Notes for CETA s Summer Bridge August 2014 Prof. Lee Townsend Townsend s Math Mantra: What s the pattern? What s the rule? Apply the rule. Repeat. Precalc Crash Course Classic Errors 1) You can cancel factors but not terms across the division line. Terms added or subtracted. Factors multiplied or divided 2 4 The 2 do not cancel. Instead, factor both numerator and denominator. 2 2 ( )( + 2) ( ) 2 4 2 2 = 2 2 ( )( + 2) ( ) 2 4 2 2 = 2 2 = + 2 The ( 2) do cancel as they are factors in both the numerator and denominator. 2) Negative eponents. The negative eponent means change the sign and put the term on the other side of the divisor line. y 3 y 3 y 3 = 1 y 3 Summer Bridge 2014 Page 1 of 14
3) Division of an equation by a common factor. In differential equations I saw this error too many times. If you have had calculus: d 2 y d dy 2 22 + 4 = e d If you have not had calculus: A 2 2 B + 4 = C There is an in each term. Most of my students divide it out of the first term and the right hand side of the equation but forget the middle terms. The safest way to do the problem is to first factor the left hand side then divide the equation by the common factor. Factor: d 2 y dy 2 2 d d + 4 = e ( A 2B + 4) = C Divide: d 2 y dy 2 2 d d + 4 = e A 2B + 4 = C 4) Trig errors The calculator is in the wrong mode radians are degrees. Inverse trig functions. Radians Degrees Inverse trig functions 0 2π 2π = 6.28 0 360 Graphs of trig functions One complete cycle: If you are setting the window as if you are in radian mode, (min=0, ma=2π, you will get essentially a straight line if your TI is in degree mode. 2π =6.28 not very many degrees. If you are setting the window as if you are in degree mode, (min=0, ma=360, you will get 360/6.28 57.3 cycles in your graph if your TI is in radian mode. Notation: sin cos tan mean nothing if there is no argument. You need something like sin(), cos(), tan(). Notation for powers of trig functions and logs: books and handwritten: sin 2 () calculator entry: ( sin() )^ 2 = sin() books and handwritten: ln 2 () calculator entry: ( ln() )^ 2 = ln() ( ) 2 ( ) 2 Summer Bridge 2014 Page 2 of 14
A) Basic Algebra Make your problem look like a rule. Here are some rules. Eponent rules: Name Eponents Product Rule u v u v b b = b + Division Rule b u b = v bu v Power Rule u uv b = b ( ) v Unity 0 b = 1 Identity u = b log b u b ν = 1 b ν Product/Factoring rules: Name Distributive Law FOIL General method: Full Product (left to right) or Factors (right to left) a(b ± c) = ab ± ac ( a + b) ( a b) = a 2 b 2 (a ± b) 2 = a 2 ± 2ab + b 2 ( a + b) c + d a + b + c ( ) = ac + ad + bc + bd ( )( d + e) = a( d + e) + b( d + e) + c( d + e) = ad + ae + bd + be + cd + ce See the handouts for sample eams with solutions. Summer Bridge 2014 Page 3 of 14
B) Matri equations n equations in n unknowns Solve: 2 + y = 1 5 2y = 11 1) You can solve by intersecting y = 1 2 and y = ( 5 + 11) 2. Type in the equations Press F2/6 (ZoomStd): min=-10, ma=10, scl=1 ymin=-10, yma=10, yscl=1 Select F5/5 (Intersection) It ask which curves. You only have Y1 and Y2 so press ENTER ENTER Lower bound is any value left of the intersection. You can scroll there or type in an value (read the tick marks). ENTER Upper bound is any value right of the intersection. You can scroll there or type in an value (read the tick marks). ENTER 2) You can solve by substitution. y = 1 2. Plug into the other equation 5 2 1 2 ( ) = 11 You need to get the s together so distribute the -2. 5 2 + 4 = 11 Add 2 to both sides to get all terms with on the left and all terms without on the right. 5 + 4 = 9 Combine the s 9 = 9 Isolate the by dividing both sides by 9. = 1 Plug this value into the y equation to find the value of y: y = 1 2 1 ( ) = 3 3) Do this math on the TI-89 Let mean you have pressed Solve(2+y=1,y) y = 1 2 5 2y=-11 then arrow up to y = 1 2 and press ENTER 5 2y=-11 y = 1 2 9 2=-11 then arrow up to 9 2=-11 and press ENTER ENTER The equation 9 2=-11 should now be highlighted on the command line. Type +2. The command line now says ans(1)+2 ans(1)+2 ENTER. 9=-9 Summer Bridge 2014 Page 4 of 14
Now type 9. The command line now says ans(1) 9. ans(1) 9 ENTER. =-1 CLEAR the command line. Scroll up to y = 1 2. ENTER to put it on the command line. Append scroll up to =-1 and press ENTER y=3 4) Use matrices First line up the equations so the s are on top of each other on the left, the y s are on top of each other on the left and the numbers are on the right. 2 + y = 1 5 2y = 11 Make a matric of the coefficients: 2 1 5 2 You can either use the data/matri app or I find it much easier just to type 2,1;5, 2 [ ] ENTER Row elements are separated by commas and the rows themselves are separated by semi-colons. Let's call this matri a. [ 2,1;5, 2 ] STO a The calculator shows [ 2,1;5, 2 ] a Let s make a vector of the right hand side and call it b. 1 [ 1; 11] b 11 Note the semi-colon as there are two rows. We now write this in traditional matri notation. 2 1 5 2 y = 1 11 i.e. A = b where both and b are vectors with representing To solve this equation multiply both sides by A 1 A = b ( ). But A 1 A = I. I is the identity matri. For the 2D case 1 0 I = - the matri equivalent of one. 0 1 Therefore = A 1 b Since you have already defined A and b, type y Summer Bridge 2014 Page 5 of 14
a ^ 1*b 1 3 but that s y Now look at three equations in three unknowns. + y + z = 2 Solve: z = 1 + y = 1 for, y, and z. 5) Use Solver F2/1 solve( 2 + y = 1 and 5 2y = 11,{,y}) =-1 and y=3 i.e. you are solving for the set {,y}) solve(+y+z=2 and z=1 and +y=1,{,y,z}) =2 and y=-1 and z=1 Summer Bridge 2014 Page 6 of 14
C) Trig in four quadrants Start with basic definitions. They are visual. Just look at the triangle. The basic trig triangle Basic Definition Definition using the diagram cos = adj cos = a hyp c c b sin = opp sin = b hyp c tan = opp tan = b a adj a The above chart leads to SOHCAHTOA. It is easier to see the meaning if we insert some spaces: SOH CAH TOA. i.e. sin = O H, cos = A H, tan = O A with O=opp, A=adj, H=hyp. Now take the same triangle and put a circle around it. The origin of the circle is at the lower left hand corner of the triangle. c b The radius of the circle is the length of the hypotenuse. O a The origin is symbolized by the letter O. Now what do we do if I change the angle so it is out of quadrant I? O The basic definitions no longer work. What s the opposite? Where s the triangle? Summer Bridge 2014 Page 7 of 14
To figure out what to do, we need to change the notation. We redraw the original triangle in a circle using the nomenclature of the aes. cos = r (,y) O r y sin = y r tan = y From here we rethink what these formulas mean. Instead of thinking of and y as the length of the legs, think of them as the coordinates of the point on the circle. Now let s look at the point on the circle (, y) = ( 4,3). (-4,3) cos = 4 5 = 0.8 5 sin = 3 5 = 0.6 O tan = 3 4 = 0.75 Summer Bridge 2014 Page 8 of 14
II I The quadrants are labeled III IV Signs of coordinates Quadrant y I + + II - + III - - IV + - Signs of trig functions (r is positive so they follow the signs of the coordinates.) Quadrant cos sin tan I + + + II - + - III - - + IV + - - Inverse Trig Functions We read sin = 3 = 0.6 as What is the angle whose sine is 0.6? 5 Let s put it in the calculator to find out the answer. I find that = sin 1 0.6 = 36.87. In radians we get = sin 1 0.6 = 0.6435. Note: 0.6435degrees is less than 1/360 th of the way around the circle. If you are looking for degrees and get really small numbers, you re in the wrong mode. Inverse Trig Functions are not functions. Why? Because there are two answers in a circle. The calculator gives you one of them. Inverse sine and tangent calculator answers are in quadrants I and IV. Inverse cosine answers are in quadrants I and II. Summer Bridge 2014 Page 9 of 14
other = cos 1 u = sin 1 u = tan 1 u Some rules for finding the second angle that satisfies a trig equation cos = u, sin = u, or tan = u and knowing the calculator angle, calculator, along with the signs of the and y coordinates follow. 1) = cos 1 u : if y < 0 then other = calculator (,y) since cos other = cos calculator = r O calculator r y (,-y) Note that to make 0 other < 360, add 360 to other : other = 360 calculator 2) = sin 1 u : if < 0 then other = 180 calculator (-,y) y - other O calculator r (,y) y since sin other = sin calculator = y r Summer Bridge 2014 Page 10 of 14
other 3) = tan 1 u : if < 0 then other = 180 + calculator (,y) since (-,-y) -y - O calculator r y tan other = y = tan calculator = y I try to use = tan 1 u as the rule if < 0 then other = 180 + calculator since is so easy to remember; i.e. if is negative, add 180 the calculator answer. (or π if you are in radian mode) to A piece of advice: When solving problems given data, always use the original data instead of calculated values which are rounded off to keep the error in your final answer to a minimum. Summer Bridge 2014 Page 11 of 14
D) Notes on TI-89 entry TI-89 Note: 7 gives you log(. Fill in the rest with log(8,2). When you press ENTER, this becomes log 2 8 which is 3. TI-89 Note: 9 gives you root(. Fill in the rest with root(8,3). When you press ENTER, this 3 becomes 8 which is 2. TI-89 Note: When you are unsure about how to enter formulas in the calculator, do an eample where you know the answers. That is what I always do with the above two notes. I am never quite sure of the order so I guess. If the answer is wrong then I guess with the numbers switched I know: 8 = 2 3 hence the numbers used above. Also check the Catalog. For eample, the integration formula (F3/2). In the Catalog got to A (press the = button) then scroll up to the integral sign. In the Catalog integration looks like ( integrate At the bottom of the screen you will see EXPR,var[,low,high] Translation: EXPR is the integrand i.e. what you are integrating var is the variable of integration i.e. the variable in d. The brackets indicate the net two arguments are optional. The are the limits. The integral: The integral: 2 d TI: ( 2 2 d TI: ( 1 I recommend that you always do the indefinite integral first to check your input. Then go back and put in the limits to get the numerical value of the definite integral. Summer Bridge 2014 Page 12 of 14
E) Logs and Eponents Transform between log and eponent form Problem 1) convert log 2 8 = 3 to eponential form. Match the form of the last column 3 = log 2 8 By pattern match find u, v, and b u = 3, v = 8,b = 2 Put the variables in the right place in column 1 8 = 2 3 v = b u v b u u = log b v 8 = 2 3 8 2 3 3 = log 2 8 Problem 2) solve for v in the following problem: 3log 4 = 9. Match the form of the last column u = log b v First we have to get rid of the coefficient of the log as the form in column 5 has no coefficient. So, divide the equation by 3. log 4 = 3 By pattern match find u, v, and b u = 3, v =, b = 4 Put the variables in the right place in column 1 v = 4 3 v = b u v b u u = log b v = 4 3 = 64 4 3 3 = log 4 Problem 3) solve for in the following problem: 16 = 2 3 5 Match the form of the first column v = b u First we have to get rid of the coefficient of the b as the form in column 1 has no coefficient. So, divide the equation by 2. 8 = 3 5 By pattern match find u, v, and b u = 5, v = 8, b = 3 Put the variables in the right place in column 5 5 = log 3 8 Divide by 5 to solve for v = b u v b u u = log b v 5 = log 3 8 8 = 3 5 8 3 5 = 1 5 log 8 3 Summer Bridge 2014 Page 13 of 14
Use solver (F2/1) to check your work. Problem 2: solve(3*log(,4)=9,) =64 Problem 3: solve(16=2*3^(5),) = 3ln(2)??? 5ln(3) Log conversion rule: log b u = log u a with a=any base. Pick e. Then our calculated log a b answer becomes = 1 5 log 8 = 1 ln8 3 5 ln 3 However 8 = 2 3. The power rule of logs says nlog b u = log b u n. Now our answer is = 1 5 ln8 ln 3 = 1 ln2 3 5 ln 3 = 3 ln2 5 ln 3 Just FYI, there are two more log rules. Product Rule: Division Rule: log b uv = log b u + log b v log b u v = log b u log b v Summer Bridge 2014 Page 14 of 14