BSc 1st Year Mathematics Books: In the last few months, we have got hundreds of requests regarding the mathematics study material for BSc. Well, we have already published another post where we have shared important questions in mathematics for BSc 1st year. Mathematics can be a subject of both honors degree or plain BSc. The plane BSc comprises three to four subjects while in an honors degree you have to study only a single subject with deep knowledge.
Talking about the syllabus of mathematics for BSc 1st year, it is divided into two semesters, i. You can also download the complete university wise syllabus for BSc Mathematics. As we have seen, the syllabus of BSc mathematics for 1st year comprises two semesters. Disclaimer: The syllabus of BSc mathematics provided on our website may vary from the syllabus of your university.
I think you have got to know enough about the syllabus, let us share the download links of mathematics books for BSc 1st year. The books for BSc 1st year mathematics are available in pdf format below. Note: This book might contain extra topics that may not be in your BSc 1st-year mathematics syllabus. So you should only download those files that are in your syllabus.
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Hi Deepak for now, BSc 6th sem notes are not available in pdf format. However, you can download question banks using the below links:. D Therefore,. Lesson 10 Supplementary Exercises 1. Which among the following functions have an inverse? Find if. Lesson Graphs of Inverse Functions PY Learning Outcome s : At the end of the lesson, the learner is able to represent an inverse function through its table of values and graph, find the domain and range of an inverse function, graph inverse functions, solve problems involving inverse functions.
O Lesson Outline: C 1. Domain and range of a one-to-one function and its inverse D Graphing Inverse Functions First we need to ascertain that the given graph corresponds to a one-to-one function E by applying the horizontal line test. If it passes the test, the corresponding function is one-to-one. EP Given the graph of a one-to-one function, the graph of its inverse can be obtained by reflecting the graph about the line.
Graph if the graph of restricted in the domain is given below. What is the range of the function? What is D the domain and range of its inverse? Take the reflection of the restricted graph of across the line. PY The range of the original function can be determined by the inspection of the graph. The range is. Verify using techniques in an earlier lesson that the inverse function is given by O.
Is this true for all one- to-one functions and their inverses? Find the inverse of using its given graph. D 68 All rights reserved.
Applying the horizontal line test, we verify that the function is one-to-one. Since the graph of is symmetric with respect to the line indicated by a dashed line , its reflection across the line is itself. Therefore the inverse of is itself or.
PY Verify that using the techniques used in the previous lesson. O Example 3. Find the inverse of using the given graph. Applying the horizontal line test, we confirm that the function is one-to-one. Reflect the graph of across the line to get the plot of the inverse function.
D The result of the reflection of the graph of is the graph of Therefore,. D b Find the equation of its asymptotes. EP a From our lessons on rational functions, we get the following results: Domain of Range of D b Using techniques from the lesson on rational functions, the equations of the asymptotes are Vertical asymptote: Horizontal asymptote: 70 All rights reserved.
In fact, the asymptotes could also be obtained by reflecting the D original asymptotes about the line. Vertical asymptote: Horizontal asymptote: E d The domain and range of the functions and its inverse are as follows: EP Domain Range We can make the observation that the domain of the inverse is the range of the D original function and the range of the inverse is the domain of the original function. In the examples above, what will happen if we plot the inverse functions of the inverse functions?
If we plot the inverse of a function, we reflect the original function about the line. If we plot the inverse of the inverse, we just reflect the graph back about the line and end up with the original function. This result implies that the original function is the inverse of its inverse, or.
Solving problems involving inverse functions We can apply the concepts of inverse functions in solving word problems involving reversible processes.
You asked a friend to think of a nonnegative number, add two to the number, square the number, multiply the result by 3 and divide the result by 2. If the result is 54, what is the original number? Construct an inverse function that will provide the original number if the result is given. We first construct the function that will compute the final number based on PY the original number.
Following the instructions, we come up with this function: O The graph is shown below, on the left. This is not a one-to-one function because the graph does not satisfy the horizontal line test.
However, the instruction indicated that the original number must be nonnegative. The domain of the function must thus be restricted to C , and its graph is shown on the right, below.
Interchange the x and y variables: 72 All rights reserved. Solve for y in terms of x: Since we do not need to consider PY Finally we evaluate the inverse function at to determine the original number: O The original number is 4. Engineers have determined that the maximum force in tons that a C particular bridge can carry is related to the distance in meters between it supports by the following function: D How far should the supports be if the bridge is to support 6.
Construct an inverse function to determine the result. EP To lessen confusion in this case, let us not interchange and as they denote specific values. Solve instead for in terms of : D The inverse function is. The supports should be placed at most 6. If is restricted on the domain , what is the domain of its inverse? The domain of the inverse of is just the range of. Therefore the domain of is 2. Given the graph of below, sketch the graph of its inverse. So we get: EP D 74 All rights reserved.
Using algebraic methods, construct the inverse of. Is the function you get the same as the sketch of the inverse in the previous number? To get , we first interchange x and y in. So we get. We then isolate y So we get. However, the graph of that will result in a parabola PY opening downwards while the sketch we have in number 2 was just half that parabola. This occurs because the function must be one-to-one to have an inverse.
Lesson 11 Supplementary Exercises O 1. Find the domain and range of the inverse of with domain restriction. Give the vertical and horizontal asymptotes of.
Give the vertical and horizontal asymptotes of its inverse. At what point will the graph of and its inverse intersect? The formula for converting Celsius to Fahrenheit is given as where C is the temperature in Celsius and F is the temperature in Fahrenheit.
Find the formula for converting Fahrenheit to Celsius. If the temperature in a EP thermometer reads Explain why the function is one-to-one, even if it is a quadratic function. Find the inverse of this function D and approximate the length of a single fish if its weight is grams. Lessons 9 — 11 Topic Test 1 1.
True or False [6] a A linear function is a one-to-one function. Identify if the following are one-to-one functions or not. Which of the following functions have an inverse function? If so, find its inverse.
Sketch the graph of the inverse of the function. Find the inverse of the following functions: [15] a E b 2. Find the domain and range of the inverse of [10] EP 3. Find the asymptotes of the inverse of [10] 4.
Lesson Representing Real-Life Situations Using Exponential Functions Learning Outcome s : At the end of the lesson, the learner is able to represent real- life situations using exponential functions. Exponential functions 2. Population, half-life, compound interest 3. A better approximation is Let b be a positive number not equal to 1. Some of the most common applications in real-life of exponential functions and their transformations are population growth, exponential decay, and compound interest.
Suppose that the bacteria doubles every hours. Give an exponential model for the bacteria as a function of t. C The half-life of a radioactive substance is the time it takes for half of the substance to decay. Suppose that the half-life of a certain radioactive substance is 10 days D and there are 10g initially, determine the amount of substance remaining after 30 days, and give an exponential model for the amount of remaining substance.
We use the fact that the mass is halved every 10 days from definition of half-life. A starting amount of money called the principal can be invested at a certain interest rate that is earned at the end of a given period of time such as one year.
If the interest rate is compounded, the interest earned at the end of the period is 78 All rights reserved. The same process is repeated for each succeeding period: interest previously earned will also earn interest in the next period. De la Cruz invested P, Define an exponential model for this situation. How much will this investment be worth at the end of each year for the next five years? The investment is worth O P, Compound Interest.
Referring to Example 5, is it possible for Mrs. De la Cruz to double her money in 8 years? The Natural Exponential Function D While an exponential function may have various bases, a frequently used based is the irrational number e, whose value is approximately 2.
The enrichment in Lesson 27 will show how the number e arises from the concept of compound interest. Because e is a commonly used based, the natural exponential function is defined having e as the base. A large slab of meat is taken from the refrigerator and placed in a pre- heated oven. Construct a table of values for the following values of t: 0, 10, 20, 30, 40, 50, 60, and interpret your results.
Round off values to the nearest integer. Robert invested P30, after graduation. If the average interest rate is 5. The money has more than doubled in 15 years. There will be bacteria after 40 days.
The half-life of a substance is years. Use this model to approximate the Philippine population during the years , , , and Round of answers to the nearest thousand. A barangay has 1, individuals and its population doubles every 60 years.
Give an exponential model for the barangay. E Give an exponential model for a sum of P10, invested under this scheme. How much money will there be in the account after 20 years? The half-life of a radioactive substance is years. If the initial amount of the substance is grams, give an exponential model for the amount remaining after t years.
What amount of substance remains after years? D 81 All rights reserved. Lesson Exponential Functions, Equations, and Inequalities Learning Outcome s : At the end of the lesson, the learner is able to distinguish among exponential functions, exponential equations and exponential inequality.
PY The definitions of exponential equations, inequalities and functions are shown below. An exponential function expresses a relationship between two variables such as x and y , and can be represented by a table of values or a graph Lessons 14 and E Solved Examples EP Determine whether the given is an exponential function, an exponential equation, an exponential inequality, or none of these.
Lesson Solving Exponential Equations and Inequalities Learning Outcome s : At the end of the lesson, the learner is able to solve exponential equations and inequalities, and solve problems involving exponential equations and inequalities Lesson Outline: 1.
Solve exponential equations 2. Write both sides with 4 as the base. Write both sides with 2 as the base. Both and 25 can be written using 5 as the base. Solve the equation. Both 9 and 3 can be written using 3 as the base. E EP Example 4. The half-life of Zn is 2. How much time has passed? Using exponential models in Lesson 12, we can determine that after t EP minutes, the amount of Zn in the substance is. We solve the equation. Solved Examples Solve for x in the following equations or inequalities.
How much time will have elapsed when only 15 grams remain? The amount of substance after t hours. Lesson 14 Supplementary Exercises In Exercises , solve for x.
How much time is needed for a sample of Pd to lose Pd has a half-life of 3. A researcher is investigating a specimen of bacteria. She finds that the original bacteria grew to 2,, in 60 hours. How fast does the bacteria a double? Lesson Graphing Exponential Functions Learning Outcome s : At the end of the lesson, the learner is able to represent an exponential function through its a table of values, b graph, and c equation, find the domain and range of an exponential function, determine the intercepts, zeroes, and asymptotes of an exponential function, and graph exponential functions Lesson Outline: 1.
Domain, range, intercepts, zeroes, and asymptotes. In the following examples, the graph is obtained by first plotting a few points. Results PY will be generalized later on. O Step 1: Construct a table of values of ordered pairs for the given function.
As x decreases without bound, the function approaches 0, i. As x increases without bound, the function approaches 0, i. The domain is the set of all real numbers. The range is the set of all positive real numbers. It is a one-to-one function. It satisfies the Horizontal Line Test. The y-intercept is 1. There is no x-intercept. There is no vertical asymptote. Solved Examples PY 1.
Indicate the domain, range, y-intercept, and horizontal asymptote. Compare the two graphs. For both these functions, the base is greater than 1. Thus, both functions are increasing. The following table of values will help complete the sketch. Thus, the function is decreasing.
Lesson Graphing Transformations of Exponential Functions Learning Outcome s : At the end of the lesson, the learner is able to graph exponential functions.
Vertical and horizontal reflection 2. Stretching and shrinking 3. Some y-values are shown on the following table. The corresponding graphs are shown below.
D 92 All rights reserved. The graphs of these functions are shown below. The domain for all three graphs is the set of all real numbers. The y-intercepts were also multiplied correspondingly. The results of Example 2 can be generalized as follows. Vertical Stretching or Shrinking Let c be a positive constant. The results of Example 3 can be generalized as follows. Vertical Shifts Let k be a real number. The y-intercepts changed. Translating a graph horizontally does not change the horizontal asymptote.
The results of Example 4 can be generalized as follows. Horizontal Shifts Let k be a real number. Step 3: The y-intercept of f x 0,1 will shift to the left by one unit and down two units towards —1, —1. This is not the y-intercept of F x. Step 3: Since the graph will be stretched by 4 units, the y-intercept of g x 0,1 will be at 0,4 , then will be shifted again by 1 unit upward to get 0, 5.
This is the y-intercept of G x. Lessons 12 — 16 Topic Test 1 1. Solve for x. The population of a certain city doubles every 50 years. Graph the following functions. Label all intercepts and asymptotes.
Indicate the domain and range. Solve for x in the following equations. Solve the inequality [10] EP 3. Maine decides to participate in an investment that yields 3. Find the D bacteria population after half an hour. Lesson Introduction to Logarithms Learning Outcome s : At the end of the lesson, the learner is able to represent real- life situations using logarithmic functions and solve problems involving logarithmic functions.
Logarithms, including common and natural logarithms 2. In both the logarithmic and exponential forms, b is the base.
In the exponential form, c is an exponent; this implies that the logarithm is actually an exponent. Hence, logarithmic and exponential functions are inverses. In the logarithmic form logbx, x cannot be negative. The value of logbx can be negative. D Definition: Common logarithms are logarithms with base 10; logx is a short E notation for log10x. Definition: Natural logarithms are logarithms to the base e approximately EP 2. In other words, lnx is another way of writing logex. Rewrite the following exponential equations in logarithmic form, D whenever possible.
Rewrite the following logarithmic equations in exponential form. Find the value of the following logarithmic expressions. Some of the most common applications in real-life of logarithms are the Richter scale, sound intensity, and pH level. O In , Charles Richter proposed a logarithmic scale to measure the intensity of an earthquake.
He defined the magnitude of an earthquake as a function of its C amplitude on a standard seismograph. The following formula produces the same results, but is based on the energy released by an earthquake. EP The formula indicates that the magnitude of an earthquake is based on the logarithm of the ratio between the energy it releases and the energy released by the reference earthquake.
Suppose that an earthquake released approximately joules of energy. New York: McGraw-Hill. A 1-liter solution contains 0.
Find its pH level. Since there are 0. The pH level is —log 10—5. College algebra 3rd ed. Precalculus: Mathematics for calculus 6th ed. Solved Examples In numbers , find the value of the following logarithmic expressions. In numbers , rewrite the following logarithmic equations in exponential form, whenever possible. What is the magnitude in the Richter scale of an earthquake that released joules of energy?
How much more energy does this earthquake release than that O of the reference earthquake? The earthquake C 14 4. What is the D corresponding sound intensity in decibels? How much more intense is this sound than the least audible sound a human can hear?
Sound intensity E decibels. EP Lesson 17 Supplementary Exercises In numbers , find the value of the following logarithmic expressions. D In numbers , rewrite the following exponential equations in logarithmic form.
A 1-liter solution contains 10—8 moles of hydrogen ions. Determine whether the solution is acidic, neutral, or basic. Lesson Logarithmic Functions, Equations, and Inequalities Learning Outcome s : At the end of the lesson, the learner is able to distinguish among logarithmic function, logarithmic equation, and logarithmic inequality.
Logarithmic equations, logarithmic inequalities, and logarithmic functions The definitions of exponential equations, inequalities and functions are shown below. A logarithmic function expresses a relationship between two variables such as x and y , and can be represented by a table of D values or a graph Lesson Solved Examples E Determine whether the given is a logarithmic function, a logarithmic equation, a logarithmic inequality or neither. Answer: Logarithmic Equation D 4.
Answer: Logarithmic Inequality 5. Lesson Basic Properties of Logarithms Learning Outcome s : At the end of the lesson, the learner is able to apply basic properties of logarithms and solve problems involving logarithmic equations Lesson Outline: 1. Basic properties of logarithms. Simplifying logarithmic expressions. Use the basic properties of logarithms to find the value of the following logarithmic expressions. Download PDF. Download PDF of Book. Download Practice Paper Book of Maths.
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