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study help
mathematics
precalculus
Questions and Answers of
Precalculus
Use a graphing utility to evaluatefor several values of n. What do you notice? sin nx lim x-0 X
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If f is undefined at x = c, then the limit of f(x) as x approaches c does not
Use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. sin 3t lim 1-0 t
Find the vertical asymptotes (if any) of the graph of the function.f(x) = 3/x
Findf(x) = x2 − 4x lim Δε - 0 f(x + Δx) = f(x) + · ΔΙ
Consider the function f(x) = √x.Is lim x→0.25 √x = 0.5 a true statement? Explain.
Describe the interval(s) on which the function is continuous. f(x) = 3 − √x
Describe the interval(s) on which the function is continuous. f(x) = sec x /4
Explain why the function has at least one zero in the given interval.Function Intervalf(x) = x2 − 2
Explain why the function has at least two zeros in the interval [1, 5].f(x) = (x − 3)2 − 2
(a) In the context of finding limits, discuss what is meant by two functions that agree at all but one point.(b) Give an example of two functions that agree at all but one point.
Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem. f(x) et - X x-4' [1,3], f(c) = 3
Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem.f(x) = √x + 7 − 2, [0, 5], f(c) = 1
Use a graphing utility to graphin the same viewing window. Compare the magnitudes of f(x) and g(x) when x is close to 0. Use the comparison to write a short paragraph explaining why lim
Write a function that is continuous on (a, b) but not continuous on [a, b].
The table lists the frequency F (in Hertz) of a musical note at various times t (in seconds). (a) Plot the data and connect the points with a curve.(b) Does there appear to be a limiting
Describe how the functionsf(x) = 3 + [[x]] and g(x) = 3 − [[−x]]differ.
A cell phone service charges $10 for the first gigabyte (GB) of data used per month and $7.50 for each additional gigabyte or fraction thereof. The cost of the data plan is given byC(t) = 10 −
Describe how to find the limit of a polynomial function p(x) as x approaches c.
Evaluate the expression and write your answer in the form a + bi.i3
Evaluate the expression and write your answer in the form a + bi.√-3 √-12
If a circle has radius 10 cm, find the length of the arc subtended by a central angle of 72o.
Find the radius of a circular sector with angle 3π/4 and arc length 6 cm.
Write the sum in sigma notation.1 + 2 + 4 + 8 + 16 + 32
Prove the following properties of complex numbers. (a) (b) (c) where n is a positive integer [Write z = a + bi, w = c + di.]
Draw, in standard position, the angle whose measure is given.-150o
Write the sum in sigma notation.x + x2 + x3 + ∙ ∙ ∙ + xn
Draw, in standard position, the angle whose measure is given.1π /3 rad
Find the value of the sum.
Find all solutions of the equation.x2 + 2x + 5 = 0
Draw, in standard position, the angle whose measure is given.-3 rad
Find the exact trigonometric ratios for the angle whose radian measure is given.11π/4
Identify the type of curve and sketch the graph. Do not plot points. Just use the standard graphs given in Figures 5, 6, 8, 10, and 11 and shift if necessary.x = 4 - y2
Find the remaining trigonometric ratios.sin θ = 3/5, 0 < θ < π/2
Prove the formula for the sum of a finite geometric series with first term a and common ratio r ≠ 1:
The latitude and longitude of a point P in the Northern Hemisphere are related to spherical coordinates p, θ, Φ as follows. We take the origin to be the center of the earth and the positive z-axis
The surfaces p = 1 + 1/5 sin mθ sin nΦ have been used as models for tumors. The “bumpy sphere” with m = 6 and n = 5 is shown. Use a computer algebra system to find the volume it encloses.
(a) Use cylindrical coordinates to show that the volume of the solid bounded above by the sphere r2 + z2 = a2 and below by the cone z = r cot Φ0 (or Φ = Φ0), where 0 0 (b) Deduce that the volume
Write the sum in expanded form.
(a) By comparing areas, show that 1/3 < ln 1.5 < 5/12(b) Use the Midpoint Rule with n = 10 to estimate ln 1.5.
Evaluate the expression and write your answer in the form a + bi.(5 - 6i) + (3 + 2i)
Find an equation of a circle that satisfies the given conditions.Center (3, -1), radius 5
Convert from degrees to radians.210o
Write the sum in expanded form.
Evaluate the expression and write your answer in the form a + bi.(4 - 1/2i) - (9 + 5/2i)
Find an equation of a circle that satisfies the given conditions.Center (-2, -8), radius 10
Convert from degrees to radians.300o
Write the sum in expanded form.
By comparing areas, show that1/2 + 1/3 + . . . + 1/n < In n < 1 + 1/2 + 1/3 + . . . + 1/n - 1
Evaluate the expression and write your answer in the form a + bi.(2 + 5i )(4 - i)
Convert from degrees to radians.9o
(a) By comparing areas, show that ln 2 < 1 < ln 3.(b) Deduce that 2 < e < 3.
Evaluate the expression and write your answer in the form a + bi.(1 - 2i)(8 - 3i)
Find an equation of a circle that satisfies the given conditions.Center (-1, 5), passes through (-4, -6)
Convert from degrees to radians.-315o
Write the sum in expanded form.
Evaluate the expression and write your answer in the form a + bi.
Show that the equation represents a circle and find the center and radius.x2 + y2 - 4x + 10y + 13 = 0
Convert from degrees to radians.900o
Write the sum in expanded form.
Evaluate the expression and write your answer in the form a + bi.
Show that the equation represents a circle and find the center and radius.x2 + y2 + 6y + 2 = 0
Convert from degrees to radians.36o
Write the sum in expanded form.
Evaluate the expression and write your answer in the form a + bi.1 + 4i/3 + 2i
Show that the equation represents a circle and find the center and radius.x2 + y2 + x = 0
Convert from radians to degrees.4π
Write the sum in expanded form.
Evaluate the expression and write your answer in the form a + bi.3 + 2i/1 - 4i
Show that the equation represents a circle and find the center and radius.16x2 + 16y2 + 8x + 32y + 1 = 0
Convert from radians to degrees.-7π/2
Write the sum in expanded form.
Evaluate the expression and write your answer in the form a + bi.1/1+ i
Show that the equation represents a circle and find the center and radius.2x2 + 2y2 - x + y = 1
Convert from radians to degrees.5π/12
Write the sum in expanded form.
Deduce the following laws of logarithms from (15):(a) logb(xy) = logbx + logby(b) logb(x/y) = logb x - logby(c) logb(xy) = y logb x
Evaluate the expression and write your answer in the form a + bi.3/4 - 3i
Under what condition on the coefficients a, b, and c does the equation x2 + y2 + ax + by + c = 0 represent a circle?When that condition is satisfied, find the center and radius of the circle.
Convert from radians to degrees.8π/3
Write the sum in sigma notation.1 + 2 + 3 + 4 + ...... + 10
Convert from radians to degrees.-3π/8
Evaluate the expression and write your answer in the form a + bi.i100
Convert from radians to degrees.5
Write the sum in sigma notation.1/2 + 2/3 + 3/4 + 4/5 + ..... + 19/20
Evaluate the expression and write your answer in the form a + bi.√-25
Find the length of a circular arc subtended by an angle of π/12 rad if the radius of the circle is 36 cm.
Write the sum in sigma notation.3/7 + 4/8 + 5/9 + 6/10 + ..... + 23/27
Write the sum in sigma notation.2 + 4 + 6 + 8 +∙ ∙ ∙ + 2n
Find the complex conjugate and the modulus of the number.12 - 5i
A circle has radius 1.5 m. What angle is subtended at the center of the circle by an arc 1 m long?
Write the sum in sigma notation.1 + 3 + 5 + 7 + ∙ ∙ ∙ + (2n - 1)
Find the complex conjugate and the modulus of the number.-1 + 2√2i
Find the complex conjugate and the modulus of the number.-4i
Draw, in standard position, the angle whose measure is given.315o
Write the sum in sigma notation.1/1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36
Find all solutions of the equation.4x2 + 9 = 0
Write the sum in sigma notation.1 - x + x2 - x3 + ∙ ∙ ∙ + (-1)nxn
Find all solutions of the equation.x4 = 1
Draw, in standard position, the angle whose measure is given.-3π/4 rad
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