Geometry 1. Two birds are flying toward a birdhouse that is halfway between them. The birds are at coordinates A (-4, 4) and B (10, -2). How far will each bird fly (to the nearest foot) to arrive at the birdhouse, if each grid represents 100 feet?: * a.580 feet b.762 feet c.1,160 feet d.1,523 feet 2. The line segment CD is the diameter of a circle whose center is the point (2, 1). If the coordinates of C are (0, -2), find the coordinates of D.: * a.(0, 4) b.(4, 0) c.(4, -2) d.(4, 4) 3. Write in standard form the equation of the line that passes through the point (3,-6) and is perpendicular to the line that passes through the points (1,3) and (6,-7).: * a.y + 6 = 1/2(x - 3) b.2x - y = 12 c.-1/2x + y = -7.5 d.x – 2y = 15 4. Which of the following groups of points are the vertices of a right triangle?: * a.(2,2) (4,1) (4,6) b.(1,1) (4,4) (7,2) c.(2,6) (-4,3) (-3,0) d.(1,1) (3,0) (0,-4) 5. Find the area of the dark blue sector shown below. The radius of the circle is 4 units and the length of the arc (the curved edge of the sector) measures 7.85 units. Express answer to the nearest tenth of a square unit.: * a.0.3 sq. units b.15.7 sq. units c.19.7 sq. units d.50.3 sq. units. 6. Find the value of x in ΔABC: * a. 4√7 b. 3√7 c. 2√30 d. √130 7. The length of a rectangle is 1 less than twice its width. A diagonal drawn in the rectangle measures 15. Find the length of the rectangle.: * a.6.64 b.9 c.13.21 d.7.2 8. In the following isosceles trapezoid AB = 3x – 5, CD = x + y, AC = 4x, BD = 3y. Find the length of the diagonal AC.: * a.10 b.30 c.17.5. d.40 9. Find the area of the rhombus MATH if HA = 6 and TA = 5. : * a.15 b.7√2 c.24 d.27 10. A side of an equilateral triangle measures 10cm. Find the area of the triangle.: * a.50 b.25√3 c. 50√3 d.There is not enough information to solve. 11. What is the surface area of a prism whose bases are right triangles with one side measuring 1.5 meters and a hypotenuse measuring 2.5 meters, if it is 2.75 meters tall?: * a. 8.25 m2 b. 16.5 m2 c. 19.5 m2 d. 22.5 m2 12. Write a converse for the statement, ''If a figure has five angles, then it is a pentagon.'':* a.''If a figure has five angles, then it is not a pentagon.'' b.''A pentagon is a figure with five angles'' c.''If a figure is a pentagon, then it has five angles'' d.''If a figure is not a pentagon, then it does not have five angles'' 13. Find the volume of the cone. (Round to the nearest tenth.): * a.15.4 cu. units b.29.5 cu. units c.38.1 cu. units d.75.4 cu. units 14. In the circle given CD = 9, BC = 3. Find the length of AB.: * a.27 b.21 c.24 d.30 15. Find the coordinates of A (3, -2) after it has been translated by reflecting it over the graph of y = 1, then reflecting it over y = -1 and then rotate it 90° around the origin.: * a.(2,-1) b.(6,3) c.(-6,-3) d.(-2,3) 16. The locus of points equidistant from the four vertices of a given rectangle, in the same plane as the rectangle is:: * a.one line b.two lines c.one point d.a circle 17. Find the area of a regular decagon with a 7 inch radius to the nearest inch.: * a.288 b.144 c.46 d.142 18. Given AD = BE, ∠CDE = ∠CED. Which way will be used to prove ΔCAD = ΔCBE?: * a.AAS b.SAS c.SSS d.ASA 19. A boat travels 150 miles directly east. It then changes course and travels 40 miles due north. Find the direction it should head to return to its point of origin and the time the trip will take if the boat travels 23 miles/hour.: * a. 75° west of south, 155.24 miles b. 67° west of south, 6 hours 45 minutes c. 67° south of west, 155.24 miles d. 75° west of south, 6 hours 45 minutes The measures of the angles of a quadrilateral are x+10, 2x+20, x+70, 2x-40. What type or types of quadrilateral could this be? I. rectangle II. trapezoid III. parallelogram: * a.II only b.III only c.I and III only d.II and III only 21. In this parallelogram find the measures of x and y.: * a.76, 37 b.42, 37 c.76, 113 d.42, 113 22. If SW=8, find RT to the nearest tenth (Assume RSTW is a kite).: * a.20.1 b.12.8 c.15.6 d.13.6 23. The volume of a spherical ball with radius of x cm is 125 cm3. Find the volume of a similar balloon with radius 2x cm.: * a. 250 cm3 b. 500 cm3 c. 1000 cm3 d. 750 cm3 24. The volume of a sphere is 4849.05 m3. What is the surface area of the sphere to the nearest hundredth?: * a. 1385.44 m2 b. 2971.82 m2 c. 346.36 m2 d. 2033.17 m2 25. Two vertices of equilateral ΔABC are A(1,7) and B(5,2). The triangle undergoes a dilation with scale factor 3.5. What is the perimeter of the image A'B'C'?: * a.22.41 b.67.23 c.19.21 d.6.40

Exponential Growth / Decay (Appreciation / Depreciation) Formulas and Example

Exponential growth/appreciation per time period 't' by a percentage (or rate) 'r' and an initial amount 'a' is given by the function f, where f(t)=a*(1+r)^t. That is:



Exponential decay/depreciation per time period 't' by a percentage 'r' and an initial amount 'a' is given by the function 'f', where f(t)=a*(1-r)^t. That is:
 

Example:
In 1967, you could by an Austin Healy 3000 Mark III sports car for $4000.00. Since then, the car has appreciated in value by 6.12% per year.

What function models the value, f, of the car t years after 1967?
Solution: a=4000, r-.0612. Thus,


What was the value of the car in 2004?
Solution: t=37. Thus,


In what year will the car's value appreciate to over $50,000? Explain the process you used to find the answer.
Solution: 
Solve 50000=4000*(1.0612)^t
by graphing
y1=50000
y2=4000*(1.0612)^x
and find the x-value of the intersection of y1 and y2.
x=42.5 year

2009.5, or 2010 by rounding up.

Simple Interest vs. Compound Interest vs Continuously Compounded Interest

Simple Interest: You get some % of the original deposit. For example, if the original deposit is $1500, and the annual interest is 10%, then every year you will earn $150 from interest.

f(x)=$1500 +150x
f(1)=1500+150
f(2)=1500+150*2, etc.
f(8)=1500+150*8=1500+1200=2700

Compound Interest: You get some % both the original deposit plus the previously earned interest per pay period. For example, if the original deposit is $1500 and the the compound interest per month is 7.75%, then you earn 7.75% of $1500 + al previously earned interest.

f(x)=1500*(1+.0775)^x
f(1)=1500*1.0775=1616.25
f(2)=1500*1.0775^2=(1500*1.0775)*1.0775
=1616.25*1.0775=1741.5, etc.

f(8)=1500*(1.0775)^8=2725.39

Continuous Compound Interest Formula:
f(x)=Initial*e^(rate*time)
f(x)=1500*e^(.0775*t)
f(8)=1500*e^(.0775*8)=2788

01:48:07

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Calculus #1: * (A) 1/3 (B) 0 (C) 3 (D) Does not exist #2: * (A) 1 (B) 0 (C) -1 (D) Does not exist #3 Find the horizontal asymptote of the following function:: * (A) 1 / 2 (B) -1 / 3 (C) 2 (D) -3 (E) 0 #4 Which of the following statements must be correct?: * (A) The function has vertical asymptotes at x = 0 and 4. (B) The function has a horizontal asymptote of y = 1 and a root at x = 3. (C) The function has roots at x = 3 and 4. (D) f(4) = 1/4 (E) B and D are correct #5 Describe the limit of f(x) as x approaches 4.: * (A) 1 / 4 (B) –1 / 4 (C) 0 (D) Does not exist Questions 6 - 7: Consider a function f(x), differentiable on [-1, 5]. Suppose f(1) = f(4) = 0 and f(3) = 6. #6 Which of the following is not necessarily true? : * (A) f ’(3) exists. (B) There is some point c in [1, 4] where f ’ (c) = 0. (C) There is some point d in [1, 3] where f(d) = 5 (D) All of these are true. #7 What is the average rate of change on [1, 3]?: * (A) 0 (B) 3 (C) 1 / 3 (D) 2 (E) 1 / 2 #8 Which is an example of a function that is continuous at x = 5 but not differentiable at x = 5?: * Choice (A) Choice (B) Choice (C) Choice (D) Choice (E) #9 What value of c will make f(x) continuous at x = 3?: * (A) -8 (B) -6 (C) -1 (D) any real number for c will work #10 Find the derivative of g(x) = (x3 – 5)(x + 4).: * (A) g’(x) = 3x2 (B) g’(x) = 4x3 + 12x2 - 5 (C) g’(x) = 4x3 + 12x2 + 5 (D) g’(x) = 4x3 - 12x2 - 5 Questions 11 - 13: Suppose f(x) and g(x) are differentiable functions on [-5, 5] and have values given by the following table:: * Choice (A) Choice (B) Choice (C) Choice (D) Choice (E) : * Choice (A) Choice (B) Choice (C) Choice (D) Choice (E) #13 Find (f o g) ’(-2), where o denotes usual functional composition.: * (A) 1 (B) 5 (C) 0 (D) 2 (E) 10 #14: * (A) sin (x) (B) –sin (x) (C) cos (x) (D) –cos (x) #15 If f(x) = sin2(e3x), find f ’ (x): * (A) 2 sin (e3x) cos (e3x) (B) sin (2e3x) (C) 3 sin (2e3x) (D) 3 sin (e3x) cos (e3x) (E) none of these are equivalent to f ’ (x). #16 Find the derivative y’ of the equation x + y = xy.: * Choice (A) Choice (B) Choice (C) Choice (D) #17 Which of the following functions satisfy f(ab) = f(a) + f(b) for all positive real values a and b?: * (A) f(x) = ex (B) f(x) = log x (C) f(x) = log 2x (D) B and C are correct. #18 Which of the following are the domain and range, respectively, of f(x) = Arctan(x) where Arctan denotes the principal inverse tangent function?: * (A) [-1, 1] and [0, π] (B) [-1, 1] and (-π / 2, π / 2) (C) (-∞, ∞) and (-π / 2, π / 2) (D) (-∞, ∞) and [0, π] Questions 19 - 21: Consider the following graph of the derivative f‘(x) of a three times differentiable function f(x), defined on the open interval (-3, 3) (thus f ’, f ’’, and f ’’’ all exist). Do not consider any behavior outside of this interval in answering.: * (A) 3 (B) 4 (C) 5 (D) 6 (E) cannot determine from the information given #20 At what value(s) of x, if any, does f(x) have a local maximum?: * (A) 1.5 (B) -1.5 (C) -1 and 1 (D) 0 #21 On which of the following intervals is f ’(x) concave up?: * (A) (-3, -2.1) (B) (-2.1, -1) (C) (-1, 0) (D) (-2.5, -1.5), (-0.5, 0.5), and (1.5, 2.5) #22: * Choice (A) Choice (B) Choice (C) Choice (D) #23: * Choice (A) Choice (B) Choice (C) Choice (D) Choice (E) #24: * Choice (A) Choice (B) Choice (C) Choice (D) #25: * Choice (A) Choice (B) Choice (C) Choice (D) #26: * (A) f(1) (B) f(1)(6) (C) -f(1) (D) -f(1)(6) For 27-28, consider the graphs y = 2x and y = 2x2 on [0, 1]. #27 Let S be the region enclosed by those graphs. What is the volume of the solid generated when S is revolved about the line y = 3? : * Choice (A) Choice (B) Choice (C) Choice (D) Choice (E) #28 The area bounded by the two curves on [0, 1] is given by: * Choice (A) Choice (B) Choice (C) Choice (D) Choice (E) #29: * (A) 1 (B) -1 (C) 0 (D) The integral diverges. #30: * (A) 2 (B) 4 (C) 1 (D) The integral diverges. #31 Find the maximum value of f(x) = 18x – 6x3 on [-2, 5].: * (A) 0 (B) 12 (C) -12 (D) 1 Question 32 uses the following slope field for a particular differential equation.: * (A) y = x2 (B) y = ex (C) y = e-x (D) y = cos x (E) y = ln x For questions 33-34, assume f(x) is twice differentiable on [1, 7], f(1) = 0, f(7) = 6, f(3) = 4, f ’(3) = 0, and f ’’ (3) = -2. #33 Which of the following must be true? : * (A) f(x) has a local maximum at x = 3. (B) f(x) has a local minimum at x = 3. (C) 9/4 (D) Cannot be found as the Mean Value Theorem assumptions are not met. #34 Where does f(x) assume its absolute maximum on [1, 7]?: * (A) At x = 1. (B) At x = 3. (C) At x = 7. (D) Cannot determine since there may be other critical points to check. #35 Two people are 50 feet apart. One of them starts walking north at a rate so that the angle θ between the two people’s paths is changing at a constant rate of 0.01 radians per minutes. At what rate is the distance between the two people changing when the θ = 0.5 radians?: * (A) 0.24 ft/min (B) 0.31 ft/min (C) 0.80 ft/min (D) 1.04 ft/min #36 Suppose the length of a diagonal of a square increases from 16 cm to 16.1 cm. What would the change in the area of the square be?: * (A) 1.61 cm2 (B) 0.07 cm2 (C) 6.42 cm2 (D) 0.01 cm2 #37: * (A) y – 2 = 3x (B) y – 2 = 6x (C) y – 2 = 1.5x (D) y = 3x #38: * (A) etan x + C (B) Cetan x (C) tan x + C (D) Cetan x + D (E) eC tan x + D #39: * (A) -5 (B) 5 (C) 9 (D) 19 (E) Cannot be determined from the given information. #40 Suppose f(x) is twice differentiable on [0, 7]. If f(3) = 5, f ’(3) = 2, and f ’’(3) = -1, which of the following statements is/are correct?: * (A) At x = 3, f(x) is increasing at an increasing rate. (B) At x = 3, f(x) is increasing at a decreasing rate. (C) f(7) ≤ 5 (D) f(x) has a local maximum at x = 3. (E) B, C, and D are correct. #41 Let x = t5 - 4t3 and y = t2 be parametric equations. Find the first derivative for this set of parametric equations at t = 1.: * (A) - 7 / 2 (B) – 1 / 3 (C) - 3 (D) – 2 / 7 #42 Let f(3) = 5, f(5) = 2, f ’ (3) = 4, and f ’(5) = 9. Let g(x) be the inverse of f(x) and assuming both f(x) and g(x) are differentiable for all real numbers. Find g’(5).: * (A) 1 / f ’(3) (B) f ‘(3) (C) 1 / f ‘(5) (D) f ‘(5) (E) Cannot be determined. #43 Find the Maclaurin series expansion of f(x) = e2x.: * Choice (A) Choice (B) Choice (C) Choice (D) #44 Let Sk = 3 - (k - 1) be a sequence starting at k = 1. To what value does the sequence Sk converge?: * (A) 1 / 3. (B) 4 / 3. (C) 3 / 2. (D) 0 #45: * Choice (A) Choice (B) Choice (C) Choice (D) #46 Which of the following series converge?: * Choice (A) Choice (B) Choice (C) Choice (D) Choice (E) #47: * Choice (A) Choice (B) Choice (C) Choice (D) Choice (E) #48: * (A) |x| = 4 (B) All non-zero values of x. (C) |x| > 4 (D) |x| ≤ 4 (E) The series diverges for all x. Questions 49 - 51: Define the two vectors a = <1, 3, -4> and b = <0, 5, 1> #49 Find the dot product. : * (A) 19 (B) 20 (C) 12 (D) 11 #50 Find the cross product.: * (A) <23, -1, 5> (B) <-23, -1, 5> (C) <23, 1, 5> (D) <23, 1, -5> #51 The angle x between vectors <-1, 1, 3> and <2, 0, -2> is given by solving which equation?: * Choice (A) Choice (B) Choice (C) Choice (D) #52 Consider the function y = x2 + 4. Compute the left endpoint, right endpoint, and midpoint approximations to the area under the curve on the interval [0, 2] with four subdivisions. Which of the following statements is correct?: * (A) Left > Mid > Right (B) Left > Right > Mid (C) Right > Mid > Left (D) Right > Left > Mid #53 Find the arc length of y = sin(√x) on [a, b] where a and b are positive.: * Choice (A) Choice (B) Choice (C) Choice (D) Choice (E) Questions 54 - 56: A particle is moving with the velocity function v(t) = t2 – 14t + 48. #54 At what point(s) does the particle have zero acceleration? : * (A) t = 6 and t = 8 (B) t = 7 (C) t = 6 (D) The acceleration is constant at 2 and never zero. #55 Where is the particle moving backwards?: * (A) t > 8 or 0 < t < 6 (B) 6 < t < 8 (C) t < 8 (D) t = 8 #56 Let s(t) represent the position of the particle at time t and suppose the initial position is 10. Find s(2) rounded to the nearest integer.: * (A) 24 (B) 34 (C) 61 (D) 71 (E) 81 #57: * Choice (A) Choice (B) Choice (C) Choice (D) #58 Which integral can be used to calculate the area enclosed by the smaller loop of the graph of the polar equation r = 1 + 2 sin θ?: * Choice (A) Choice (B) Choice (C) Choice (D) #59 Which of the following is not necessarily true about a real-valued function f(x)?: * (A) If f(x) is differentiable at x = 3, then f(x) is continuous at x = 3. (B) If f ’(3) = 0, then f(x) has an absolute maximum or absolute minimum at x = 3. (C) If f(x) is differentiable at x = 3, then g(x) = (f(x))3 is differentiable at x = 2. (D) All of the above are always true #60 All of the following statements are always correct for functions f(x) and g(x) except:: * Choice (A) Choice (B) Choice (C) Choice (D) Choice (E)