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study help
mathematics
precalculus
Questions and Answers of
Precalculus
Find the limit of each rational function (a) As x→ ∞ and (b) As x→ -∞. f(x) = x + 1 2 x² + 3
How can looking at the graph of a function help you tell where the function is continuous?
Use the method in Example 3 to find (a) The slope of the curve at the given point P, and (b) An equation of the tangent line at P.y = x3 - 12x, P(1, -11)Example 3Find the slope of the
The accompanying figure shows the plot of distance fallen versus time for an object that fell from the lunar landing module a distance 80 m to the surface of the moon.a. Estimate the slopes of the
If limx→0+ ƒ(x) = A and limx→0- ƒ(x) = B, finda. limx→0+ ƒ(x3 - x) b. limx→0- ƒ(x3 - x)c. limx→0+ ƒ(x2 - x4) d. limx→0- ƒ(x2 - x4)
Find the limit of each rational function (a) As x→ ∞ and (b) As x→ -∞. f(x) = 3x 2 X + 7 - 2
Leta. Show that ƒ is continuous at x = 0.b. Use the fact that every nonempty open interval of real numbers contains both rational and irrational numbers to show that ƒ is not continuous at any
What does it mean for a function to be right-continuous at a point? Left-continuous? How are continuity and one-sided continuity related?
Use the method in Example 3 to find (a) The slope of the curve at the given point P, and (b) An equation of the tangent line at P.y = x3 - 3x2 + 4, P(2, 0)Example 3Find the slope of the
Use the formal definition of limit to prove that the function has a continuous extension to the given value of x. g(x) x² - 2x - 2x - 6 3 x = 3
Find the limit of each rational function (a) As x→ ∞ and (b) As x→ -∞. h(x) = 9x² + x 2x4+ 5x² x + 6 -
What does it mean for a function to be continuous on an interval? Give examples to illustrate the fact that a function that is not continuous on its entire domain may still be continuous on selected
Which of the following statements are true, and which are false? If true, say why; if false, give a counterexample (that is, an example confirming the falsehood).a. If limx→c ƒ(x) exists but
What are the basic types of discontinuity? Give an example of each. What is a removable discontinuity? Give an example.
If x is a rational number, then x can be written in a unique way as a quotient of integers m/n where n > 0 and m and n have no common factors greater than 1. (We say that such a fraction is in
What does it mean for a function to have the Intermediate Value Property? What conditions guarantee that a function has this property over an interval? What are the consequences for graphing and
Find the limit of each rational function (a) as x→ ∞ and (b) as x→ -∞. g(x) = 10x° + x+ + 31 0
Find the limit of each rational function (a) as x→ ∞ and (b) as x→ -∞. f(x) = 3x7 + 5x² - 1 6x³ - 7x + 3
Let limx→1 h(x) = 5, limx→1 p(x) = 1, and limx→1 r(x) = 2. Name the rules in Theorem 1 that are used to accomplish steps (a), (b), and (c) of the following calculation.(a)(b)(c)
Find the limit of each rational function (a) As x→ ∞ and (b) As x→ -∞. g(x) = zXL + EX 2 x² = x + 1
Under what circumstances can you extend a function ƒ(x) to be continuous at a point x = c? Give an example.
The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x: Divide numerator and denominator by the highest power of x
What exactly do limx→∞ ƒ(x) = L and limx→-∞ ƒ(x) = L mean? Give examples.
a. Show that the expressionequals a if a ≥ b and equals b if b ≥ a. In other words, max {a, b} gives the larger of the two numbers a and b.b. Find a similar expression for min {a, b}, the smaller
Find the limit of each rational function (a) As x→ ∞ and (b) As x→ -∞. h(x) 5x8 - 2x³ + 9 3 + x - 4x5
The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x: Divide numerator and denominator by the highest power of x
Use limits to determine the equations for all horizontal asymptotes.a.b.c.d. y = 1 - x² 2 x² + 1
What are limx→±∞ k (k a constant) and limx→±∞ (1/x)? How do you extend these results to other functions? Give examples.
Is there any reason to believe that there is always a pair of antipodal (diametrically opposite) points on Earth’s equator where the temperatures are the same? Explain.
Let ƒ(x) = x3 - x - 1.a. Use the Intermediate Value Theorem to show that ƒ has a zero between -1 and 2.b. Solve the equation ƒ(x) = 0 graphically with an error of magnitude at most 10-8.c. It can
The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x: Divide numerator and denominator by the highest power of x
Use limits to determine the equations for all vertical asymptotes.a.b.c. y = x² + 4 X x - 3
How do you find the limit of a rational function as x→±∞? Give examples.
Let ƒ(x) = (x2 - 9)/(x + 3).a. Make a table of the values of ƒ at the points x = -3.1, -3.01, -3.001, and so on as far as your calculator can go. Then estimate limx→-3 ƒ(x). What estimate do you
If limx→c (ƒ(x) + g(x)) = 3 and limx→c (ƒ(x) - g(x)) = -1, find limxSc ƒ(x)g(x).
The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x: Divide numerator and denominator by the highest power of x
What are horizontal and vertical asymptotes? Give examples.
Show that the equation x + 2 cos x = 0 has at least one solution.
Graph the rational function. Include the graphs and equations of the asymptotes and dominant terms. y 1 2x + 4
A real-valued function ƒ is bounded from above on a set D if there exists a number N such that ƒ(x) ≤ N for all x in D. We call N, when it exists, an upper bound for ƒ on D and say that ƒ is
Graph the rational function. Include the graphs and equations of the asymptotes and dominant terms. y 1 x - 1 X
Graph the rational function. Include the graphs and equations of the asymptotes and dominant terms. y = -3 3
Graph the rational function. Include the graphs and equations of the asymptotes and dominant terms. y = 1 x + 1
Graph the rational function. Include the graphs and equations of the asymptotes and dominant terms. y = x + 3 x + 2
Sketch the graph of a function y = ƒ(x) that satisfies the given conditions. No formulas are required—just label the coordinate axes and sketch an appropriate graph. f(0) = 0, lim f(x) = 0, lim
Can ƒ(x) = x(x2 - 1)/ | x2 - 1| be extended to be continuous at x = 1 or -1? Give reasons for your answers. (Graph the function— you will find the graph interesting.)
Sketch the graph of a function y = ƒ(x) that satisfies the given conditions. No formulas are required—just label the coordinate axes and sketch an appropriate graph. f(0) = 0, f(1) = 2, f(−1) =
Explain why the function ƒ(x) = sin (1/x) has no continuous extension to x = 0.
Find a function that satisfies the given conditions and sketch its graph. h(x) = -1, lim h(x) = 1, lim h(x) = -1, and X78 x-0 X--X lim h(x) = 1 x-0+
Graph the rational function. Include the graphs and equations of the asymptotes and dominant terms. y || 2x x + 1
Find a function that satisfies the given conditions and sketch its graph. lim f(x) = 0, lim f(x) = ∞, and lim f(x) = 8 x→ +∞ x-2 x-2+
Show by example that the following statement is wrong.The number L is the limit of ƒ(x) as x approaches c if ƒ(x) gets closer to L as x approaches c.Explain why the function in your example does
Sketch the graph of a function y = ƒ(x) that satisfies the given conditions. No formulas are required—just label the coordinate axes and sketch an appropriate graph. f(0) = 0, lim f(x) = 0, lim
Find a function that satisfies the given conditions and sketch its graph. lim k(x) x→ +∞ = 1, lim k(x) x-1 = ∞, and lim k(x) = -∞ x-1+
Sketch the graph of a function y = ƒ(x) that satisfies the given conditions. No formulas are required—just label the coordinate axes and sketch an appropriate graph. ƒ(2) = 1, ƒ(−1) = 0, lim
Find a function that satisfies the given conditions and sketch its graph. lim g(x) x-+∞ = 0, lim g(x) x-3¯ = -∞, and lim g(x) x-3+ = ∞
Here is the definition of infinite right-hand limit.Modify the definition to cover the following cases.a. limx→cƒ(x) = ∞b. limx→c+ ƒ(x) = -∞c. limx→c-ƒ(x) = -∞ We say that f(x)
Use formal definitions to prove the limit statement. x = zł 0
Use the formal definitions from Exercise 93 to prove the limit statement.Here is the definition of infinite right-hand limit.Modify the definition to cover the following cases.a. limx→cƒ(x) =
Use the formal definitions from Exercise 93 to prove the limit statement.Here is the definition of infinite right-hand limit.Modify the definition to cover the following cases.a. limx→cƒ(x) =
Use formal definitions to prove the limit statement. -2 lim x-3 (x - 3)² = -
Use the formal definitions from Exercise 93 to prove the limit statement.Here is the definition of infinite right-hand limit.Modify the definition to cover the following cases.a. limx→cƒ(x) =
Use formal definitions to prove the limit statement. 8 || X 0
Graph the rational function. Include the graphs and equations of the asymptotes. y = z.X x - 1
Use the formal definitions from Exercise 93 to prove the limit statement.Here is the definition of infinite right-hand limit.Modify the definition to cover the following cases.a. limx→cƒ(x) =
Use formal definitions to prove the limit statement. 1 lim x-5 (x + 5)² = ∞
Use the formal definitions from Exercise 93 to prove the limit statement.Here is the definition of infinite right-hand limit.Modify the definition to cover the following cases.a. limx→cƒ(x) =
Suppose that ƒ(x) and g(x) are polynomials in x and that limx→ ∞ (ƒ(x)/g(x)) = 2. Can you conclude anything about limx→ -∞ (ƒ(x)/g(x))? Give reasons for your answer.
Suppose that ƒ(x) and g(x) are polynomials in x. Can the graph of ƒ(x)/g(x) have an asymptote if g(x) is never zero? Give reasons for your answer.
Graph the rational function. Include the graphs and equations of the asymptotes. y = X 1
Graph the rational function. Include the graphs and equations of the asymptotes. y x² - 4 x - 1
How many horizontal asymptotes can the graph of a given rational function have? Give reasons for your answer.
Use the formal definitions of limits as x→ ±∞ to establish the limit.If ƒ has the constant value ƒ(x) = k, then limx→∞ ƒ(x) = k.
Use the formal definitions of limits as x→ ±∞ to establish the limit.If ƒ has the constant value ƒ(x) = k, then limx→-∞ ƒ(x) = k.
Graph the rational function. Include the graphs and equations of the asymptotes. y x² + 1 x - 1
Graph the curves Explain the relationship between the curve’s formula and what you see. y X -2 で
Graph the curves Explain the relationship between the curve’s formula and what you see. y - 1 √4 - x² 2
Graph the rational function. Include the graphs and equations of the asymptotes. y = 1 2x + 4
Graph the curves Explain the relationship between the curve’s formula and what you see. y = x²/3 + 1 x1/3
Graph the rational function. Include the graphs and equations of the asymptotes. y x3 x³ + 1 -2 X
Graph the curves Explain the relationship between the curve’s formula and what you see. y = sin TT 1.1.² + 1/
Graph the function. Then answer the following questions.a. How does the graph behave as x→0+?b. How does the graph behave as x→±∞?c. How does the graph behave near x = 1 and x = -1?Give
Graph the function. Then answer the following questions.a. How does the graph behave as x→0+?b. How does the graph behave as x→±∞?c. How does the graph behave near x = 1 and x = -1?Give
Match the function with its graph using horizontal asymptotes as an aid. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f) -2 -1 3 1 y 1 2 X
Match the function with its graph using horizontal asymptotes as an aid. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f) -2 -1 3 1 y 1 2 X
Describe the x-values at which f is differentiable. f(x) -4 = I [x²2 - 4, 14- x², 2 4 y x ≤ 0 x > 0 4 X
Find an equation of the tangent line to the graph at the given point. f(x) = 432- X 1 4x 2+6 (2,3) 1234 X
Match the function with its graph using horizontal asymptotes as an aid. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f) -2 -1 3 1 y 1 2 X
Match the graph of the function with the graph of its derivative. [The graphs of the derivatives are labeled (a), (b), (c), and (d).](a)(b)(c)(d) y 3 VAA 2 -2 -3+ X
Match the graph of the function with the graph of its derivative. [The graphs of the derivatives are labeled (a), (b), (c), and (d).](a)(b)(c)(d) y 3 VAA 2 -2 -3+ 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 y is a differentiable function of u, u is a differentiable function of v,
Match the function with its graph using horizontal asymptotes as an aid. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f) -2 -1 3 1 y 1 2 X
Match the function with its graph using horizontal asymptotes as an aid. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f) -2 -1 3 1 y 1 2 X
Match the function with its graph using horizontal asymptotes as an aid. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f) -2 -1 3 1 y 1 2 X
Match the graph of the function with the graph of its derivative. [The graphs of the derivatives are labeled (a), (b), (c), and (d).](a)(b)(c)(d) y 3 VAA 2 -2 -3+ X
Match the graph of the function with the graph of its derivative. [The graphs of the derivatives are labeled (a), (b), (c), and (d).](a)(b)(c)(d) y 3 VAA 2 -2 -3+ X
Use the rules of differentiation to find the derivative of the function.y = x7
Use the rules of differentiation to find the derivative of the function.y = 2x3 + 6x2 − 1
The graph of f is shown. State the signs of f′ and f″ on the interval (0, 2). y f 1 2 X
Use the graph of f to find(a) The largest open interval on which f is increasing and(b) The largest open interval on which f is decreasing. TI -6- 4 2 -2 -2 y -4 2 4 4 ign X
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