Algebra/Trigonometry Review Notes

Transcription

1 Algebra/Trigonometry Review Notes MAC 41 Calculus for Life Sciences Instructor: Brooke Quinlan Hillsborough Community College

2 ALGEBRA REVIEW FOR CALCULUS 1 TOPIC 1: POLYNOMIAL BASICS, POLYNOMIAL END BEHAVIOR, AND LINEAR EQUATIONS I. FEATURES OF A POLYNOMIAL Terms are separated by addition or subtraction. In calculus, it is very important to be able to distinguish terms. Eample 1: How many terms are in each epression? d) A polynomial is a sum or difference of terms in which the eponents of the variables are positive whole numbers. Eample : Determine if each epression is a polynomial or not. If not, eplain why. d) The degree of a term is the sum of the eponents of the variables of that term. Eample : Find the degree of each term y 5 d) 16 A polynomial is in standard form when the terms are written in decreasing order of their eponents (degrees). The leading term of a polynomial is the first term when the polynomial is in standard form. The degree of a polynomial is the highest degree of any single term in the polynomial. Assuming the polynomial is in standard form, the degree of the polynomial is the same as the degree of the leading term. Eample 4: Write each polynomial in standard form, then underline the leading term and indicate the degree of the polynomial Algebra Review Topic 1: Polynomials & Linear Equations Page 1 Page 1 of 0

3 II. POLYNOMIAL END BEHAVIOR When graphing a polynomial, its end behavior depends on its leading term. In particular, you have to determine if the leading term is positive or negative, and if the degree of the leading term is even or odd. There are only 4 possible ways a polynomial can end, which are shown in the chart below. even +a 4 E:... E: even a 4... E: odd +a 5... E: odd a 5... Remember that a polynomial s end behavior depends entirely on its leading term!!! Eample 5: Find the leading term in order to determine the end behavior of each polynomial. d) f 4 ( ) 6 7 f 5 ( ) 7 f 4 5 ( ) f 10 ( ) 6 You will need to remember these rules for polynomial end behavior when we start doing limits. III. LINEAR EQUATIONS Whenever you are asked to write the equation of a line, you need to put you answer in slope-intercept form, which is y m b, where "m" is the slope and "b" is the y-intercept. In calculus, you will frequently be asked to find the equation of a tangent line (we will learn what this is later) given only an -value. First, you will have to find the corresponding y-value by plugging the given -value into the function. Net you will have to calculate the slope using derivatives (we will learn how to do this later also). Then you will use y y m. the point-slope form to find the equation of the tangent line. Point-slope form is 1 1 Algebra Review Topic 1: Polynomials & Linear Equations Page Page of 0

4 Eample 6: Using the given function, -value, and slope, write the equation of the tangent line in slope-intercept form. f 5; ; m 1 The corresponding y-value to will be f The problem gives us the slope (although once we are doing actual calculus we will have to find the slope using derivatives). So using the point-slope form, we have y 1 1 y y 1. y m f 5 4, 1, m 4 y 1 The bo below summarizes equations of lines. Make note of the equations for vertical and horizontal lines because these will be used often throughout the course. Algebra Review Topic 1: Polynomials & Linear Equations Page Page of 0

5 ALGEBRA REVIEW FOR CALCULUS 1 TOPIC : EXPONENTS I. POSITIVE EXPONENTS Eample 1: You can also see this by writing out the 's: 6 Total of 8 's You can also see this by writing out the 's: 4 's left 4 d) 9 e) 4 f) 6 You can also see this by writing out the 's: y y y y g) 4 h) Total of 6 's u v w i) j) y 6 9 y y 4uv 5w 4 5 Algebra Review Topic : Eponents Page 1 Page 4 of 0

6 II. NEGATIVE EXPONENTS To get rid of a negative eponent on a factor, move the factor to wherever it is not currently residing and change the eponent to a positive. (i.e. if it's in the numerator, move it to the denominator, and if it's in the denominator, move it to the numerator) Eample : Rewrite the epression with only positive eponents. d) Move to the bottom & make eponent positive 4 Only the has negative eponent 5 5 y y Move 8,, and y, then y Evaluate 8 y 1 4 make their eponents positive 8 y e) 5 4 y 6 z Eample : Combine the negative eponent rules with the rules from Page 1 to simplify the epressions so that your answer contains only positive eponents. 1 4 Apply the outside eponent Move any factor that y 4 5 to every factor inside the ( ) 4 5 has a negative eponent y z 4 y z z Simplify y z u Apply the outside eponent u Move any factor that u w 4 5 to every factor inside the ( ) 8 10 has a negative eponent 8 v w v w v 4 10 Simplify u w 9v Apply the outside eponent 4 to every factor inside the ( ) 5 y z Algebra Review Topic : Eponents Page Page 5 of 0

7 Often in Calculus, you actually have to start a problem by writing it WITH a negative eponent (even though we will always write our answers without negative eponents). So for Eample 4 we will practice rewriting the epression WITH negative eponents. Eample 4: Write the epression with negative eponents. 1 Move the up & make 1 the eponent negative 4 7 It is important to understand how to separate a fraction into a coefficient times the variable. We will practice this in Eample 5. Eample 5: Separate the coefficient from the variable in each epression and write the variable with a negative eponent if it was originally in the denominator. 1 Same as d) and only move the up Leave the 1 & where they are 4 e) 4 f) 6 8 g) 10 5 Algebra Review Topic : Eponents Page Page 6 of 0

8 III. RATIONAL EXPONENTS & RADICALS Rational (fractional) eponents translate to radicals and vice versa using the following relationship: p power r r, where " " stands for "power" and " " stands for "root", so we have root root p p r power Eample 6: Rewrite each radical so that it has a rational eponent. same as 1 1 is the power 1 and is the root same as 1 is the power 1 1 is the root d) e) 5 1 f) 7 IV. MIXING RATIONAL AND NEGATIVE EXPONENTS Recall that a factor gets a negative eponent if it came from a denominator, and it gets a rational (fractional) eponent if it came from a radical (root). DO NOT CONFUSE THESE TWO CONCEPTS! Eample 7: Rewrite the epression so that there are no radicals and also no variables in denominators. d) e) 1 (negative eponent because the was in the denominator) 1 1 (negative eponent because the was in the denominator, rational eponent because the was in a radical) (separate the coefficient ¼ from the variable. Then make the variable's eponent rational because the was in a radical) 5 f) Algebra Review Topic : Eponents Page 4 Page 7 of 0

9 ALGEBRA REVIEW FOR CALCULUS 1 TOPIC : FACTORING I. FACTORING QUADRATIC EXPRESSIONS A polynomial is a quadratic if its degree is. If you are given a quadratic epression of the form a b c, then it can be factored if the discriminant, b 4ac, is a perfect square: 0, 1, 4, 9, 16, 5, 6, 49, 64, 81, 100, etc. (Thinking of this a slightly different way, the quadratic can be factored if the square root of the discriminant is a real whole number, including zero.) Then to find the factorization, you have to figure out the factor pair u, v add up to equal b (so u v a c and u v b ). Eample 1: Factor the quadratic epression, if possible. that multiply together to equal ac but First we calculate the square root of the discriminant to see if the quadratic is factorable. b 4ac (the quadratic epression is factorable since 17 is a real whole number) Net we multiply a times c to get Now make a list of all of the pairs of numbers that multiply together to equal 60. Because a c is negative, u and v must have opposite signs. Make the larger number in each pair negative since the sign of b is negative. The "magic" pair is 5 and 1. These are the numbers that multiply to equal a c but add up to equal b. Factor pairs u, v that multiply together to equal 60 a c u v Choose the pair that has a sum of 7 b u 1, v u, v 0 8 u, v 0 17 u 4, v u 5, v 1 7 u 6, v 10 4 So now you need to rewrite the original quadratic by splitting the original middle term b into u v : b uv Finish using the "factoring by grouping" technique: Pull out of Group 1 and - out of Group Group 1 Group Term 1 Term Now Term 1 and Term both contain a common factor: 6 it leaves an from Term 1 and a from Term : When you pull out this GCF, Algebra Review Topic : Factoring Page 8 of 0 Page 1

10 Eample 1 (continued): Factor the quadratic epression, if possible. d) 4 5 First we calculate the discriminant to see if the quadratic is factorable. b 4ac Imaginary! Because the square root of the discriminant is not a real number, this quadratic epression is NOT factorable. 4 5 Discriminant b 4ac Because the square root of the discriminant is not a whole number, this quadratic epression is NOT factorable First we calculate the discriminant to see if the quadratic is factorable. b 4ac (the quadratic epression is factorable since 1 is a real whole number) Net we multiply a times c to get Now make a list of all of the pairs of numbers that multiply together to equal 7. Because a c is positive, u and v must have the same sign. Both signs will be positive because the sign of b is positive. The "magic" pair is 8 and 9. These are the numbers that multiply to equal a c but add up to equal b. Factor pairs u, v that multiply together to equal 7 a c u v Choose the pair that has a sum of 17 b u 1, v 7 7 u, v 6 8 u, v 4 7 u 4, v 18 u 6, v 1 18 u 8, v 9 17 Now rewrite the original equation, leaving the first and last terms the alone but splitting the middle term into u v : Finish using the "factoring by grouping" technique: Algebra Review Topic : Factoring Page 9 of 0 Page

11 II. FACTORING A DIFFERENCE OF TWO SQUARES A "difference of two squares" is a special type of quadratic epression that we will come across often. This special quadratic has no middle term; it only has the first and last terms of the quadratic epression, and they both must be perfect squares to use this formula. The factorization of a difference of two squares a b a b a b. There is NO factorization for a sum of two squares a b does not factor. Eample : Factor if possible. is: Factors to a b ab ab Factors to a b ab ab 5 64 d) 9 III. FACTORING SUMS AND DIFFERENCES OF CUBES If you have either a sum or a difference of perfect cubes, the epression can be factored into a binomial times a trinomial. Let's start with a sum of cubes: a b Factors to a b a ab b Then there is the difference of cubes: a b Factors to a b a ab b What you should notice is that the binomial contains the same sign as the original epression. The first sign in the trinomial is opposite of the sign in the binomial, and the second sign in the trinomial is always positive. The factorization can be summarized as: same sign b opposite sign always positive Factors to a b a a ab b Eample : Factor the sum or difference of cubes. Factors to a b ab a abb Factors to a b ab a abb Algebra Review Topic : Factoring Page 10 of 0 Page

12 Eample (continued): Factor the sum or difference of cubes d) 16 IV. FACTORING ON STEROIDS In calculus, you will often find that the hardest part of the problem is the algebra (specifically the factoring) that must be done to finish a problem. I call this "factoring on steroids" because it is taking all of the factoring skills you have ever learned up to a whole other level! The basic concept for these problems is to identify the GCF of all terms and factor it out, which in theory is very easy. Here is an eample of a problem you may have done in a previous algebra class: Factor 4 y 1 y 6 y 60 y. You can see that there are terms and that the coefficient of every term is divisible by, so is a "common factor" of all the terms. You also notice that every term has an and a y, so and y are "common factors" of all terms. But the greatest common factor is. We chose these eponents for and y because they are the. When you "pull out a GCF" you are actually dividing it out. So imagine going through each term in the original epression and dividing out the GCF. The problem would look like this: y 1 y 6 y 60 y 1 y. 1 y 1 y 1 y 1 y Then you could simplify each term by using eponent rules to get: 1 y y 1 y 5y. Don t forget that the second term becomes a 1! If you start with 4 terms, then you have to finish with 4 terms inside the parentheses! Now we are going to look at a problem that you might come across in calculus. Eample 4: Factor out the GCF and simplify your answer You can see that this epression has terms. The coefficient of each term is divisible by. The other common factors in each term are and. The greatest common factor (GCF) is. Algebra Review Topic : Factoring Page 11 of 0 Page 4

13 Eample 4 continued: Factor out the GCF and simplify your answer. 4 Pull out and by the GCF Term 1 Term 4 Distribute the Combine Term 1 like terms Term Eample 5: Factor out the GCF and simplify the answer This epression has terms. The coefficient of each term is divisible by. Although the second term has an, the first term does not, so is not a common factor. The GCF is. Now pull out the GCF and divide each term by it. Simplify what is left in the parentheses. Algebra Review Topic : Factoring Page 1 of 0 Page 5

14 Eample 6: Factor out the GCF and simplify the answer The GCF is (make sure you choose the smallest eponents!) Eample 7: Factor out the GCF and simplify the answer Algebra Review Topic : Factoring Page 1 of 0 Page 6

15 ALGEBRA REVIEW FOR CALCULUS 1 TOPIC 4: SOLVING QUADRATIC EQUATIONS I. METHODS FOR SOLVING QUADRATIC EQUATIONS You learned 4 methods for solving quadratic equations in your algebra classes: factoring/zero product principle, the quadratic formula, the square root method, and completing the square. The last one will not be used much (if at all) in this class, so we will focus on the first three. II. FACTORING AND THE ZERO PRODUCT PRINCIPLE To solve a quadratic equation using either factoring or the quadratic formula, the equation must be set equal to zero. Once it is, then factor it using your amazing factoring skills, set each factor equal to zero (that's the "zero product principle" part), and solve for. Eample 1: Solve the equation by factoring Set the equation equal to zero: So the equation becomes We know this is factorable because b 4ac , and 9 is a real whole number. Find factors of 60 a c that add up to 9 The magic pair is 15 and 4, so split the middle term using these numbers OR OR OR 6 b. Factor pairs u, v that multiply together to equal 60 a c u v Choose the pair that has a sum of 9 b u 1, v u, v u, v u 4, v u 5, v 7 67 u 6, v u 8, v 45 7 u 9, v 40 1 u 10, v 6 6 u 1, v 0 18 u 15, v 4 9 u 18, v 0 1 Algebra Review Topic 4: Quadratic Equations Page 14 of 0 Page 1

16 III. QUADRATIC FORMULA The quadratic formula is b b 4ac. a Typically you will use the quadratic formula to solve a quadratic equation if it has three terms but will not factor. In order to use the quadratic formula, the equation MUST be set equal to ZERO!!!! Eample: We encountered this quadratic epression in the factoring review and found that it was not factorable. So let s use the quadratic formula to solve the equation. First we have to identify a, b, and c. a, b 4, c So the two solutions to the quadratic equation are 19 19, Eample : 8 9 Algebra Review Topic 4: Quadratic Equations Page 15 of 0 Page

17 IV. SQUARE ROOT METHOD You should use the Square Root Method whenever you have only a squared term and a constant term (no term) but when you set it equal to zero it does not create a difference of two squares (so it can t be factored). To use the square root method, you just isolate the squared term then take the square root of both sides, remembering to put a ± symbol in front of the square root of the constant. Eample 4: Solve the quadratic equation. 4 0 This is missing the middle term, but it is not a difference of two squares, so we need to isolate the squared term and use the square root method But you can't leave a radical in a denominator, Rationalize so you have to rationalize it: Denominator, Algebra Review Topic 4: Quadratic Equations Page 16 of 0 Page

18 ALGEBRA REVIEW FOR CALCULUS 1 TOPIC 5: RATIONAL EXPRESSIONS AND FUNCTIONS I. SIMPLIFYING RATIONAL EXPRESSIONS Recall that the word "rational" means the same as "fractional" in math. So in this Topic, we are going to be looking eclusively at epressions that are fractions, and just like with number-only fractions, our goal is cancel out the greatest common factor that appears in both the numerator and denominator. For eample, when you "reduce" the fraction 10 1 to 5, what you are doing is dividing the numerator and 6 denominator by the greatest common factor of 10 and 1, which is. Let's look at the problem this way: Factors to 5 5 Cancel out the 's Leaves When you are asked to simplify rational epressions (meaning now there are variables in the fraction), you really are 7 1 doing the same thing. If you are given a problem that looks like this:, where the numerator and 1 denominator are already factored, then it is easy to see that the common factor is 1. When you cancel 7 1 (divide) this factor out, you get: 1 have just a single factor left on top or bottom). 7 or just 7 (it is OK to drop the parentheses if you But usually rational epressions are not given to you in their factored form. Instead, they will have polynomials on top and bottom that you will have to factor. Only once you have changed the numerator and denominator from polynomials to their factored forms can you actually cancel out common factors. 6 7 So typically, the previous problem would have been given to you as. There is a polynomial in the 4 numerator with terms and another polynomial in the denominator with terms. Students often get confused and think they can cancel terms (NO NO NO!!!!), and would proceed to cancel out the terms, reduce the 6 and 4 by, and cancel out the 's, and their work would look something like this: This is UTTER NONSENSE!!!! You have to resist the temptation to ever cancel out terms. You can not cancel terms. You can only cancel FACTORS. So the work for this problem should actually look like this: 6 7 Factor top & bottom Cancel common factors Algebra Review Topic 5: Rationals Page 1 Page 17 of 0 7

19 The important (and often overlooked) component of the GCF is the "F" the idea that you can only cancel out a FACTOR. Let's look at another eample that shows the difference between cancelling terms (totally illegal) and cancelling factors (LEGAL). Eample 1: Simplify the epression. INCORRECT: 1 A lot of people would (incorrectly) say that the TERMS and have an "" in common, and that the TERMS 1 and have a "" in common (leaving positive 6), and they would proceed like this: +6 1 Appears to leave Combine like terms 6 8 This is SO WRONG because you CAN'T CANCEL TERMS!!! CORRECT: 1 6 Pull out GCF Factor the from numerator numerator Now the epression is completely factored, and the only common FACTOR in both the numerator and denominator is, so we can divide (cancel) that factor out: Leaves CORRECT!!! Eample : Simplify the epressions Algebra Review Topic 5: Rationals Page 18 of 0 Page

20 II. DOMAINS AND VERTICAL ASYMPTOTES To find the domain of a rational function, just find which -values would make the denominator zero, then eclude those -values from the domain. Eample 4: Find the domain of each function. f ( ) 9 f ( ) 5 4 Each number that is ecluded from the domain of a rational function must be accounted for graphically by either being a hole or a vertical asymptote (VA). If a number, n, is ecluded from the domain, and it came from a factor n in the denominator, then there is a hole at n if there is also a factor n in the numerator (i.e. if the n factor can cancel out). If the factor can not cancel out, then there is a Vertical Asymptote at n. (Just remember, "if it cancels it leaves a hole, if not there is a VA".) Eample 5: Determine if there is a hole or a VA at each number you eclude from the domain. f 9 f f 4 Algebra Review Topic 5: Rationals Page 19 of 0 Page

21 III. HA'S AND END BEHAVIOR You may remember from Topic 1 that the end behavior of a polynomial was controlled by its leading term. Similarly, the end behavior of a rational function is controlled by the leading terms of both the numerator and denominator. For any rational function R a b n......, we will assume that the leading term of the numerator is n a (so "n" is d b (so "d" is the degree of the d the degree of the numerator) and the leading term of the denominator is denominator). The end behavior of such a rational function is controlled by either horizontal or oblique asymptotes (HA's or OA's). To know whether (and where) you have an HA or an OA, you have to compare the degree of the numerator (n) to the degree of the denominator (d). There are 4 cases that you need to memorize: Case 1 n d : There is an HA at y 0. Thus, as you trace the function all the way to the right or the left, it is approaching a y-value of 0. a Case n d : There is an HA at the ratio of the leading coefficients. So the HA is at y. Thus, as you trace the b function all the way to the right or the left, it is approaching a y-value of a b. Case n d 1: There is an Oblique Asymptote. To find the equation of the OA, divide the denominator into the numerator using polynomial long division, and the result will be the equation of a line m b. Thus, as you trace the function all the way to the right or the left, it is being guided by that line y m b. Case 4 n d 1: The function does not have an HA or an OA. We will learn methods using calculus to determine the end behavior of these functions. Eample 6: Find the HA or OA for each function. f 5 f f Algebra Review Topic 5: Rationals Page 0 of 0 Page 4

22 IV. COMPLEX FRACTIONS A comple fraction is a fraction that has one or more fractions in either the numerator or the denominator or both. You can think of them as "fractions containing fractions". In order to simplify a comple fraction, you have to remove the "mini-fractions" from the numerator or denominator so that it just looks like a regular fraction or rational epression. You do this by multiplying each term in the fraction by the LCM of the mini-fractions. Eample : Simplify each comple fraction This comple fraction has terms, and of the terms are "mini-fractions". The denominators of the mini-fractions are and 5, and the least common multiple of and 5 is 15. So multiply each term by 15 and simplify. Don't forget to multiply the 6 by 15 also!! Cancel Leaves 5 Multiply We can see that all three terms of this epression are divisible by, but we know that we can't cancel terms, we can only cancel factors, so we factor out a from the numerator and denominator: Factor out a 5 Cancel the 's Leaves There are 4 total terms in this epression ( in the numerator and in the denominator). Of these terms, are mini-fractions, with denominators of we multiply all four terms by,, and and simplify. Don't forget to multiply the 8 by.the LCM of these denominators is. So also!! 8 7 Cancel 8 7 Leaves Algebra Review Topic 5: Rationals Page 1 of 0 Page 5

23 d) 1 16 This one is a little tricky because it does not appear to be a comple fraction. 4 But remember that a negative eponent on a factor means to move the factor then make the eponent 1 16 positive. So 16 becomes, 1 becomes, and 4 4 becomes. So when we rewrite this rational epression it becomes: 16 Rewrite Multiple of the mini-fractions is 16 4 Cancel. It has mini-fractions, and the Least Common, so multiply all terms by this LCM: Like Terms 4 16 Like Terms 4 4 Leaves 16 Distribute Combine e) 1 8 y y y 1 Rewrite The key to correctly simplifying comple fractions is to make sure you multiply every term of the epression by the Least Common Multiple of all the mini-fractions. Do NOT multiply some terms by one thing and other terms by something else. ALL TERMS MUST BE MULTIPLIED BY THE SAME THING. Algebra Review Topic 5: Rationals Page of 0 Page 6

24 ALGEBRA REVIEW FOR CALCULUS 1 TOPIC 6: EXPONENTIALS AND LOGARITHMS I. EXPONENTIAL EXPRESSIONS An eponential epression is one where the variable is in the eponent and the base is a number, b, where b is either between 0 and 1 or greater than 1. 0 b 1 or b 1 5 1,,, and e are eamples of eponential epressions. The last eample shows the natural base, which is encountered frequently in nature. This number is approimately equal to It is a non-terminating number, and it has been assigned the letter "e" (similar to the way the number has been assigned the Greek letter ""). II. LOGARITHMIC EXPRESSIONS A logarithmic epression is one that contains a "log" of some base, b, where b has the same requirements that it did for eponential epressions (so 0 b 1 or b 1 ) and is written as a subscript net to the word "log". The "inside" of log base inside. the log usually contains a variable epression. So a logarithm typically looks like this: Eamples of logarithmic epressions are log 4, log 5, log 7, and ln don't have subscripts actually have special names.. The last two eamples that When you see the word "log" without a base subscript, then it is a common log and the base is understood to be 10. Most calculators have a "LOG" button. This button can calculate common (base 10) logs only. log inside. You can rewrite any epression that looks like log inside as When you see the letters "ln" (or "LN") without a base subscript, then it is a natural log and the base is understood to be the natural base, e. All calculators have a "LN" button. This button can calculate natural (base e) logs only. You can rewrite any epression that looks like ln inside as log e inside. Because students often mistake the lowercase "l" for an uppercase "i", I tend to use cursive in class, so my natural log epressions will look like: ln inside. 10 III. RELATIONSHIP BETWEEN EXPONENTIALS AND LOGARITHMS Eponential b and logarithmic log b functions are inverse functions. As with all inverse functions, you can use one to "undo" the other. Recall from your algebra class that composing a function with its inverse always 1 equals : f f and The same goes with eponentials logarithm gives: log b 1 f f. b and logarithms log b : plugging the eponential in for the "inside" of the power log b and plugging the logarithm in for the power of the eponential gives: b b inside. Algebra Review Topic 6: Eponentials & Logs Page 1 Page of 0

25 IV. PROPERTIES OF LOGARITHMS In order to succeed in calculus, it will be very important for you to remember the properties of logarithms that you learned in your algebra class. In particular, you need to realize that these are in fact algebra properties, because once we start doing calculus to logarithms, students often confuse which skills come from calculus and which ones come from algebra. A. The Most Important Property of Logs In my opinion, the absolute most important property of logs was presented on the bottom of the previous page: log b b. In other words, if you can make the "inside" of the logarithm be written as the stated base (the subscript) to some power, then the logarithm just equals the power. Eample 1: log Since the base of the logarithm (the subscript) matches the base of the inside, the answer is just the power. Rewrite 5 5 log log 5 as log 81 d) log 100 e) ln e 1 f) log 4 64 B. Product and Quotient Properties of Logs This eample demonstrates both the product and quotient properties of logarithms: came came came came came from from from from from num. num. denom. denom. denom. AB Epands log b log b log b log b log b log to b CDE A B C D E Positive Logs Negative Logs In other words, if the INSIDE of a logarithm contains a product or quotient (not a sum or difference!!!) of n factors, then it can be epanded into n individual logarithms, where the sign of the log will be positive if the inside came from a numerator, and the sign of the log will be negative if the inside came from a denominator. Algebra Review Topic 6: Eponentials & Logs Page 4 of 0 Page

26 Eample : Epand each logarithm and simplify if possible. log log e ln 1 C. Power Property of Logs If the inside of a logarithm is raised to a power, then you can move the power in front of the logarithm. Eample : Apply the power property of logs. 4 log D. The Logarithm of 1 p Move power p b log log b in front of log ln 7 No matter what base a logarithm is, the logarithm of 1 is always zero! ln log b 1 0 and 1 0 Now we will do some eamples showing how we combine all of these properties to epand logarithms. Eample 4: Epand and simplify each logarithm. log 8 y log log d) 14 4 e 5 1 ln Algebra Review Topic 6: Eponentials & Logs Page 5 of 0 Page

27 V. SOLVING EXPONENTIAL EQUATIONS An eponential equation is any equation where the variable is in the eponent. To "free" the variable from the eponent, follow these steps: 1. Isolate the eponential (whatever is being raised to the power).. Take the natural log (LN) of both sides of the equation.. Apply the power property of logs to move the power in front of the log. 4. Solve for the variable. Eample 5: Solve the equation 1 7. ln 1 1 ln Do This 7 Take LN of both sides Do This 7 Move power in front 1 ln ln 7 Do This Divide both sides by ln ln 7 1 ln Do This Add 1 to both sides ln ln 7 1 Eample 6: Solve the equations ,000 5, t Algebra Review Topic 6: Eponentials & Logs Page 6 of 0 Page 4

28 TRIGONOMETRY REVIEW FOR CALCULUS 1 I. DEFINING THE 6 BASIC TRIG FUNCTIONS USING A RIGHT TRIANGLE An acute angle is any angle that is between 0 and 90 (or 0 and radians). Drawing a vertical line between the terminal side and the initial side of the angle creates a right triangle. The side labeled c is the hypotenuse of the right triangle. (The hypotenuse is the side directly across from the right angle.) The leg of the triangle that is directly opposite from the angle θ is called the opposite side. And the leg of the triangle that is attached to the angle θ is called the adjacent side. Assuming θ is an acute angle (θ can never be the right angle!), we define the si trigonometric functions based on the ratios between any two sides of this triangle. The si trig functions are Sine (sin), Cosine (cos), Tangent (tan), Cosecant (cs, Secant (se, and Cotangent (cot), and their definitions are given below: Notice that csc θ is the reciprocal of sin θ, sec θ is the reciprocal of cos θ, and cot θ is the reciprocal of tan θ. These are the Reciprocal Identities. The easiest way to learn the trig values of certain angles is to learn only sine, cosine, and tangent of the angles, and then remember that cosecant, secant, and cotangent are merely the reciprocals of those values. Note that the s of sin pairs with the c of csc and the c of cos pairs with the s of sec. So, for instance, if you are ever trying to remember which one (sec or cs is the reciprocal of cosine, remember that a c is always paired with an s, so it must be sec that is the reciprocal of cosine. It is easy to remember these definitions by remembering: Soh Cah Toa! SOH means: Sine Opposite Hypotenuse CAH means: Cosine Adjacent Hypotenuse TOA means: Tangent O pposite A djacent Then, remember: Cosecant is the reciprocal of Sine. Secant is the reciprocal of Cosine. Cotangent is the reciprocal of Tangent. The Quotient Identities state: The Pythagorean Identities (these come from using the Pythagorean Theorem: a + b = c ) are: The first and second ones are the most important and will be used often! Trigonometry Review Page 7 of 0 Page 1

29 II. FINDING SINE AND COSINE OF THE QUADRANTAL ANGLES The and y aes split the coordinate plane into 4 quadrants. The angles that lie on the aes are thus called quadrantal angles. Starting on the positive -ais and moving in a counterclockwise direction (as we always do for positive angles), the quadrantal angles are as follows: Positive -ais = 0 = 0 radians Positive y-ais = 90 = radians Negative -ais = 180 = radians Negative y-ais = 70 = radians The net move would take us to back to the positive -ais and would be 60 = radians, and we could keep going around and around in this manner forever. The unit circle is a circle, centered at the origin, with a radius of one. To draw this circle, put points on the and y aes that are 1 away from the origin. On the positive -ais, that point would have the coordinates (1, 0) because you have moved right 1 on the -ais but have not moved up or down on the y-ais. On the positive y-ais, the point would be (0, 1) because you have not moved left or right on the -ais but you moved up 1 on the y-ais. On the negative -ais the point would be (-1, 0) because you moved left 1 on the -ais but have not moved up or down on the y-ais. And on the negative y-ais the point would be (0, -1) because you have not moved left or right on the -ais but you moved down 1 on the y-ais. Once you have labeled the points on the aes of the unit circle, it is easy to see what the values of cosine and sine are for various angles. You must remember that cos = the -value and sin = the y-value of any point on the unit circle. Eample 1: Looking at the labeled unit circle below, fill in the following values. sin cos cos d) sin e) cos0 f) cos g) sin h) sin sin Then, to find tangent, you just remember the Quotient Identity: tan. For which quadrantal angles is cos tangent undefined? Trigonometry Review Page 8 of 0 Page

30 III. THE MOST COMMONLY-USED ANGLES IN TRIG The three acute angles in Quadrant 1 that are frequently used in trig are 0 radians 6, 45 radians 4, and 60 radians. We will also commonly use the angles in the other quadrants that have these angles as their reference angles. The circle below is labeled with all of these angles, plus the quadrantal angles. Notice how the denominators match up as you go around. The angles closest to the -ais have a denominator of 6; the angles closest to the y-ais have a denominator of, and the angles in the center of each quadrant have a denominator of 4. If you have trouble memorizing these angles, then one simple thing to do is to start by drawing the and y aes. Net, draw 5 lines that are evenly-spaced between the aes in each quadrant. Starting at the first line (the dashed one between 0 and 6 in the figure to the left), label all of the lines (including the aes) in increments of 1. So you 4 would have,,,,..., Then reduce any fractions that can be reduced (for instance, reduces to, 15 reduces to 5, etc.) Try this now! In calculus, we will almost eclusively use angles measured in radians, so you need to become very familiar with radian measurement if you aren't already. Another very important concept to recall is which trig functions are positive in each quadrant. The graphic to the right shows that All trig functions are positive in Quadrant 1, Sine and its reciprocal (cs are positive in Quadrant, Tangent and its reciprocal (cot) are positive in Quadrant, and Cosine and its reciprocal (se are positive in Quadrant 4. You can remember the phrase "All Students Take Calculus" to aid in labeling the quadrants correctly. S T A C Trigonometry Review Page 9 of 0 Page

31 IV. FINDING THE EXACT VALUES OF THE TRIG FUNCTIONS FOR THE COMMONLY-USED ANGLES We are going to learn how to create a table that shows sine, cosine, and tangent of the main Quadrant 1 angles. It will be very valuable for you to learn how to recreate this table for tests!!! Start by creating a 4 4 table. Label the nd, rd, and 4 th boes of the top row with,, and (that's 0, 45, 6 4 and 60, respectively). Also label the nd, rd, and 4 th boes of the left column with sin θ, cos θ, and tan θ. Your table should look like this: θ 0 rad 6 45 rad 4 60 rad sin cos tan Steps to fill in the boes 1) Near the top of the boes in the first row, write the numbers 1,, and. ) Near the top of the boes in the second row, write the numbers,, and 1. ) Take the square root of each number you have written. 4) Divide every number by. sin 5) To fill in the tangent row, recall that tan, so divide the number in the first bo by the number in cos the second bo and that will give you the tangent value. Even though the table only shows the values of the Quadrant 1 angles, as long as you know your "ASTC" rules, you can actually use the table to find the trig values of any angle from any quadrant that has one of these as its reference angle. 7 Eample : Find the eact value of sin. 6 First, note that the Q1 reference angle for 7 6 is 6 (the reference angle always has the same denominator as the 7 given angle and a single in its numerator). Net, find sin in your table:. Now locate : it is in 6 6 Quadrant. Note from your "ASTC" rules that only Tangent (and cot) are positive in Quadrant ; therefore, sine is negative in that quadrant. Thus, your final answer will be since sine is negative in Quadrant. Trigonometry Review Page 0 of 0 Page 4