Algebra/Pre-calc Review The following pages contain various algebra and pre-calculus topics that are used in the stud of calculus. These pages were designed so that students can refresh their knowledge of topics to help better prepare themselves for college math courses. Each section contains practice problems for ou to tr, but please do not feel ou need to do all of the problems. Contents Eponents Radicals and Rational Eponents Adding Fractions..6 Functions..7 Logarithms 9 Trigonometr Factoring...8 Quadratic Equations. Graphs...
Eponents Laws of Eponents Assume a and b are real numbers and m and n are integers.. m n m n a a a. ( ab) n a n b n Eamples using the Laws of Eponents. a.. a. ( ) 7 5 b. 5 b. (5) 5 85. a a m n a mn. a. 5 7 b. 7 6 m. a n a mn. a. z z b. 6 5 5 565 6 n n a a n b b 5. b 5. a. 8 b. 5 5 5 Zero Eponent If a is a nonzero real number, then Eamples of Zero Eponents 6. a 6. a. b. Negative Eponent If a is a nonzero real number and n is a nonzero integer, then 7. a n n a Eamples of Negative Eponents 7. a. b. 6 Watch out for the following common eponent mistakes!. Eponents applied to polnomials: remember, these need to be multiplied out! ab a b. The correct wa is a b a ab b. Parentheses and negative signs: a. 6 b. 6 c. 8 In part (a), the eponent is applied to the number ; notice the even eponent makes the outcome positive. In part (b), the eponent is applied onl to the number, so the outcome is negative. In part (c) the eponent is applied to the number, but since the eponent is odd the outcome is negative.. Negative eponents: remember, a negative eponent requires a reciprocal to make it positive! Note:. The correct wa is.
Practice Problems Use the properties of eponents to simplif each eponential epression... ( ) ( ) 5 7. r. s 8. 6 ( 5) ( 5) 5. 5m 9 8. m n. 8. 6. 5 9. p q. Simplif each of the following so that no negative eponents remain.. 6. 5 9.. z. 5 7... 5 5. 5 8. 5.. 5m n 5 Perform the indicated operations. Write answers using onl positive eponents. 5. 6. 5 5 6 7. 9 9 8. 9.. 77r 7 r. r s 5. a a a. 7. 7 5 7 6 5. 6. (6 z) z
Radicals and Rational Eponents Definitions of n a and n a Rational Eponents For an positive integer n, n n a a Note that n a is not a real number if a and n is even. Assume m and n are integers with n m a a a mn n m or equivalentl mn n m n m a a a n Properties of Radicals Eamples using the Properties of Radicals Assume a, b, n a, and n b are real numbers. n a a a. n n n. a. b. 8 mn n This is a special case of a a. m a ba b ab ab n n.. n n n n This is a special case of the rule m m m ab a b (p.). n n n a a a a n n b from the eponents section where b b b b b. 8 8 6. 5 5 7 Watch out for the following... 8 8 8 ; remember, if there is no inde given it means square root. ; when switching from radical notation to eponential notation, remember that the root goes on the bottom of the fractional eponent. Think of it like a tree; the roots are alwas at the bottom!
5 Practice Problems Simplif each of the following.. 5. 8. 6. 8 5 5. 6. 9 7. 8 6. 6r 6 8. 7. 6a 56 9. 5. 6 Perform the indicated operations. Write answers using onl positive eponents. 5. m m. z z 5. a b a b 6. 6 z z 6 5 7. 8. 5
6 Adding Fractions In arithmetic we know that the sum or difference of fractions with same denominator is given b the sum or difference of the numerators divided b the common denominator. If fractions do not have a common denominator one must be obtained before addition or subtraction can take place. Eample: Find the sum: 7 6 6 6 get denomintor of 6 for both fractions add numerators of fractions together When we add or subtract algebraic fractions the method is eactl the same. First a common denominator must be obtaineded. To find the common denominator:. Factor completel each denominator.. The least common denominator is the product of all the different factors with each factor raised to the highest power to which it appears in an one factorization. Eample: Find the difference: h h h h h h h h the common denominator is h get denominator of h for both fractions h subtract the numerators h h h hh h Find the sum or difference. = simplif and distribute the negative sign h h Practice Problems.. 5. z z. 5 8 5. 9 6 6 6. 9 7 5n 7. n n 8. a a 9.
7 Functions Functions are most commonl written as f, where is the input value and is the output value. In other words, the value for is completel determined b the value of. Sometimes is referred to as the independent variable and as the dependent variable. The domain of a function is the set of all the values that can be plugged into the function. The range of a function is the set of all the possible outputs of the function. Consider the function f. Find f a To do this, ou will need to plug a into the function wherever there is an. So, f a a a aa a. No matter what the input value is, it will alwas be plugged in wherever there is an. For eample, find f ( ).. f ( ) ; the is plugged in for the values. Eamples: a. Let f Find f (). ( ). b. Let f ( ). Find f( a ). f () a a a f( a) a a a a a 6 9 a a 6 a a a c. Let g. 6 Find g h. d. Let g. Find. g a h g a h g h 6 h 6 6h g a h g a a h a h h a ahh a h ah h h
8 Practice Problems Let h. h. h a 7. if 9. Find: h. h a 5. ha h 5 8. if. h 6. ha 9. h a Let f. f. f p. Find:. f. f p. f 6 5. f p f p 6. f a h 7. f h 8. f a h f a h
9 Logarithms Logarithms are closel related to eponents. In general: log b is equivalent to b. For eample the equation can be written as log. This is read as log base of equals. To solve log6, we want to think: what power do we raise to in order to give us 6? Or, if 6, what is? We know 6, so. Logarithmic functions For each b, there is a function called log-base-b defined b f log for all. b Properties of Logarithms. log ( ) log log b b b Eamples Using Logarithm Properties. log log (5) log log 5. log log log b b b log log log. k. log ( ) klog. log b b b b.. log 5 log5 log5 5 log 5 5. log b b log 5. 6. log b 6. log5 Note: The properties of logarithms can be performed in either direction. In certain cases it might be necessar to write epressions as a single logarithm. E: logb logb logb logb In other cases it might be necessar write single logarithm as the sum/difference of logs. E: logb logb logb Natural Log and Base e A common base that pops up in man applications is base e, where e.78... We call log e the natural log, which is often written as ln. The two functions ln and e are commonl used in the stud of calculus, and it is a good idea to familiarize ourself with them. The function ln has domain and the function e has domain all real numbers.
Practice Problems Epress in eponential form.. log 7. log 8. log55 Epress in logarithmic form.. Evaluate. 8 5.. 6. 8 7. log 6 8. log 9. log 5 5 5 7 Solve.. log 7 6. log6. log 8 Use the properties of logarithms to epress the following as the sum/difference of logs.. log 6. log 5. log b z 7. log z 5. log 6 8. log 9. log b. log b( ). log b 5. log b. log b. 7 log b z Epress as a single logarithm. 5. log log 6. log log6 7. 5log log9 8. log log log z 9. log5 log5 5. log log t
Trigonometr Radians Like degrees, radians give us a wa to measure angles. One radian is the measure of the angle on the unit circle, where the arc it intercepts is equal to. The circumference of a circle is given b r, where r is the length of the radius. B using this definition it is eas to see that there are radians in a complete circle. radians 6 and radians 8 Therefore, radians and 8 8 radian = In calculus radians will ALWAYS be used for measuring angles.
Right Triangle Definition For this definition we assume that. opposite sin hpotenuse opposite hpotenuse adjacent adjacent cos hpotenuse opposite tan adjacent Trick: A wa to remember this is SOH CAH TOA, where SOH refers to Sine Opp Hp, CAH refers to Cosine Adj Hp, and TOA refers to Tangent Opp Adj. Unit Circle Definition For this definition, is an angle and the circle has a radius of unit. (, ) sin csc = cos sec = tan cot Note: The Right Triangle Definition and the Unit Circle Definition are equivalent for. Both definitions will be useful to know when studing calculus.
The Unit Circle The unit circle can be helpful in remembering certain values of sin and cos. The image below depicts the important values along the unit circle. Note: It is most helpful to know the values of sine and cosine at each of the intercepts,,, and and in the first quadrant. From these values all the other values of sine and cosine can be determined using knowledge of the signs of and in other quadrants. For eample, if 5 5 ou need the value of sin, it is determined b the value of sin, with a sign adjustment, since is in the nd quadrant, where is negative.
Practice Problems Without use of a calculator, find the value for each of the following.. sin. cos 7. cos. sin. tan. cos 5. sin 8. tan. cos 6. cos. tan 6. cos 9. tan. sin 5. sin 6. cos 7. sin 8. sin 6
Formulas and Identities Tangent and Cotangent Identities sin cos tan cot cos sin Reciprocal Identities csc sin sin csc sec cos cos sec cot tan = tan cot Pthagorean Identities sin cos tan sec cot csc Double Angle Formulas sin sincos cos cos sin cos sin Other Identities sin ( cos ) cos ( cos ) 5 Practice Problems Use trigonometric identities to rewrite each of the following.. sin. tan. cos tan. sin cos 5. cos sin tan 6. sec cos 7. cos sin tan 8. sec 9. tan
6 Inverse Trigonometric Functions In general the inverse function undoes whatever the function does. For eample, if inverse function of f and f 7then f 7. f is the As another eample, consider the function f( ). This function takes the input value, multiplies it b, and then adds to get a -value f. The inverse of this function, f, will do just the opposite to the input value, subtracting from it and then dividing it b. The easiest wa to find the inverse of a given function is to switch the variables and solve for. f f. E.. Find (switching and ) (solving for ) f ( ) is the inverse function Inverse functions have the propert that its inverse results in simpl. Note in our eample: f f f f ; is, that composing a function with f f and f f
7 Inverse Trigonometric Functions (Continued) Finding inverse trigonometric functions using the method above would be a bit difficult. For eample consider the function sin, which means is an angle whose sine is. If we switch the variables we get sin, but we cannot use algebra to solve for as we did in the eample above. This is wh we define special names for inverse trigonometric functions. arcsin is the inverse function of sin for. In other words, arcsin is the angle between and whose sine is. E : Find the value of arccos. cos (using the values from the unit circle) so, arccos E : Find the value of arctan. sin tan cos (using identities) sin cos (using values from the unit circle) so, tan so, arctan We can define inverse functions for all the trigonometric functions. Here are the most commonl used inverse trigonometric functions. Function arcsin arccos arctan Domain Range
8 Practice Problems Without a calculator find each of the following values.. arcsin. arctan. arcsin. arcsin 5. arctan 6. arccos 7. arcsin 8. arcsin 9. arctan
9 Factoring Being able to factor polnomials is an essential skill needed in calculus. Below are some of the techniques used to factor polnomials. Factoring out the Greatest Common Factor (GCF) Eamples: ( ) GCF of and is The Greatest Common Factor (GCF) is the largest factor that divides into ever term in a given polnomial. 6 9 ( ) GCF is Perfect Square Trinomials Difference of Two Squares ( )( ) General Trinomials ( ( ab) ab) ( a)( b) Difference of Two Cubes ( )( ) Sum of Two Cubes ( )( ) Perfect Square Trinomial Eample 6 9 6 6 Difference of two Squares Eample 9 Trinomial Factoring Eamples 7 5 6 7 Difference of Two Cubes Eample 8 Sum of Two Cubes Eample 7 9 Remember! Alwas tr to factor out the greatest common factor first! A polnomial ma look like it is not factorable, but b taking out a common factor ou ma be able to factor it with ease.
Practice Problems Factor each trinomial.. 7. 6 8. 5. 5 5. 7 6. 8 6 7. 6a 8a 8. 8 9. 9.. h h 8. Factor the difference of two squares.. 9a 6. 6 5 5. 8 6. 5 9 7. m 6 8. 56 65
Solving Quadratic Equations Give a quadratic equation, a b c, there are two basic methods that one can use to solve for the value of : factoring or using the quadratic formula. Factoring Eample: 5 factor as much as possible 5 or set each factor equal to zero 5 or solve each equation Thus, the solutions are 5 or. Quadratic Formula Eample: Recall the quadratic formula: a b b ac, so here 9 9 7 7 5 or The solutions are 5 or. Often note that the quadratic formula is used to solve an equation that cannot easil be solved b the method of factoring. When a factorable quadratic is given (as in this eample), so the quadratic formula is not necessar, but the solutions do end up the same whichever method is used. Solve for. Practice Problems. 57. 5 5. 8. 7 5. 6 8 6. 7. 5 8. 5 6 9. 9 6. 5. 6. 9
Equations of Lines The equation of a line is a function that can be written in the form a b c. For a line that passes through the points m., and the slope of the line, m, is given b,, Note that horizontal lines have a slope of, and vertical lines have an undefined slope. The slope-intercept form of a line is m b where m is the slope of the line and b is the -intercept, or the -value at the point where the line crosses the -ais. The point-slope form of the line passing through the point m, with slope m is Practice Problems Find the slope of the line that goes through the following points.. (,5) and (,7). (,6) and (5, ). (, ) and (,) Write an equation of the line using either point-slope or slope-intercept form. with slope m and through the point (, ) 5. through the points (, 7) and ( 5, ) 6. through the points (, 6) and (9, 6) 7. with slope m and through the point (, )
Graphs \ In the stud of calculus, graphs are often used. It will be helpfull to be familiar with the following graphs. Take note of their general shape, intercepts, and where the graph is positive and negative. Graph of Graphh of Graph of Graphh of Graph of e Graph of ln
Graphs Continued Graph of sin Graph of cos Graph of tan