PreCalculus Second Semester Review Ch. P to Ch. 3 (1st Semester) ~ No Calculator

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1 PreCalculus Second Semester Review Ch. P to Ch. 3 (1st Semester) ~ No Calculator Solve. Express answer using interval notation where appropriate. Check for extraneous solutions. P3 1. x x x = P = 3 3 x+ 1 x x x.4 3. Find the zeros of the function.8 4. x 3 x + x 0 f (x) = x 4 + 3x 3 3x + 3x 4 given zeros x = 1 and x = 4 P1 Simplify. Express your answer without negative exponents. ( ) 3 5. uv 6. 4a3 b u 5 v a b 3b 3 a b 4 P4 7. Write the equation of a line a) parallel to and b) perpendicular to 5x y = 7 and passing through the point (3, 4). 1. Prove algebraically whether the function is even, odd, or neither. 8. f (x) = 3x 3 x 9. f (x) = x 4 4x + 7 p. 1

2 1. Find the domain. Express the answer in interval notation. 10. gx ( ) 6 5x = 11. f( x) = log ( x+ 3) 1.5 Identify the parent function, as well as the transformation. Then, graph the function including key points and/or asymptotes. 1. f x 3 x ( ) 1 = gx ( ) = x Graph the piecewise-defined function. State whether the function is continuous or discontinuous at x = 0. x if x f ( x) = x 1 if x > 0 x if x < f ( x) = 3 x if x Find the inverse of hx 3 ( ) = 6x 1 p.

3 Simplify. Express the answer in a + bi form. + 3i 1 i.7 Find (if it exists) the a) asymptotes, b) intercepts, and c) domain of the function. Be sure to list any holes. 18. g( x) = x 9 x x Simplify each expression. 4 (a) log 5 1 (b) log 10 (c) 3 log Expand each logarithm: (a) log 8 5 y x (b) log x (c) ln ( 6x e ) Calculator Allowed A radioactive isotope decays at a rate of 3% per day. A scientist has an initial amount of 50 g. Write a model for this situation. Determine approximately how many days it will take for half the isotope to decay. p. 3

4 3.5. Solve algebraically: (a) log 3 x + log 3 (x + 8) = (b) ( x ) log + 5 log x = 7 x (c) 3 6 = 4 (d) 1 7 = 3 4 6x Ch. 4, Ch. 5, 9., 9.4, 6.1and 6.3 ( nd Semester) ~ No Calculator Find each exact value π (a) cos (b) 4 7π sin 6 (c) 3π tan (d) 7π cos Find one positive angle and one negative angle that are coterminal with: 3 π Given: sin θ = functions of θ. and cos θ > 0. Find the values of the remaining five trigonometric 3 p. 4

5 State the amplitude, period, phase shift, and vertical shift, and then graph each function. Clearly label all tick marks on the axes. (a) y tan ( 4x) = (b) y = sec x 1 sin 6 3 π = (d) y = 3sin x (c) y ( x) π (e) y = tan x + (f) 1 π y = cos x π π Write an equation of the cosine function with amplitude =, period =, phase shift = 8 and vertical shift = Use the given graph 5.3 (a) Write a sine function that fits the graph. (b) Write a cosine function that fits the graph. (c) Using identities, prove that the two equations you wrote are equal. p. 5

6 Find each value. 1 (a) arccos (b) cos 1 1 (c) 1 3 sec sin (d) 1 3 sin tan 3 (e) cos[arcsin( 1)] (f) sin 1 5π sin Write an algebraic expression equivalent to each expression below. Hint: Draw a. (a) 1 tan(sin x) 1 (b) sec( tan x) 1 (c) cot ( csc 3x) 5.1- Verify that each of the following is an identity. Be sure to show all steps. 5.4 π 31. sin θ( csc θ 1) + tan ( θ) cosθ + cos θ = cos θ 3. sin β cos β + = 1 csc β sec β tan x 33. sin x 1 + tan x = 34. cos x cos x + = sec x 1+ sin x 1 sin x 35. sin (π x) = sin x p. 6

7 Find the exact value of cos Rewrite using identities, then simplify if possible sin 150 π π π π 38. sin cos + cos sin If cos x = 1 and x is in the second quadrant, find 13 (a) sin (x) (b) tan (x) If cot x = and x is in the fourth quadrant, find 5 (a) sin (x) (b) cos (x) Ch5 41. Solve each equation for [0, π). (a) sin x = 3 sin x (b) 8cos x = 4 (c) cos( x) + sin ( x) = 0 (d) ( x) ( x) sin cos = 0 p. 7

8 Ch5 4. Solve for all values of x. (a) cos x sin x 0 + = (b) 3π sin x = 1 Calculator Allowed The wheel (including the tire) of a sports car under development by an auto company has an eleven inch radius. How many rpm s does the wheel make at 55 mph? Find the measure of the intercepted arc in terms of π in a circle with diameter 60 inches and central angle of The angle of depression from the top of a building to a point 55 feet away from the building (on level ground) is 39. Determine the height of the building A shoreline runs north-south, and a boat is due east of the shoreline. The bearings of the boat 5.6 from two points on the shore are 110 and 100. Assume the two points are 550 feet apart. How far is the boat from the shore? Find the area of each triangle. (a) a = 7, b = 1, c = 13 (b) A = 47, b = 3, c = 19 p. 8

9 Solve each triangle. If there are two s, solve both! 5.6 (a) A = 36, b = 13, C = 48 (b) a = 15, A = 5, b = 150 (c) a = 1.5, b =.3, c = The sequence {, 6, 18, 54,...} is geometric. Find (a) a recursive rule for the nth term. (b) an explicit formula for the nth term Suppose an arithmetic sequence contains a = 49 and a = Find 18 5 (a) the common difference (b) a 1 (c) a recursive formula The fourth and ninth terms of a geometric sequence are 18 and respectively. Find a) the common ratio b) a 1 c) an explicit formula p. 9

10 Write each series using sigma notation. (a) (b) Find the sum of the terms of the arithmetic sequence: {8,, 16, 10,..., 38} For #54 and #55, tell whether the series converges or diverges. If the series converges, find its sum. If the series diverges, explain why Expand: (a) (x ) 4 (b) (x + y) Find the coefficient of the x 8 y 3 term in (x + y) A vector u has a magnitude of 3.4 cm and a direction of 7. Write the component form for this vector. p. 10

11 Given points A( 6, ) and B(1, 3). Find the component form and magnitude of AB Given v = 3,4. Find v and give the direction angle in [0, 360 ) A force of 45 lb acts on an object at an angle of 65. A second force of 85 lb acts on the object at an angle of 55. Find the direction and magnitude of the resultant force. 45 lb Eliminate the parameter and identify the graph of the curve. a) x = t 4, y = t² b) x = 4 sin t, y = 4 cos t Create a table of values and the graph of the pair of parametric equations x = t 1, y = t p. 11

12 Gretchen Austgen, an outfielder for the West Chicago Wildcats, is 15 feet from home plate after catching a fly ball. The runner tags third and heads for home. Gretchen releases the ball at an initial velocity of 75 ft/s at an angle of 5 with the horizontal. Assume Gretchen releases the ball 5 feet above the ground and aims it directly in line with the plate. (a) Write two parametric equations that represent the path of the ball. (b) Use a calculator to graph the path of the ball. Sketch the graph shown on the screen. (c) How far will the ball travel horizontally before hitting the ground? (d) When will the ball hit the ground? (e) What is the maximum height of the trajectory? (f) When will the object reach its maximum height? p. 1