Pre-Calculus Semester 2 Final Exam

Find the fourth roots of 256(cos 280° + i sin 280°).

360 degrees / 4 = 90 degrees First Root (1/4)(280 degrees) = 70 degrees 4(cos 70 degrees + i sin 70 degrees) Second Root 70 degrees + 90 degrees = 160 degrees 4(cos 160 degrees + i sin 160 degrees) Third Root 160 degrees + 90 degrees = 250 degrees 4(cos 250 degrees + i sin 250 degrees) Fourth Root 250 degrees + 90 degrees = 340 degrees 4(cos 340 degrees + i sin 340 degrees)

If a person puts 1 cent in a piggy bank on the first day, 2 cents on the second day, 3 cents on the third day, and so forth, how much money will be in the bank after 40 days? A. $8.20 B. $16.40 C. $0.40 D. $4.10

A. $8.20

Find an equation in standard form for the hyperbola with vertices at (0, ±4) and foci at (0, ±5). A. (y^2/16) - (x^2/9)= 1 B. (y^2/25) - (x^2/16)= 1 C. (y^2/16) - (x^2/25)= 1 D. (y^2/9) - (x^2/16)= 1

A. (y^2/16) - (x^2/9)= 1

Find the limit of the function algebraically. lim (x^2 - 4) / (x + 2) x->-2 A. -4 B. -2 C. Does not exist D. 1

A. -4

Find the limit of the function by using direct substitution. lim (x^2 + 3x - 1) x->3 A. 17 B. 0 C. -17 D. Does not exist

A. 17

Find the derivative of f(x) = 4x + 7 at x = 5. A. 4 B. 1 C. 5 D. 7

A. 4

Given that P = (-4, 11) and Q = (-5, 8), find the component form and magnitude of vector QP. A. <1, 3>, sqrt 10 B. <-9, 3>, sqrt 0 C. <1, 3>, 10 D. <-1, -3>, sqrt 10

A. <1, 3>, sqrt 10

Let u = <6, -9>, v = <1, -4>. Find u - v. A. <5, -5> B. <10, -10> C. <15, 5> D. <7, -13>

A. <5, -5>

The given measurements may or may not determine a triangle. If not, then state that no triangle is formed. If a triangle is formed, then use the Law of Sines to solve the triangle, if it is possible, or state that the Law of Sines cannot be used. B = 111°, c = 8, b = 12 A. C = 38.5°, A = 30.5°, a ≈ 6.5 B. no triangle is formed C. C = 30.5°, A = 38.5°, a ≈ 6.5 D. The triangle cannot be solved with the Law of Sines.

A. C = 38.5°, A = 30.5°, a ≈ 6.5

Find the center, vertices, and foci of the ellipse with equation (x^2/36) + (y^2/100) = 1. A. Center: (0, 0); Vertices: (0, -10), (0, 10); Foci: (0, -8), (0, 8) B. Center: (0, 0); Vertices: (-10, 0), (10, 0); Foci: (0, -6), (0, 6) C. Center: (0, 0); Vertices: (-10, 0), (10, 0); Foci: (-8, 0), (8, 0) D. Center: (0, 0); Vertices: (0, -10), (0, 10); Foci: (-6, 0), (6, 0)

A. Center: (0, 0); Vertices: (0, -10), (0, 10); Foci: (0, -8), (0, 8)

Find the vertices and foci of the hyperbola with equation (((x + 5)^2)/81) - (((y - 3)^2)/144)= 1 A. Vertices: (4, 3), (-14, 3); Foci: (-20, 3), (10, 3) B. Vertices: (3, 7), (3, -17); Foci: (3, -17), (3, 7) C. Vertices: (7, 3), (-17, 3); Foci: (-17, 3), (7, 3) D. Vertices: (3, 4), (3, -14); Foci: (3, -20), (3, 10)

A. Vertices: (4, 3), (-14, 3); Foci: (-20, 3), (10, 3)

Find an equation for the nth term of a geometric sequence where the second and fifth terms are -2 and 16, respectively. A. an = 1 • (-2)^n - 1 B. an = 1 • 2^n C. an = 1 • (-2)^n + 1 D. an = 1 • 2n - 1

A. an = 1 • (-2)^n - 1

Determine whether the vectors u and v are parallel, orthogonal, or neither. u = <3, 0>, v = <0, -6> A. orthogonal B. neither C. parallel

A. orthogonal

Eliminate the parameter. x = t - 3, y = t^2 + 5 A. y = x^2 + 6x + 14 B. y = x^2 - 14 C. y = x^2 - 6x - 14 D. y = x2 + 14

A. y = x^2 + 6x + 14

State whether the given measurements determine zero, one, or two triangles. A = 36°, a = 3, b = 9 A. zero B. two C. one

A. zero

Give an example of a function with both a removable and a non-removable discontinuity.

An example of a function with both a removable and a non-removable discontinuity is f(x)= (x + 1)(x + 2) / (x + 2)

Find the rectangular coordinates of the point with the polar coordinates. (-5, (5pi/3)) A. ((-5/2), (5/2)) B. ((-5/2), ((5 sqrt 3)/2)) C. (((5 sqrt 3)/2), (-5/2)) D. ((5/2), (-5/2))

B. ((-5/2), ((5 sqrt 3)/2))

Find all polar coordinates of point P where P = (5, (-pi/6)) A. (5, (-pi/6) + 2nπ) or (-5, (-pi/6) + 2nπ) B. (5, (-pi/6) + 2nπ) or (-5, (-pi/6) + (2n + 1)π) C. (5, (-pi/6) + 2nπ) or (-5, (pi/6) + (2n + 1)π) D. (5, (-pi/6) + (2n + 1)π) or (-5, (-pi/6) + 2nπ)

B. (5, (-pi/6) + 2nπ) or (-5, (-pi/6) + (2n + 1)π)

Find the derivative of f(x) = 4/x at x = 2. A. -4 B. -1 C. 1 D. 4

B. -1

Evaluate the expression. r = <9, -7, -1>, v = <2, 2, -2>, w = <-5, -2, 6> v ⋅ w A. <-18, 14, -2> B. -26 C. 1 D. <-10, -4, -12>

B. -26

Find the limit of the function by using direct substitution. lim (x^2 - 6) x->0 A. Does not exist B. -6 C. 6 D. 0

B. -6

Find the derivative of f(x) = -4x^2 + 11x at x = 10. A. -19.5 B. -69 C. 30 D. -300

B. -69

Use graphs and tables to find the limit and identify any vertical asymptotes of lim (1)/(x - 5) x->5^- A. 1 ; no vertical asymptotes B. -∞ ; x = 5 C. ∞ ; x = -5 D. -∞ ; x = -5

B. -∞ ; x = 5

Find the sum of the arithmetic sequence. 3, 5, 7, 9, ..., 21 A. 39 B. 120 C. 20 D. 23

B. 120

Find the limit of the function algebraically. lim (x^2 - 4) / (x - 2) x->2 A. 1 B. 4 C. 2 D. Does not exist

B. 4

Find the indicated limit, if it exists. {7 - x^2 x<0 lim f(x), f(x)= {7 x=0 x->0 {-10x + 7 x>0 A. The limit does not exist B. 7 C. -10 D. -17

B. 7

Find the limit of the function by using direct substitution. lim (x^2 + 8x - 2) x->1 A. -7 B. 7 C. Does not exist D. 0

B. 7

Two triangles can be formed with the given information. Use the Law of Sines to solve the triangles. B = 40°, b = 25, c = 26 A. A = 98°, C = 42°, a = 16.2; A = 2°, C = 138°, a = 16.2 B. A = 98°, C = 42°, a = 38.5; A = 2°, C = 138°, a = 1.4 C. A = 92°, C = 48°, a = 38.9; A = 88°, C = 132°, a = 38.9 D. A = 92°, C = 48°, a = 16.1; A = 88°, C = 132°, a = 16.1

B. A = 98°, C = 42°, a = 38.5; A = 2°, C = 138°, a = 1.4

Determine whether the sequence converges or diverges. If it converges, give the limit. 108, -18, 3, -1/2, ... A. Converges; 19980 B. Converges; 0 C. diverges D. Converges; 648/5

B. Converges; 0

Find the indicated limit, if it exists. lim f(x), f(x)= {x + 4 x<-5 x->-5 {4 - x x is greater than or equal to -5 A. -5 B. The limit does not exist C. 9 D. -1

B. The limit does not exist

Find the vertex, focus, directrix, and focal width of the parabola. (-1/20)x^2= y A. Vertex: (0, 0); Focus: (-10, 0); Directrix: x = 5; Focal width: 80 B. Vertex: (0, 0); Focus: (0, -5); Directrix: y = 5; Focal width: 20 C. Vertex: (0, 0); Focus: (0, -5); Directrix: y = 5; Focal width: 80 D. Vertex: (0, 0); Focus: (0, 5); Directrix: y = -5; Focal width: 5

B. Vertex: (0, 0); Focus: (0, -5); Directrix: y = 5; Focal width: 20

Find an equation for the nth term of the sequence. -2, -8, -32, -128, ... A. an = 4 • -2^n B. an = -2 • 4^n - 1 C. an = -2 • 4^n D. an = 4 • -2^n + 1

B. an = -2 • 4^n - 1

Find the standard form of the equation of the parabola with a focus at (0, -10) and a directrix at y = 10. A. y= (-1/10)x^2 B. y= (-1/40)x^2 C. y^2= -40x D. y^2= -10x

B. y= (-1/40)x^2

Determine two pairs of polar coordinates for the point (3, 3) with 0° ≤ θ < 360°. A. (3 square root 2 , 315°), (-3 square root 2 , 135°) B. (3 square root 2 , 135°), (-3 square root 2 , 315°) C. (3 square root 2 , 45°), (-3 square root 2 , 225°) D. (3 square root 2 , 225°), (-3 square root 2 , 45°)

C. (3 square root 2 , 45°), (-3 square root 2 , 225°)

Find an equation in standard form for the hyperbola with vertices at (0, ±4) and asymptotes at y = ±1/3 x A. (y^2/144) - (x^2/16)= 1 B. (y^2/16) - (x^2/36)= 1 C. (y^2/16) - (x^2/144)= 1 D. (y^2/36) - (x^2/4)= 1

C. (y^2/16) - (x^2/144)= 1

Find the limit of the function by using direct substitution. lim (x^2 - 3) x->0 A. 3 B. Does not exist C. -3 D. 0

C. -3

Use graphs and tables to find the limit and identify any vertical asymptotes of lim (1)/(x - 3) x->3^- A. ∞; x = -3 B. -∞; x = -3 C. -∞; x = 3 D. 1 ; no vertical asymptotes

C. -∞; x = 3

Find the limit of the function by using direct substitution. lim (3e^x cos x) x->pi/2 A. 1 B. pi/2 C. 0 D. 5e^pi/2

C. 0

Determine whether a triangle can be formed with the given side lengths. If so, use Heron's formula to find the area of the triangle. (5 points) a = 240 b = 132 c = 330 A. no triangle is formed B. 30,776.84 C. 13,385.87 D. 30,790.94

C. 13,385.87

Find the sum of the geometric sequence. 4/3, 16/3, 64/3,256/3, 1024/3 A. 1363/3 B. 1364/15 C. 1364/3 D. 1363/15

C. 1364/3

Find the area of the triangle with the given measurements. Round the solution to the nearest hundredth if necessary. A = 50°, b = 31 ft, c = 18 ft A. 427.45 ft^2 B. 179.34 ft^2 C. 213.73 ft^2 D. 558 ft^2

C. 213.73 ft^2

Find the derivative of f(x) = 8x + 4 at x = 9. A. 0 B. 4 C. 8 D. 9

C. 8

Find the derivative of f(x) = -9/x at x = -4. A. 4/9 B. 16/9 C. 9/16 D. 9/4

C. 9/16

Let u = <-6, 3>, v = <-1, -6>. Find 4u + 2v. A. <-26, -3> B. <-22, 24> C. <-26, 0> D. <-28, -12>

C. <-26, 0>

Find the indicated limit, if it exists. lim f(x), f(x)= {x + 10 x<8 x->8 {10 - x x is greater than or equal to 8 A. 8 B. 18 C. The limit does not exist D. 2

C. The limit does not exist

Solve the triangle. A = 51°, b = 14, c = 6 A. no triangles possible B. a ≈ 14.9, C ≈ 28.1, B ≈ 100.9 C. a ≈ 11.2, C ≈ 24.1, B ≈ 104.9 D. a ≈ 14.9, C ≈ 24.1, B ≈ 104.9

C. a ≈ 11.2, C ≈ 24.1, B ≈ 104.9

Find an equation for the nth term of the arithmetic sequence. -17, -13, -9, -5, ... A. an = -17 + 4(n + 2) B. an = -17 x 4(n - 1) C. an = -17 + 4(n - 1) D. an = -17 + 4(n + 1)

C. an = -17 + 4(n - 1)

Find the standard form of the equation of the parabola with a focus at (-2, 0) and a directrix at x = 2. A. y^2= 4x B. 8y= x^2 C. x= (-1/8)y^2 D. y= (-1/8)x^2

C. x= (-1/8)y^2

Find an equation in standard form for the ellipse with the vertical major axis of length 16 and minor axis of length 4. A. (x^2/64) + (y^2/4)= 1 B. (x^2/2) + (y^2/8)= 1 C. (x^2/8) + (y^2/2)= 1 D. (x^2/4) + (y^2/64)= 1

D. (x^2/4) + (y^2/64)= 1

Find the derivative of f(x) = 8/x at x= -1. A. 4 B. 0 C. 8 D. -8

D. -8

Find the limit of the function algebraically. lim (x^2 - 9) / (x^3 + 3) x->-3 A. -6 B. -3 C. Does not exist D. 0

D. 0

Find the first six terms of the sequence. a1 = 1, an = 4 • an-1 A. 0, 4, 4, 8, 12, 16 B. 4, 16, 64, 256, 1024, 4096 C. 1, 4, 8, 12, 16, 20 D. 1, 4, 16, 64, 256, 1024

D. 1, 4, 16, 64, 256, 1024

Find the derivative of f(x) = -11/x at x= 9. A. 11/9 B. 81/11 C. 9/11 D. 11/81

D. 11/81

A building has a ramp to its front doors to accommodate the handicapped. If the distance from the building to the end of the ramp is 19 feet and the height from the ground to the front doors is 4 feet, how long is the ramp? (Round to the nearest tenth.) A. 5.7 ft B. 18.6 ft C. 4.8 ft D. 19.4 ft

D. 19.4 ft

Find the derivative of f(x) = 12x^2 + 8x at x = 9. A. 256 B. -243 C. 288 D. 224

D. 224

Write the complex number in the form a + bi. 6(cos 330° + i sin 330°) A. -3 sqrt 3 - 3i B. -3 sqrt 3 + 3i C. 3 sqrt 3 + 3i D. 3 sqrt 3 - 3i

D. 3 sqrt 3 - 3i

Express the complex number in trigonometric form. 4i A. 4(cos 270° + i sin 270°) B. 4(cos 180° + i sin 180°) C. 4(cos 0° + i sin 0°) D. 4(cos 90° + i sin 90°)

D. 4(cos 90° + i sin 90°)

Find the angle between the given vectors to the nearest tenth of a degree. u = <8, 4>, v = <9, -9> A. 81.6° B. 25.8° C. 35.8° D. 71.6°

D. 71.6°

Solve the triangle. A = 46°, a = 34, b = 27 A. B = 34.8°, C = 99.2°, c ≈ 28 B. B = 34.8°, C = 119.2°, c ≈ 37.3 C. Cannot be solved D. B = 34.8°, C = 99.2°, c ≈ 46.7

D. B = 34.8°, C = 99.2°, c ≈ 46.7

Find the center, vertices, and foci of the ellipse with equation 5x^2 + 9y^2= 45. A. Center: (0, 0); Vertices: (-9, 0), (9, 0); Foci: (-2 sqrt 14, 0), (2 sqrt 14, 0) B. Center: (0, 0); Vertices: (0, -9), (0, 9); Foci: (0, -2 sqrt 14), (0, 2 sqrt 14) C. Center: (0, 0); Vertices: (0, -3), (0, -3); Foci: (0, -2), (0, 2) D. Center: (0, 0); Vertices: (-3, 0), (3, 0); Foci: (-2, 0), (2, 0)

D. Center: (0, 0); Vertices: (-3, 0), (3, 0); Foci: (-2, 0), (2, 0)

Find the indicated limit, if it exists. {5 - x x<5 lim f(x), f(x)= {8 x=5 x->5 {x + 3 x>5 A. 0 B. 8 C. 3 D. The limit does not exist

D. The limit does not exist

Find the indicated limit, if it exists. lim f(x), f(x)= {5x - 9 x<0 x->0 {|2 - x| x is greater than or equal to 0 A. -9 B. -7 C. 2 D. The limit does not exist

D. The limit does not exist

Find an equation for the nth term of the arithmetic sequence. a16 = 21, a17 = -1 A. an = 351 + 22(n + 1) B. an = 351 - 22(n + 1) C. an = 351 + 22(n - 1) D. an = 351 - 22(n - 1)

D. an = 351 - 22(n - 1)

Use graphs and tables to find the limit and identify any vertical asymptotes of the function. lim (1)/((x - 10)^2) x->10

Limit: infinity Vertical Asymptote: 10

Use mathematical induction to prove the statement is true for all positive integers n. 6 + 12 + 18 + ... + 6n = 3n(n + 1)

Step 1: Show that statement is true for n=1 6 = 3(1)(1 + 1) 6 = 3(2) 6 = 6 Step 2: Substitute k for n 6 + 12 + 18 + ... + 6k = 3k(k + 1) Step 3: Show that statement is true for k + 1 6 + 12 + 18 + ... + 6(k + 1) = 3(k + 1)((k + 1) + 1) 3k(k + 1) + 6(k + 1) 3(k + 1) (k + 2) 3(k + 1)((k + 1) + 1) Therefore, the statement is true for all positive integers n.

Determine if the graph is symmetric about the x-axis, the y-axis, or the origin. r = 8 cos 3θ

cos(3theta) = cos(-3theta) The graph is symmetric about the x-axis.

Find the limit of the function algebraically. lim (x^2 - 81) / (x - 9) x->9

lim (x^2 - 81) / (x - 9) x->9 lim (x - 9) (x + 9) / (x - 9) x->9 lim (x + 9) x->9 lim (9 + 9) x->9 lim = 18 x->9

A certain species of tree grows an average of 3.1 cm per week. Write an equation for the sequence that represents the weekly height of this tree in centimeters if the measurements begin when the tree is 600 centimeters tall.

n= number of weeks h= height of tree (cm) h = 600 cm + 3.1 cm(n - 1)

The position of an object at time t is given by s(t) = -1 - 13t. Find the instantaneous velocity at t = 8 by finding the derivative.

s(t)= -1 - 13t s(t)= 0 - 13(t) s(t) = -13 s(8)= -13

The position of an object at time t is given by s(t) = 1 - 10t. Find the instantaneous velocity at t = 10 by finding the derivative.

s(t)= 1-10t s(t + h) = 1 - 10(t + h) s(t + h) = 1 - 10t - 10t lim = 1 - 10t - 10h - (1 - 10t) h->0 = 1 - 10t - 10h - 1 + 10t = -10h -10h/h = -10 Because there is no t remaining the derivative of s(t) = 1 - 10t is -10.

Two forces with magnitudes of 90 and 50 pounds act on an object at angles of 30° and 160°, respectively. Find the direction and magnitude of the resultant force. Round to two decimal places in all intermediate steps and in your final answer.

u= 90[cos(30 degrees),sin(30 degrees)] = <77.94,45> v= 50[cos(160 degrees),sin(160 degrees)] = <-46.98,17.10> w = u + v = <77.94 + -46.98,45 + 17.10> = <30.96,62.1> ||w||= sqrt 30.96^2 + 62.1^2 = sqrt 958.52 + 3856.41 = sqrt 4814.93 = 69.39 pounds Magnitude of Resultant Force= 69.39 pounds theta = tan^-1(62.1 / 30.96) = tan^-1(2.01) = 63.55 degrees Direction of Resultant Force= 63.55 degrees