Law of Sines, Law of Cosines, Heron s Formula:

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Roderick Newton
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1 PreAP Math Analsis nd Semester Review Law of Sines, Law of Cosines, Heron s Formula:. Determine how man solutions the triangle has and eplain our reasoning. (FIND YOUR FLOW CHART) a. A = 4, a = 4 ards, b = 6 ards b. R = 08, r = 8 ft, s = ft c. P = 6, Q = 8, p = 6 in. d. A = 87, a = 8 feet, b = feet. Solve the triangle. a. a = 6, b = 4, C = 40 b. b = 4, c =, B = 40 c. a = 6, b = 9, c = 4. Two observers are 000 ft apart and sight an airplane between them. Observer A spots the plane at a 40 angle and Observer B spots the same plane at a angle. Draw a picture. How high is the airplane? 4. The dimensions of a triangular lot are 00 feet b 0 feet b 7 feet. If the price of such land is $ a square foot, how much does the lot cost? Page of 0

2 Limits: Evaluate its approaching a number graphicall, numericall, and analticall using direct substitution, cancellation techniques and rationalizing techniques. (not in tetbook). Evaluate each it. a) b) ( ) c) 7 d.. Graph the function on our calculator and tell if the specified it eists. a) f ( ), f ( ) b) f ( ), f ( ) c. f ( ), f ( ). Find the it if it eists. Verif b describing what ou see on our calculator a. b. c. 6 d. t t t 4. Evaluate the it algebraicall. a. b. c. 0 Page of 0

3 Average Rate of Change, Slope of Secant & Tangent Lines: The instantaneous rate of change of the graph of f() at the point (, f()) is the slope m of the tangent line f ( h) f ( ) at (, f()) given b m, provided the it eists. h 0 h If the distance an object travels is given as a function of time f(t), then the instantaneous velocit v(t) at a f ( t h) f ( t) time t is given b vt ( ). h 0 h. Find and equation for the slope of the graph of at an point.. The distance d of an object is above the ground t seconds after it is dropped is given b d(t). Find the instantaneous velocit of an object at the given value for t. d( t) 8t 6 t ; t.8 If position is given as a function of time f(t), for an two points in time a and b, the average velocit v is changeindis tan ce f ( b) f ( a) given b vavg change int ime b a. Find the average velocit of the object in miles per hour for the given interval of time. s( t).08t 0 for 4 t 8 Page of 0

4 Quadratic, Polnomials, & and other misc. functions:. Determine the domain, range and intercepts for each of the following. A) = 9 B) f() = C) f ( ) 4 C) f ( ) D) f ( ) E) f ( ). Given the equation f() = a c, the -intercept will be.. How do ou find the discriminant and what does the discriminant determine? 4. A rectangular plaing field with a perimeter of 00 meters is to have an area of at least 00 square meters. Within what bounds must the length be? Eplain our answer.. A farmer has 946 meters of fencing available to enclose a rectangular portion of his land. One side of the rectangle being fenced lies along a river, so onl three sides require fencing. (a) Epress the area A of the rectangle as a function of, where is the length of the side parallel to the river. (b) For what value of is the area largest? 6. Determine if + is a factor of Find the polnomial, in standard form that has, - and as zeros. 8. List the possible rational zeros for the function f() = + The possible values are found b listing all the p/q values. To solve: use snthetic division to see when ones give Solve: + 6 = 0 Page 4 of 0

5 0. At most, how man solutions will solution. f ( ) have? Eplain how ou determined our. If one of the zero s is i, find the remaining zeros of the function f ( ) Write the piecewise function for the graph. State the intervals where the function is increasing, decreasing, and constant.. Given the function f ( ) ( 4)( ) ( ), determine the following: a. X and intercepts: b. What happens at each -intercept: c. End Behavior (using it notation): d. Maimum number of turning points: e. Graph the function. - - Page of 0

6 Rational Functions:. Determine the horizontal, vertical and slant asmptotes for the graph of each equation. A) f ( ) B) f ( ) C) f ( ) 0 Horiz: If n = d then = A/B If n > d, then none If n < d, then = 0. Find the domain of the function. f b g Vert: Simplif fraction and then set denominator = 0 Slant: If n is larget than d, then perform long division to get slant asmpt. Find all the asmptotes, make a table and graph a) f( ) b) f( ) Do problems 4-8 on our own paper. (-8 are not on the 04 test) 4. Solve the inequalit: 7. Find the partial fraction decomposition of 6. Write the partial fraction decomposition of f( ) 4 A B 7. If find A Write the from of the partial fraction decomposition of Page 6 of 0

7 Composition of Funtions, Inverses, Odd/Even Funtions, & Imaginar Numbers:. Classif 8 f ( ) 7 as even, odd or neither.. An odd function has smmetr with respect to the. Sketch the remain portion of the function if a) The function is odd b) The function is even If ou substitute for and get the same equation, the function is considered even. If ou substitute for and get an equation totall opposite in signs as compared to the original, then the function is odd. 4. If f ( ) and g( ) 8, find: A) ( f g)( ) B) ( f * g )( ). f ( ) g( ), find f ( g( )). 6. Confirm that f and g are inverses b showing that f ( g( )) and g( f ( )) if f ( ) g( ) 7. Find f ( ) graphicall and algebraicall if f ( ) - - Page 7 of 0

8 Eponential & Logarithmic Functions:. Graph: A) f ( ) log ( ) B) f() = A certain population grows according to the equation = 40 e.0t. Find the initial population and the population when t = 0.. Write log 7 in eponential form. 4. Write e = 4 in logarithmic form.. Write the following as a single logarithm: log 4 log log 8log z 6. Solve: A) log log 4 = log 4 B) = 7. Suppose $7 is deposited in a savings account. The interest rate is 9.% and is compounded continuousl. When will the original deposit be quadrupled? 8. Find the balance in an account at the end of ears if $400 is invested at an interest rate of 6% rt compounded continuousl. ( A Pe ) Calculator read form: Answer: 9. The population of an endangered animal species is. If the population is increasing at an annual rate of 8%, write an equation that can be used to predict future populations. General Equation: Find the population in ears: Page 8 of 0

9 Matrices: A 4 4 B C D E 4. What is the final dimensions of A*B and C*E?. How come ou cannot multipl B*E?. Which of the matrices above DO NOT have an inverse? Eplain wh: 4. Use the matrices above to find the following without using our calculator. a. A = b. B = c. B*C d. A D e. D*E f. D - C*B. Find the determinant a 0 4 z 6. Solve for : 4 7. Write the following as a sstem as an augmented matri and used row reduced Eschelon form to find and 6 4 : 4 Page 9 of 0

10 Conics: (not on 04 eam) Page 0 of 0

Name Please print your name as it appears on the class roster.

Name Please print your name as it appears on the class roster. Berkele Cit College Practice Problems Math 1 Precalculus - Final Eam Preparation Name Please print our name as it appears on the class roster. SHORT ANSWER. Write the word or phrase that best completes