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Not Enough Time? This Schaum's Outline gives you: Practice problems with full explanations that reinforce knowledge Coverage of the most up-to-date developments in your course field In-depth review of practices and applications Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Schaum's Outlines-Problem Solved. To every point Q on 9to the left of the origin 0, - we assign a negative real number as its coordinate; the number -Q0,the negative of the distance between Q and 0. For example, in the diagram the point U is assumed to be a distance of one unit from the origin 0; therefore, the coordinate of U is - 1.
Clearly, every negative real number is the coordinate of a unique point on 9 to the left of the origin. The origin 0 is assigned the number 0 as its coordinate. Choosing a different origin, a different direction along the line, or a different unit distance would result in a different coordinate system. But if b is negative, its absolute value I b I is the corresponding positive number - b. U lul -c 1. Subtracting 3, -8 1 I 5 x 5 2.
To see this, notice that a 0. A product is positive if and only if both factors are of like sign. Similarly for part b. Choose two perpendicular lines in the plane of Fig. Let us assume for the sake of simplicity that one of the lines is horizontal and the other vertical. The horizontal line will be called the x-axis and the vertical line will be called the y-axis.
The origin for both coordinate systems is taken to be the point 0,where the axes intersect. The x-axis is directed from left to right, the y-axis from bottom to top. The part of the x-axis with positive coordinates is called the positioe x-axis, and the part of the y-axis with positive coordinates the positive y-axis. Consider any point P in the plane. Take the vertical line through the point P, and let a be the coordinate of the point where the line intersects the x-axis.
This number a is called the x-coordinate of P or the a6scissa of P. Now take the horizontal line through P, and let 6 be the coordinate of the point where the line intersects the y-axis. The number 6 is called the y-coordinate of P or the ordinate of P. Every point has a unique pair a, b of coordinates associated with it. We have limited ourselves to integer coordinates only for simplicity. Conversely, every pair a, 6 of real numbers is associated with a unique point in the plane.
To find the point with coordinates -2, 4 , start at the origin 0, move two units to the left and then four units upward. To find the point with coordinates - 1, - 3 , start from the origin, move one unit to the left and then three units downward. Given a coordinate system, the entire plane except for the points on the coordinate axes can be divided into four equal parts, called quadrants. All points with both coordinates positive form the first quadrant, quadrant I, in the upper right-hand corner see Fig. Quadrant I1 consists of all points with negative x-coordinate and positive y-coordinate; quadrants I11 and IV are also shown in Fig.
The points having coordinates a, 0 are the points on the x-axis. We wish to find a formula for the distance PIP2. Let R be the point where the vertical line through P, intersects the horizontal line through P,. Clearly, the x-coordinate of R is x2 ,the same as that of P,;and the y-coordinate of R is y,, the same as that of P,. By the Pythagorean theorem, 2. The x-coordinates of A, B, C are x,, x, x 2 , respectively. Solved Problems 2.
Use 2. Let the origin of a coordinate system be located at the right angle C; let the positive x-axis contain leg CA and the positive y-axis leg CB [see Fig. Let M be the midpoint of the hypotenuse. By the midpoint formulas 2. Find the area of each right triangle. Find the mid oints of the line segments with the following endpoints: a 1, - 1 and 7, 5 ; b 3, 4 and and The graph of i consists of all points a, b that satisfy the equation when a is substituted for x and b is substituted for y.
We tabulate some points that satisfy i in Fig. It is apparent that these points all lie on a straight line. In fact, it will be shown later that the graph of i actually is a straight line. These points suggest that the graph looks like what would be obtained by filling in the dashed curve. This graph is of the type known as a parabola.
As shown in Fig. The points on the hyperbola get closer and closer to the axes as they move farther and farther from the origin. Sometimes the equation of a circle will appear in a disguised form. Solved Problems 3. These points form a vertical line [Fig. These points form a horizontal line [Fig. Plotting several points suggests the curve shown in Fig.
Hence, the graph consists x - 4 2 of the single point 4, - 8. Hence, the graph consists of no points at all, or, as we shall say, the graph is the null set. But a, b is obtained by moving a - 2, b two units to the right. The reasoning is as in part c. Parts c and d can be generalized as follows. The argument is analogous to that for Problem 3 3 4. So - 2 1 and, therefore, I x I 2 3. Check on a graphing calculator. How is this graph related to that of y calculator. Sketch the graphs of the following equations. Obtain part e from part a. See Fig. Prove this. Solve the three resulting equations for D, E, and F.
It is the ratio of the change y, - y , -in the y-coordinate to the change x2 - x, in the x-coordinate. Y Y Fig. If another pair P3 x3,y3 and P4 x4,y4 is chosen, the same value of m is obtained. In fact, in Fig. Hence, AP, P, S is similar to M,P, R because two angles of the first triangle are equal to two corresponding angles of the second triangle.
Let us see what the sign of the slope indicates.
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Consider a line 9 that extends upward as it extends to the right. From Fig. Consider a horizontal line 9. Consider a vertical line 9. Hence, the expression Y, -Yl - x1 is undefined. The concept of slope is not defined for 9. Sometimes we express this situation by saying that the slope of 9is infinite.
First let us consider lines with positive slopes, passing through a fixed point P , x , , yJ. One such line is shown in Fig. Take another point, - P 2 x 2 ,y, , on A? Then, -by definition, the slope rn is equal to the distance RP,. Now as the steepness of the line increases, RP, increases without limit [see Fig. By a similar construction we can show that as a negatively sloped line becomes steeper and steeper, the slope steadily decreases from 0 when the line is horizontal to - CQ when the line is vertical [see Fig.
For any other point P x, y on the line, the slope rn is, by definition, the ratio of y - y, to x - xl. Hence, On the other hand, if P x, y is not on line 9 [Fig.
Schaum's Outline of Beginning Calculus, Third Edition | Angus & Robertson
Equation 4. So a point x, y is on line 14 if and only if it satisfies 4. Hence, the point 0, b lies on 9. Thus, b is the y-coordinate of the point where 9 intersects the y-axis see Fig. The number b is called the y-intercept of 9,and 4. Let R , be one unit to the right of P,, and R, one unit to the right of P,. Conversely, if different lines Yl and Y 2are not parallel, then their slopes must be different.
For if Yl and 9, meet at the point P [see Fig. Thus, we have proved: Theorem 4. Hence, the slope of A is 3, and the slope of the parallel line 9 also must be 3. For a proof, see Problem 4. Draw the line and determine whether the points 10,23 and 6, 12 are on the line. The line goes through the point 0, To draw the line, we need another point on the line. Hence, 2, 3 is a point on the line see Fig. We could have found other points on the line by substituting numbers other than 2 for x.
The two sides turn out to be equal; so 10, 23 is on the line. A similar check shows that 6, 12 is not on the line. By the midpoint formula, the coordinates of M are 3,2. Assume that 9, and 9, are perpendicular nonvertical lines of respective slopes m, and m,. Let be the line through the origin 0 and parallel to 9,, and let be the line through the origin and parallel to 14, [see Fig. Also is perpendicular to 9;since 9, is perpendicular to 14,. Let R be the point on 9fwith x-coordinate 1, and let Q be the point on 9 3 with x-coordinate 1 [see Fig.
Similarly, the y-coordinate of Q is m,. Then 9, is not parallel to 9,. Otherwise,by Theorem 4. Let Yl intersect Y 2at point P see Fig. Let LY3 be the line through P and perpendicular to Y1. Drop a perpendicular PQ to the positive x-axis. Consider each quadrant separately. This yields the line with slope - 1 and y-intercept 1. This line intersects the x-axis at 1,O. Hence, in the first quadrant our graph consists of the line segment connecting 1, 0 and 0, 1 see Fig. This yields the line with slope 1 and t' t' Fig. Likewise, in the third quadrant we obtain the segment connecting - 1,O and 0, - l , and in the fourth quadrant the segment connecting 0,- 1 and 1,O.
Also find the coordinates of a point other than 0, b on each line.
The relation between Fahrenheit and Celsius temperaures is given by a linear equation. The freezing point of water is 0" Celsius or 32" Fahrenheit, and the boiling point of water is " Celsius or " Fahrenheit. Find equations of: a the median from A to the midpoint of the opposite side; b the altitude from B to the opposite side; c the perpendicular bisector of side AB. Find out whether the points 12,9 and 6,3 lie on this line. Draw a diagram indicating the set.
If you expect to drive x miles per day, for what values of x would it cost less to rent the car from Heart? Solve part a using a graphing calculator. Prove the following geometric theorems by using coordinates. A parallelogram with perpendicular diagonals is equilateral a rhombus. Thus, it is the number U for which a, 0 lies on 9. Which lines do not have x-intercepts? These points can be found by solving simultaneously the equations that determine the graphs. Thus, there are two intersection points, 1, 3 and - 2. Thus, the intersection points are 2,4 and - 1, 1 see Fig.
Let the perpendicular from P to. If we can find the coordinates can be computed by the distance formula 2. Therefore, by Theorem 4. Let y represent the number of dollars per pound that buyers are willing to pay for mutton. Find the intersection of the graphs of the supply and demand equations. Therefore, the Pythagorean theorem may be used to give a second equation for the coordinates of the point of contact. Chapter 6 Symmetry 6. Although 1, 1 is on the parabola, 1, - 1 is not on the parabola.
The point Q symmetric to the point P x, y with respect to the origin has coordinates -x, -y. Hence, Symmetry of a graph about a point is defined in the expected manner. In particular, a graph Y is said to be symmetric with respect to the origin if, whenever a point P lies on Y, the point Q symmetric to P with respect to the origin also lies on 9. Hence, any straight line through the origin is symmetric with respect to the origin.
The line is not symmetric with respect to the x-axis, since -1, 1 is on the line, but -1, -l , the reflection of - 1, 1 in the x-axis, is not on the line. However, the converse is false, as is shown by Problem 6. If P has coordinates a, b , show that Q has coordinates 6, a. B bisects PQ; hence, its coordinates are given by 2. Thus, Q has coordinates b, U. Supplementary Problems 6. For example, the volume V of a cube is a function of the length s of a side.
Such a specific association of a number s3 with a given number s is what mathematicians usually mean by a function. In Fig. This is a straight line, with slope 1 and y-intercept 1 see Fig. This is the parabola of Fig. The numbers x for which a functionfproduces a valuef x form a collection of numbers, called the domain off.
On the other hand, the domain of the doubling function consists of all real numbers. The numbers that are the values of a function form the range of the function. The domain and the range of a functionfoften can be determined easily by looking at the graph offi The domain consists of all x-coordinates of points of the graph, and the range consists of all y-coordinates of points of the graph.
The domain consists of all real numbers, but the range is made up of all nonnegative real numbers. The graph is shown in Fig. When x bY Fig. It is a little harder to find the range of g. This is clear from the graph of g [see Fig. The first formula applies when x is negative, and the second when 0 5 x s 1. The domain consists of all numbers x such that x 5 1. The range turns out to be all positive real numbers. This can be seen from Fig. In other words, a function is defined to be a setfof ordered pairs such thatfdoes not contain two pairs a, b and a, c with b c.
The solid dots on the line at a and b means that a and b are included in the closed interval [a, b]. Hence, the range offis the closed interval [0, This is a semicircle. The domain of the function graphed in Fig. More generally, a function that is defined for all x and involves only even powers of x is an even function. Hence,fis not even. A function f is even fi and only if the graph off is symmetric with respect to the y-axis.
More generally, iff x is defined as a polynomial that involves only odd powers of x and lacks a constant term , thenf x is an odd function. By actual substitution, it is found that 1, -2, and 3 are roots. Theorem 7. By Theorem 7. Calculation reveals that 2 is a root. Hence Theorem 7. Thus, - 1 and 3 are the roots off x ; 3 is called a repeated root because x - 3 ' is a factor of x3 - 5x2 3x 9. Since the complex roots of a polynomial with real coefficients occur in pairs, a f b n , the polynomial can have only an even number possibly zero of complex roots.
Hence, a polynomial of odd degree must have at least one real root. Solved Problems 7. Since -x2 is defined for every real number x, the domain offconsists of all real numbers. To find the range, notice that x2 2 0 for all x and, therefore, - x2 I0 for all x. Find the domain and the range, and draw the graph off: Since [x] is defined for all x, the domain is the set of all real numbers. The range offconsists of all integers. Part of the graph is shown in Fig. It consists of a sequence of horizontal, half-open unit intervals.
A function whose graph consists of horizontal segments is called a stepfunction. Hence, the domain offis the set of all real numbers different from 1. Thus, the range consists of all real numbers except 2. Conversely, if a set d of points intersects each vertical line in at most one point, define a functionfas follows. In each case, we substitute the specified argument for all occurrences of x in the formula forf x.
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The integral roots if any must be divisors of f 1, f 2, f 4, f 5, f 10, f Calculation shows that 4 is a root. Hence, x - 4 must be a factor off x see Fig. Check your answers by means of a graphing calculator. It will be necessary to see how the valuef x of a functionfbehaves as the argument x approaches a given number.
This leads to the idea of limit. Thus,fis defined for all x for which the denominator x - 3 is not 0; that is, for x 3. What happens to the value f x as x approaches 3? Well, x2 approaches 9, and so x2 - 9 approaches 0; moreover, x - 3 approaches 0. For instance, when x approaches 4, x2 - 9 approaches 7 and x - 3 approaches 1. Let us write them down explicitly. If c is a constant andfis a function, lim c. If x lim x x-ro does not exist. As x approaches 1 from the left that is, through values of x f x approaches 1. The value of L - o r the very existence of L-is determined by the behavior offnear a, not by its value at a if such a value even exists.
Solved Problems 8. So, by Property V, b Here it is necessary to proceed indirectly. But that is not the case. However, as x approaches 2 from the left that is, with x 8. This limit will be important in the h study of differential calculus. However, since 1 is a root of - 1 produces the factorization x3 - 1, x - 1 is a factor of x 3 - 1 Theorem 7.
Remember that I x - a I 0. The number 6 depends on the given number E; if is made smaller, then 6 may also have to be chosen smaller. The precise version just given for the limit concept is called an epsilon-delta definition because of the traditional use of the Greek letters 6 and 6. Let 6 be the minimum of 6, and 6,. As x approaches 0 from the left, f x approaches - 1. As x approaches 0 from the right,f x approaches 2. The graph off is shown in Fig.
For the function graphed in Fig. Of course, in that case, the absolute value Ix I becomes larger without bound. Similarly, lim f x is undefined. Iff x and g x are polynomials and the degree of g is smaller than the degree Solved Problems 9. If a, and bk arc the leading cocfiuents off and g, respectively, then the limit is equal to lim?!
Thus, increases without bound. Thus, by Rule B of Section 9. The horizontal asymptotes may be found by Rule B of Section 9. Which of the following holds? This can be made precise in the following way. Definition: A functionfis said tcD be continuous at a if the following three conditions hold: i lirn f x exists. However, condition iii fails: 1 0. The function is continuous at every point different from 0. Notice that there are no gaps or jumps in the graph off see Fig. The discontinuities show up as jumps in the graph of the function.
If a functionfis not continuous at a, thenfis said to have a remooable discontinuity at a if a suitable change in the definition offat Q can make the resulting function continuous at a. I I Fig. The discontinuities of the functions in examples c and d above are not removable.
A discontinuity of a function f a t a is removable if and only if lim f x exists. In that case, the value x-a of the function at a can be changed to limf x. I f f i s continuous at every number of its domain, then we simply say thatfis continuous or thatfis a continuousfunction. This follows from example b of Property V in Section 8. This follows from Property VI in Section 8. There are certain properties of continuity that follow directly from the standard properties of limits 8.
Assumefand g are continuous at a. If c is a constant, the function cfis continuous at a. Definition: A functionfis continuous ouer [a, b] if: i fis continuous at each point of the open interval a, b. For x - 1,fis continuous, sincefis the quotient of two continuous functions with nonzero denominator. See the graph off in Fig. Find the points at whichf is discontinuous.
At those points, determine whether f is continuous on the right or continuous on the left or neither. It follows thatfis discontinuous at each integer. Therefore, there are no points of discontinuity other than the integers. At each point of discontinuity, determine whether the function is continuous on the right or on the left or neither.
Continuity on the left holds at 0, since the value at 0 is the number approached by the values assumed to the left of 0. At 1 the function is continuous neither on the left nor on the right, since neither the limit on the left nor the limit on the right equalsf 1. In fact, neither limit exists. Continuity on the left holds at 0, but neither continuity on the left nor on the right holds at 1.
Isfcontinuous over: for k x s 2 Yes, sincefis continuous on the right at 0 and on the left at 1. For each point of discontinuity, determine whether it is removable. X CHAP. This discontinuity is x2 - 1 removable. So, if we define the funcx-1 x-1 x-. Draw the graphs of the functions.
Determine whether the discontinuities are removable. If it exists, find its x-0 value. Conversely, any fixed irrational number can be approached arbitrarily closely through rational numbers. At each point P of a circle, there is a line 9 such that the circle touches the line at P and lies on one side of the line entirely on one side in the case of a circle.
For the curve of Fig. Let us develop a definition that corresponds to these intuitive ideas about tangent lines. Let P be a point of the graph having abscissa x. Q will be close to P if and only if h is close to 0 becausefis a continuous function. Some of the positions of Q have been designated as Q1,Q2,Q 3 , These lines are getting closer and closer to what we think of as the tangent line 5 to the graph at P.
Hence, the slope of the line PQ will approach the slope of the tangent line at P; that is, the slope of the tangent line at P will be given by What we have just said about tangent lines leads to the following precise definition. Definition: Let a function f be continuous at x. By the tangent line to the graph off at P x, f x is meant that line which passes through P and has slope Solved Problems Find the slope-intercept equation of the tangent line at the point 0,4 of the graph. Draw the graph offand show the tangent line at 0,4. Since the line passes through 0,4 , the y-intercept b is 4.
I By Problem Thus, the expression defines a function, called the derivative off. The derivative is so important in all parts of pure and applied mathematics that we must devote a great deal of effort to finding formulas for the derivatives of various kinds of functions. In this case, the derivative is independent of x.
For this purpose, several rules of differentiation will be proved. For proofs of i and ii , see Problem For a proof, see Problem RULE 4. To differentiate a polynomial, change each nonconstant term akxk to kakxk-' and drop the constant term if any. We can prove the rule by mathematical induction. For example, The tangent line F to the graph at xo, yo goes through 0, 1 if and only if F is the line 9 that connects xo, y o and 0, 1.
But that is true if and only if the slope m y of. T is the same as the slope my of 9. Supplementary Problems Find D, 3x7 - - x5 d Find Findf' x for arbitrary x. Letfbe differentiable that is,f' exists. Hint: f x This proves: Tkorem In other words, the converse of Theorem The sharp corner in the graph is a tip-off. Where there is no unique tangent line, there can be no derivative. X Fig. Iffand g are differentiable at x and if g x 0, then For a proof, see Problem Moreover, since g is continuous at x by Theorem Dx g x [by Property VI of limits and differentiability of g] [by Property I1 of limits and continuity of g] [g x I2 Dx g x Having thus proved that we may substitute in the product rule proved in Problem When k is positive, this is just Rule 4 Chapter Thus, in Fig.
The value at a relative minimum need not be the smallest value of the function; for example, in Fig. By a relative extrernum is meant either a relative maximum or a relative minimum.
Points at which a relative extremum exists possess the following characteristic property. The theorem is intuitively obvious. But at a relative maximum or relative minimum, the tangent line is horizontal see Fig. For a rigorous proof, see Problem The converse of Theorem But from the graph offin Fig. Letfbe a function defined on a set d and possibly at other points, too , and let c belong to b. Then f is said to achieve an absolute maximum on Q at c if f x f d for all x in 8. If the set d is a closed interval [a, b ] , and if the functionfis continuous over [a, b] see Section Read more about the condition.
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