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mathematics
calculus
Questions and Answers of
Calculus
(a) Use the Intermediate Value Theorem and the table feature of a graphing utility to find intervals one unit in length in which the polynomial function is guaranteed to have a zero. (b) Adjust the
Use long division to divide. 1. (30x2 − 3x + 8) / (5x - 3) 2. (4x + 7) / (3x - 2) 3. (5x3 - 21x2 - 25x - 4) / (x2 - 5x - 1)
Use synthetic division to divide. 1. (2x3 − 25x2 + 66x + 48) / (x - 8) 2. (5x3 + 33x2 + 50x - 8) / (x + 4) 3. (x4 - 2x2 + 9x) / (x + 3) 4. 6x4 - 4x3 - 27x2 + 18x) / (x - 2)
Use the Remainder Theorem and synthetic division to find each function value. f (x) = x4 + 10x3 - 24x2 + 20x + 44 (a) f (-3), (b) f (-1)
Use synthetic division to determine whether the given values of x are zeros of the function. f (x) = 20x4 + 9x3 − 14x2 − 3x (a) x = −1 (b) x = 34 (c) x = 0 (d) x = 1
(a) verify the given factor(s) of f (x), (b) find the remaining factors of f (x), (c)use your results to write the complete factorization of f (x), (d)list all real zeros of f, and (e) confirm
Determine the number of zeros of the polynomial function. 1. f (x) = x - 6 2. g (x) = x2 − 2x − 8 3. h (t) = t2 − t5 4. f (x) = x8 + x9 5. f (x) = (x − 8)3 6. g (t) = (2t − 1)2 − t4
Find the rational zeros of the function. 1. f (x) = x3 + 3x2 − 28x − 60 2. f (x) = x3 − 10x2 + 17x − 8 3. f (x) = 3x3 + 8x2 − 4x - 16
Find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.) 1. 2/3, 4, √3i 2. 2, -3, 1 - 2i
Use the given zero to find all the zeros of the function. Function ........................................................................... Zero 1. h (x) = −x3 + 2x2 − 16x + 32
Write the polynomial as the product of linear factors and list all the zeros of the function. 1. f (x) = x3 + 4x2 − 5x 2. g (x) = x3 − 7x2 + 36 3. g (x) = x4 + 4x3 − 3x2 + 40x + 208 4. f (x) =
Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros
Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. 1. g (x) = 5x3 + 3x2 − 6x + 9 2. h (x) = −2x5 + 4x3 − 2x+ + 5
Use synthetic division to verify the upper and lower bounds of the real zeros of f. 1. f (x) = 4x3 − 3x2 + 4x - 3 (a) Upper: x = 1 (b) Lower: x = −1/4 2. f (x) = 2x3 − 5x2 − 14x + 8 (a)
A right cylindrical water bottle has a volume of 36 cubic inches and a height nine inches greater than its radius. Find the dimensions of the water bottle.
A kitchen has a volume of 60 cubic meters. The width of the room is one meter greater than the length and the height is one meter less than the length. Find the dimensions of the room.
(a) Find the zeros of each quadratic function g(x). (i) g(x) = x2 − 4x − 12 (ii) g(x) = x2 + 5x (iii) g(x) = x2 + 3x − 10 (iv) g(x) = x2 − 4x + 4 (v) g(x) = x2 − 2x − 6 (vi) g(x) = x2 +
One of the fundamental themes of calculus is to find the slope of the tangent line to a curve at a point. To see how this can be done, consider the point (2, 4) on the graph of the quadratic function
A rancher plans to fence a rectangular pasture adjacent to a river (see figure). The rancher has 100 meters of fencing, and no fencing is needed along the river.(a) Write the area A of the pasture as
A wire 100 centimeters in length is cut into two pieces. One piece is bent to form a square and the other to form a circle. Let x equal the length of the wire used to form the square.Figure(a) Write
At a glassware factory, molten cobalt glass is poured into molds to make paperweights. Each mold is a rectangular prism whose height is 3inches greater than the length of each side of the square
(a) Find the zeros of each quadratic function g(x). (i) g(x) = 2x2 + 5x − 3 (ii) g(x) = − x2 − 3x − 2 (b) For each function in part (a), find the zeros of f(x) = g(1/2 x). (c) Describe the
Quonset huts were developed during World War II. They were temporary housing structures that could be assembled quickly and easily. A Quonset hut is shaped like a half cylinder. A manufacturer has
Show that if f (x) = ax3 + bx2 + cx + d, then f(k) = r, where r = ak3 + bk2 + ck + d, using long division. In other words, verify the Remainder Theorem for a third-degree polynomial function?
In 2000 B.C., the Babylonians solved polynomial equations by referring to tables of values. One such table gave the values of y3 + y2. To be able to use this table, the Babylonians sometimes used the
Can a cubic function with real coefficients have two real zeros and one complex zero? Explain.
(a) Complete the table.(b) Use the table to make a conjecture relating the sum of the zeros of a polynomial function to the coefficients of the polynomial function. (c) Use the table to make a
Determine whether the statement is true or false. If false, provide one or more reasons why the statement is false and correct the statement. Letf x) = ax3 + bx2 + cx + d, a ( 0And let f (2) =
The parabola shown in the figure has an equation of the form y = ax2 + bx + c. Find the equation of this parabola using each method.(a) Find the equation analytically.(b) Use the regression feature
Is it possible for the graph of a rational function to have all three types of asymptotes? Why or why not? To work an extended application analyzing the total numbers of military personnel on active
1. A ________ is the intersection of a plane and a double-napped cone. 2. The equation (x − h)2 + (y − k)2 = r2 is the standard form of the equation of a ________ with center ________ and radius
Find the focus and directrix of the parabola. Then sketch the parabola. 1. y = 12x2 2. y = −4x2 3. y2 = −6x 4. y2 = 3x
Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. 1. Focus: (3, 0) 2. Focus: (0, 1/2) 3. Directrix: y = 2 4. Directrix: x = −4
Find the standard form of the equation of the parabola and determine the coordinates of the focus.1.2.
1. The light bulb in a flashlight is at the focus of the parabolic reflector, 1.5 centimeters from the vertex of the reflector (see figure). Write an equation for a cross section of the flashlight's
Each cable of the Golden Gate Bridge is suspended (in the shape of a parabola) between two towers that are 1280 meters apart. The top of each tower is 152 meters above the roadway.The cables touch
A simply supported beam (see figure) is 64 feet long and has a load at the center. The deflection of the beam at its center is 1 inch. The shape of the deflected beam is parabolic.(a) Write an
Find the standard form of the equation of the ellipse with the given characteristics and center at the origin.1.2. 3. Vertices: (±5, 0); foci: (±2, 0) 4. Vertices: (0, ±8);
Find the vertices and eccentricity of the ellipse. Then sketch the ellipse.1.2. 3. 4.
Find the standard form of the equation of the ellipse with the given vertices, eccentricity e, and center at the origin. 1. Vertices: ((5, 0); e = 4/5 2. Vertices: (0, (8); e = 1/2
A mason is building a semielliptical fireplace arch that has a height of 2 feet at the center and a width of 6 feet along the base (see figure). The mason draws the semi ellipse on the wall by the
A semielliptical arch over a tunnel for a one-way road through a mountain has a major axis of 50 feet and a height at the center of 10 feet. (a) Sketch the arch of the tunnel on a rectangular
Repeat Exercise 56 for a semielliptical arch with a major axis of 40 feet and a height at the center of 15 feet. The dimensions of the truck are 10 feet wide by 14 feet high. Refer to Exercise 56, A
A line segment through a focus of an ellipse with endpoints on the ellipse and perpendicular to the major axis is called a latus rectum of the ellipse. An ellipse has two latera recta. Knowing the
Sketch the ellipse using the latera recta (see Exercise 58).1.2. 3. 9x2 + 4y2 = 36 4. 3x2 + 6y2 = 30
Find the standard form of the equation of the hyperbola with the given characteristics and center at the origin. 1. Vertices: (0, ±2); foci: (0, ±6) 2. Vertices: (±4, 0); foci: (±5, 0) 3.
Find the vertices of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid.1.2. 3. 4.
A cross section of a sculpture can be modeled by a hyperbola (see figure).(a) Write an equation that models the curved sides of the sculpture. (b) Each unit on the coordinate plane represents 1 foot.
A hyperbolic mirror (used in some telescopes) has the property that a light ray directed at focus A is reflected to focus B. Find the vertex of the mirror when its mount at the top edge of the mirror
When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane. If the airplane is flying parallel to the ground, then the sound waves intersect the ground
Long-distance radio navigation for aircraft and ships uses synchronized pulses transmitted by widely separated transmitting stations. These pulses travel at the speed of light (186,000 miles per
1. The equation x2 − y2 = 144 represents a circle. 2. The major axis of the ellipse y2 + 16x2 = 64 is vertical? Determine whether the statement is true or false. Justify your answer.
1. It is possible for a parabola to intersect its directrix? 2. When the vertex and focus of a parabola are on a horizontal line, the directrix of the parabola is vertical?
Consider the ellipse(a) The area of the ellipse is given by A = ab. Write the area of the ellipse as a function of a. (b) Find the equation of an ellipse with an area of 264 square centimeters. (c)
Match the equation with its graph. [The graphs are labeled (a)-(f).](a)(b) (c) (d) (e) (f) 1. x2 = - 2y 2. y2 = 2x 3. x2 / y + y2 = 1 4. x2 - y2 / y = 1
In parts (a)-(d), describe how a plane could intersect the double-napped cone to form each conic section (see figure).(a) Circle (b) Ellipse (c) Parabola (d) Hyperbola
Explain how to use a graphing utility to check your graph in Exercise 43. What equation(s) would you enter into the graphing utility?Refer in Exercise 43,
1. How can you tell whether an ellipse is a circle from the equation? 2. Is the graph of x2 − 4y4 = 4 a hyperbola? Explain.
1. The graph of x2 − y2 = 0 is a degenerate conic. Sketch this graph and identify the degenerate conic? 2. Which part of the graph of the ellipse 4x2 + 9y2 = 36 does each equation represent? Answer
1. Write a paragraph discussing the changes in the shape and orientation of the graph of the ellipseas a increases from 1 to 8. 2. Use two thumbtacks, a string, and a pencil to draw an ellipse, as
Use the definition of an ellipse to derive the standard form of the equation of an ellipse. (The sum of the distances from a point (x, y) to the foci is 2a.)
Use the definition of a hyperbola to derive the standard form of the equation of a hyperbola. (The absolute value of the difference of the distances from a point (x, y) to the foci is 2a.)
Match the description of the conic with its standard equation. The equations are labeled (a)-(f). a. (x - h)2 / a2 + (y - 5)2 / b2 = 1 b. (x - h)2 / a2 - (y - k)2 / b2 = 1 c. (y - 5)2 / a2 - (x -
In Exercises 1-4, find the center and radius of the circle. 1. x2 + y2 = 49 2. x2 + y2 = 1 3. (x − 4)2 + (y − 5)2 = 36 4. (x + 8)2 + (y + 1)2 = 144
In Exercises 1-4, write the equation of the circle in standard form, and then find its center and radius. 1. x2 + y2 − 8y = 0 2. x2 + y2 − 10x + 16 = 0 3. x2 + y2 − 2x + 6y + 9 = 0 4. 2x2 + 2y2
In Exercises 1-4, find the vertex, focus, and Directix of the parabola. Then sketch the parabola. 1. (x − 1)2 + 8(y + 2) = 0 2. (x + 2) + (y − 4)2 = 0 3. (y + 1 / 2)2 = 2(x − 5) 4. (x + 1 / 2)2
In Exercises 1-4, find the standard form of the equation of the parabola with the given characteristics. 1. Vertex: (3, 2); Focus: (1, 2) 2. Vertex: (-1, 2); Focus: (-1, 2) 3. Vertex: (0, 4);
A satellite in a 100-mile-high circular orbit around Earth has a velocity of approximately 17,500 miles per hour (see figure). When this velocity is multiplied by 2, the satellite has the
Water flowing from a horizontal pipe 48 feet above the ground has the shape of a parabola whose vertex (0, 48) is at the end of the pipe (seefigure). The water strikes the ocean at the point (10
A ball is thrown from the top of a 100-foot tower with a velocity of 28 feet per second. (a) Find the equation that represents the parabolic path. (b) How far does the ball travel horizontally before
A cargo plane is flying at an altitude of 500 feet and a speed of 135 miles per hour. A supply crate is dropped from the plane. How many feet will the crate travel will the crate travel horizontally
In Exercises 1-4, find the center, foci, and vertices of the ellipse. Then sketch the ellipse 1. (x - 1)2 / 9 + (y - 5)2 / 25 = 1 2. (x - 6)2 / 4 + (y + 7)2 / 16 = 1 3. (x + 2)2 + (y + 4)2 / 1/4 =
In Exercises 1-4, find the standard form of the equation of the ellipse with the given characteristics. 1. Vertices: (3, −3), (3, 3); minor axis of length 2 2. Vertices: (−2, 3), (6, 3); minor
The dwarf planet Pluto moves in an elliptical orbit with the sun at one of the foci, as shown in the figure. The length of half of the major axis, a, is 3.67 Ã 109 miles, and the
In Australia, football by Australian Rules is played on elliptical fields. The field can be a maximum of 155 meters wide and a maximum of 185 meters long. Let the center of a field of maximum size be
In Exercises 1-2, find the center, foci, and vertices of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid 1. (x - 2)2 / 16 - (y + 1)2 / 9 = 1 2. (x - 1)2 / 144 (7 - 4)2 / 25 = 1
In Exercises 1-4, identify the conic. Then describe the translation of the conic from standard position.1. (x + 2)2 + (y - 1)2 = 42. (y 1)2 = 4(2)(x + 2) 3. (y + 3)2 / 4 - (x -1)2 = 1 4.
In Exercises 1-4, find the standard form of the equation of the hyperbola with the given characteristics. 1. Vertices: (0, 2), (0, 0); foci: (0, 3), (0, −1) 2. Vertices: (1, 2), (5, 2); foci: (0,
In Exercises 1-4, identify the conic by writing its equation in standard form. Then sketch its graph and describe the translation from standard position. 1. y2 - x2 + y = 0 2. x2 + y2 - 6x + 4y + 9
In Exercises 1-3, determine whether the statement is true or false. Justify your answer. 1. The conic represented by the equation 3x2 + 2y2 − 18x − 16y + 58 = 0 is an ellipse. 2. The graphs of x2
Consider the ellipse (x2 / a2) + (y2 / b2) = 1.a. Show that the equation of the ellipse can be written as (xh)2 / a2 + (yk)2 / a2(1 - e2) = 1 Where e is the eccentricity.b. Use a graphing utility to
Find the domain of the function and discuss the behavior of f near any excluded x-values. 1. f(x) = 3x / x+10 2. f(x) = 4x3 / 2 + 5x 3. f(x) = 8 / x2-10x+24 4. f(x) = x2+ x - 2 / x2 -4x +4
1. The cost C (in dollars) of producing x units of a product is given by C = 0.5x + 500 and the average cost per unit C̅ is given by C̅ = C / x = 0.5x + 500 / x, x>0. Determine the average cost
(a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the
(a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational
The cost C (in dollars) of producing x units of a product is given by C = 100,000 + 0.9x and the average cost per unit C̅ is given by C̅ = C / x = 100,000 + 0.9x / x, x>0. (a) Sketch the graph of
A page that is x inches wide and y inches high contains 30 square inches of print. The top and bottom margins are each 2 inches deep and the margins on each side are 2 inches wide. (a) Show that the
A parks and wildlife commission releases 80,000 fish into a lake. After t years, the population years, the population N of the fish (in thousands) is given by N = 20(3t +4) / 0.05t +1, t ≥ 0. (a)
1. The concentration C of a chemical in the bloodstream t hours after injection into muscle tissue is hours after injection into muscle tissue is given by C (t) = (2t + 1) / (t2 + 4), t > 0. (a)
Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. 1. Focus: (−6, 0) 2. Focus: (0, 7) 3. Directrix: y = −3 4. Directrix: x = 3 5.
1. A cross section of a large parabolic satellite dish is modeled by y = x2 / 200, 100 ¤ x ¤ 100 (see figure). The receiving and transmitting equipment is
Find all vertical and horizontal asymptotes of the graph of the function. 1. f (x) = 6x2 / x + 3 2. f (x) = 2x2 + 5x -3 / x2 + 2 3. g (x) = x2 / x2 - 4 4. g (x) = x + 1 / x2 - 1 5. h (x) = 5x + 20 /
Find the standard form of the equation of the ellipse with the given characteristics and center at the origin.1.2. 3. Vertices: (0, ±7); foci: (0, ±6)
1. A semielliptical archway is formed over the entrance to an estate. The arch is set on pillars that are 10 feet apart and has a height (atop the pillars) of 4 feet (see figure). Describe the
Find the standard form of the equation of the hyperbola with the given characteristics and center at the origin. 1. Vertices: (0, ±1); foci: (0, ±5) 2. Vertices: (±4, 0); foci: (±6, 0) 3.
Find the standard form of the equation of the parabola with the given characteristics. 1. Vertex: (4, 2); focus: (4, 0) 2. Vertex: (2, 0); focus: (0, 0) 3. Vertex: (8, −8); directrix: x = 1 4.
Find the standard form of the equation of the ellipse with the given characteristics. 1. Vertices: (0, 2), (4, 2); minor axis of length 2 2. Vertices: (5, 0), (5, 12); minor axis of length 10 3.
Find the standard form of the Equation of the hyperbola with the given characteristics. 1. Vertices: (−10, 3), (6, 3); foci: (−12, 3), (8, 3) 2. Vertices: (2, −2), (2, 2); foci: (2, −4), (2,
Identify the conic by writing its equation in standard form. Then sketch its graph and describe the translation from standard position. 1. x2 − 6x + 2y + 9 = 0 2. y2 − 12y − 8x + 20 = 0 3. x2 +
1. A parabolic archway is 12 meters high at the vertex. At a height of 10 meters, the width of the archway is 8 meters (see figure). How wide is the archway at ground level?2. A church window is
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