think is the appropriate limit of these sequences? Explain a process for finding a limit. Communication 3. Write a concise description of the meaning of lim f x Calculate each limit. Part B 6. For the function y f x in the graph below, find the following: a. Use the graph to find the limit, if it exists. Evaluate each limit.

Communication In each of the following, find the indicated limit if it exists. Sketch the graph of the function. Sketch the graph of any function that satisfies the given conditions in each case. Let f x mx b, where m and b are constants. The fish population in thousands in a lake at time t, in years, is modelled by the function p t 3 11 t2, 0 2 t t2, 6 t 2 11 8 6 This function describes a sudden change in the population at time t 6, due to a chemical spill.

Sketch the graph of p t. Evaluate lim p t and lim p t. Determine how many fish were killed by the spill. At what time did the population recover to the level before the spill? This means that in finding the limit of f x as x approaches a, there is no need to consider x a. In fact, f a need not even be defined.

The only thing that matters is the behaviour of f x near x a. Therefore, it appears that lim 3x2 4x — 1 12 8 1 It is possible to evaluate limits using the following properties of limits, which can be proved using the formal definition of limits. This is left for more advanced courses. Properties of Limits For any real number a, suppose f and g both have limits at x a.

This is of special interest when direct substitution results in an 0 indeterminate form. In such cases, we look for an equivalent function that 0 agrees with f for all values except the troublesome value x a. Here are some examples. The next step is to simplify the function by factoring and reducing to see if the limit of the reduced form can be evaluated. Therefore, x 2x 3 lim x 1 4. Find lim by rationalizing. Let u x, and rewrite lim in terms of u.

Rationalizing the term x 8 3 is not so easy. However, the expression can be simplified by substitution. As x approaches the value 0, u approaches the value 2. Illustrate with a graph. Evaluate lim 9 x2. Explain why the limit as x approaches 3 cannot be determined. The graph of f x below. Therefore, lim 9 x2 0. The function is not defined for x 3. In this section, we have learned the properties of limits and developed algebraic methods for evaluating limits. The examples in this section have complemented the table of values and graphing techniques introduced in previous sections.

How do you find the limit of a rational function? Give reasons for your answer. Use a graphing calculator to graph the function and to estimate the limit. Then find the limit by substitution. Evaluate the limit of each indeterminate quotient. Evaluate the limit by change of variable.

Evaluate each limit, if it exists, using any appropriate technique. By using one-sided limits, determine whether the limit exists. Illustrate the results geometrically by sketching the graph of each function. In the table, one mole of hydrogen is held at a constant pressure of one atmosphere. The volume V is measured in litres and the temperature T is measured in degrees Celsius. By finding a difference row, show that T and V are related by a linear relation. Find the linear equation V in terms of T.

Solve for T in terms of V for the equation in part b. Show that limT is approximately Using the information found in parts b and d, draw a graph of V versus T. Show, using the properties of limits, that if x 4 7. If lim f x 3, use the properties of limits to evaluate each limit.

If lim x 1 and lim x 2, then evaluate each limit. Evaluate lim of the indeterminate quotient. Does lim exist? For what values of m and b is the statement lim 1? A graph that is not continuous at a point sometimes referred to as being discontinuous at a point has a break of some type at the point. The following graphs illustrate these ideas.

Discontinuous at x 1 a. Discontinuous at x 1 2 4 6 d. Discontinuous at x 1 y 10 8 8 6 6 4 4 y Vertical asymptote 2 2 —2 Hole x —2 10 y x x 2 4 —1 6 1 2 3 What conditions must be satisfied for a function f to be continuous at a? First, f a must be defined. The curves in figure b and figure d are not continuous at x 1 because they are not defined at x 1.

A second condition for continuity at a point x a is that the function makes no jumps there. This condition is satisfied if lim f x f a. Graph the function f x b. Find lim f x. Find f 1. From the graph lim f x 2. If a function is not continuous at x 2, give a reason why it is not continuous. The function f is continuous at x 2 since f 2 6 lim f x. The function g is not continuous at x 2 because g is not defined at this point.

The function F is not continuous at x 2 because F 2 is not defined. Draw the graph for each function in Example 2. Which of the graphs are continuous, contain a hole or a jump, or have a vertical asymptote? Given only the defining sentence of a function y f x such as 8x 9x 5 , explain why the graphing technique to test for continuity f x x2 x 3 on an interval may be less suitable. Find where f x x2 x continuous. How can looking at a graph of a function help you tell where the function is continuous?

What does it mean for a function to be continuous over a given domain? What are the basic types of discontinuity? Give an example of each. Find the value s of x at which the functions are discontinuous. Find all values of x for which the given functions are continuous.

Examine the continuity of g x x 3 at the point x 2. Sketch a graph of h x x 1, x 3 5 x, x 3 and determine if the function is continuous everywhere. Sketch a graph of f x x2, x 0. The sides of the half-pipe are very steep S but it is not very steep near the base B.

Then we find the gradient between the two. We can use the formula for the curve to produce a formula for the gradient.

Higher Outcome 3. Special Points Higher Outcome 3. This can be used as follows ….. Leibniz Notation Leibniz Notation is an alternative way of expressing derivatives to f' x , g' x , etc.

So you will never obtain a negative by squaring any real number. It is not an accurate scale drawing. Process a Find where the curve cuts the co-ordinate axes. In this section we shall concentrate on the important details to be found in a small section of graph. In commerce or industry production costs and profits can often be given by a mathematical formula.

Optimum profit is as high as possible so we would look for a max value or max TP. Optimum production cost is as low as possible so we would look for a min value or min TP.