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mathematics
calculus

Questions and Answers of Calculus

Determine whether f is even, odd, or neither even nor odd. (a) f(x) = 2x5 - 3x2 + 2 (b) f(x) = x3 - x7 (c) f(x) = e-x2 (d) f(x) = 1 + sin x
If and f(x) = ln x and g(x) = x2 - 9 find the functions (a) f o g, (b) g o f, (c) f o f, (d) g o g, and their domains.
Life expectancy improved dramatically in the 20th century. The table gives the life expectancy at birth (in years) of males born in the United States. Use a scatter plot to choose an appropriate type
Find the exact value of each expression. (a) e2ln3 (b) log10 25 + log10 4 (c) tan (arcsin ½) (d) sin (cos-1 (4/5))
The population of a certain species in a limited environment with initial population 100 and carrying capacity 1000 is P(t) = 100,000/100 + 900e-t Where t is measured in years. (a) Graph this
If f(x) = x2 - 2x + 3, evaluate the difference quotient f (a + h) - f (a) / h
Find the domain and range of the function. Write your answer in interval notation. (a) F(x) = 2/(3x - 1) (b) h(x) = ln (x + 6)
Suppose that the graph of is given. Describe how the graphs of the following functions can be obtained from the graph of (a) y = f(x) + 8 (b) y = f(x + 8) (c) y = 1 + 2 f (x) (d) y = f(x - 2) - 2 (e)
One of the legs of a right triangle has length 4 cm. Express the length of the altitude perpendicular to the hypotenuse as a function of the length of the hypotenuse.
Evaluate (log2 3) (log3 4) (log4 5) ··· (log31 32).
Solve the inequality ln (x2 - 2x - 2) ≤ 0.
A driver sets out on a journey. For the first half of the distance she drives at the leisurely pace of 30 mi/h; she drives the second half at 60 mi/h. What is her average speed on this trip?
Prove that if n is a positive integer, then 7n - 1 is divisible by 6.
If f0 (x) = x2 and fn + 1(x) = f0 (fn (x)) for n = 0, 1, 2,..., find a formula for fn(x).
Solve the equation |2x - 1| - |x + 5| = 3.
Sketch the graph of the function f(x) = |x2 - 4| x| + 3|.
Draw the graph of the equation x + |x| = y + |y|.
The notation max{a, b,...} means the largest of the numbers a, b,... Sketch the graph of each function. (a) f(x) = max{x, 1/x} (b) f(x) = max {sin x, cos x} (c) f(x) = max {x2, 2 + x, 2 - x}
(a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How can you tell whether a given curve is the graph of a function?
How is the composite function defined? What is its domain?
Suppose the graph of f is given. Write an equation for each of the graphs that are obtained from the graph of f as follows. (a) Shift 2 units upward. (b) Shift 2 units downward. (c) Shift 2 units to
(a) What is a one-to-one function? How can you tell if a function is one-to-one by looking at its graph? (b) If f is a one-to-one function, how is its inverse function f-1 defined? How do you obtain
(a) How is the inverse sine function f(x) = sin-1x defined? What are its domain and range? (b) How is the inverse cosine function f(x) = cos-1x defined? What are its domain and range? (c) How is the
Discuss four ways of representing a function. Illustrate your discussion with examples.
(a) What is an even function? How can you tell if a function is even by looking at its graph? Give three examples of an even function. (b) What is an odd function? How can you tell if a function is
Give an example of each type of function. (a) Linear function (b) Power function (c) Exponential function (d) Quadratic function (e) Polynomial of degree 5 (f) Rational function
Sketch by hand, on the same axes, the graphs of the following functions. (a) f(x) = x (b) g(x) = x2 (c) h(x) = x3 (d) j(x) = x4
Draw, by hand, a rough sketch of the graph of each function. (a) y = sin x (b) y = tan x (c) y = ex (d) y = ln x (e) y = 1/x (f) y = | x | (g) y = sx (h) y = tan-1 x
Suppose that f has domain A and has domain . (a) What is the domain of f + g? (b) What is the domain of fg? (c) What is the domain of f/g?
A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume V of water remaining in the tank (in gallons) after t minutes.(a)
The point P(2, -1) lies on the curve y = 1/(1 - x). (a) If Q is the point (x, 1/(1 - x)), use your calculator to find the slope of the secant line PQ (correct to six decimal places) for the following
If a ball is thrown into the air with a velocity of 40 ft/s, its height in feet seconds later is given by y = 40 t - 16t2. (a) Find the average velocity for the time period beginning when t = 2 and
The table shows the position of a cyclist.(a) Find the average velocity for each time period: (i) [1, 3] (ii) [2, 3] (iii) [3, 5] (iv) [3, 4] (b) Use the graph of as a function of to estimate the
The point P(1, 0) lies on the curve y = sin(10 π /x).. (a) If Q is the point (x, sin(10π/x)), find the slope of the secant line (correct to four decimal places) for x = 2, 1.5, 1.4, 1.3, 1.2, 1.1,
Explain in your own words what is meant by the equationIs it possible for this statement to be true and yet f(2) = 3? Explain.
Sketch the graph of the function and use it to determine the values of for which limx†’a exists.
Use the graph of the function to state the value of each limit, if it exists. If it does not exist, explain why.f(x) = 1/ 1+e1/x
Sketch the graph of an example of a function f that satisfies all of the given conditions.
Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places).X = 2.5, 2.1, 2.05, 2.01, 2.005, 2.001, 1.9, 1.95, 1.99, 1.995, 1.999x = 0,
Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically.
(a) By graphing the function f(x) = (cos 2x - cos x) / x2 and zooming in toward the point where the graph crosses the -axis, estimate the value of limx→0 f(x).(b) Check your answer in part (a) by
Determine the infinite limit.
Explain the meaning of each of the following.(a)(b)
Determine(a) By evaluating f(x) = 1/(x3 - 1) for values of that approach 1 from the left and from the right,(b) By reasoning as in Example 9, and(c) From a graph of f.
(a) Estimate the value of the limit limit limx→0 to five decimal places. Does this number look familiar?(b) Illustrate part (a) by graphing the function y = (1 + x)1/x.
(a) Evaluate the function f(x) = x2 - (2*/1000) for x = 1, 0.8, 0.6, 0.4, 0.2, 0.1, and 0.05, and guess the value of(b) Evaluate f(x) for x = 0.04, 0.02, 0.01, 0.005, 0.003, and 0.001. Guess again.
Graph the function f(x) = sin (π/x) of Example 4 in the viewing rectangle [-1, 1], by [-1, 1]. Then zoom in toward the origin several times. Comment on the behavior of this function.
Use a graph to estimate the equations of all the vertical asymptotes of the curve Y = tan (2 sin x) ................ - π ≤ x ≤ π Then find the exact equations of these asymptotes.
For the function f whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why.(a)(b)(c)(d)(e)f(3)
For the function whose graph is given, state the value of eachquantity, if it exists. If it does not exist, explain why.(g) g(2)
For the function whose graph is shown, state the following.(f) The equations of the vertical asymptotes.
Given thatfind the limits that exist. If the limit does not exist, explain why.
Evaluate the limit, if it exists.
Evaluate the limit and justify each step by indicating the appropriate Limit Law(s).(a) (5x3 - 3x2 + x - 6)(b)(x4 €“ 3x) (x2 + 5x + 3)
(a) Estimate the value ofby graphing the function f(x) = x/(ˆš1 + 3x - 1). (b) Make a table of f(x) values of for x close to 0 and guess the value of the limit. (c) Use the Limit Laws to
Use the Squeeze Theorem to show that limx→0 (x2 cos 20 πx) = 0. Illustrate by graphing the functions f(x) = -x, g(x) = x2 cos 20πx, and h(x) = x2 on the same screen.
If 4x - 9 ‰¤ f(x) ‰¤ x2 - 4X + 7 for x ‰¥ 0, find f(x).
Prove that cos 2/x = 0.
Find the limit, if it exists. If the limit does not exist, explain why.
The signum (or sign) function, denoted by sgn, is defined by(a) Sketch the graph of this function.(b) Find each of the following limits or explain why it does not exist.
Let g(x) = x2 + x - 6/ |x - 2|.(a) Find(b) Does lim x †’ 2 g(x) exist?(c) Sketch the graph of t.
(a) If the symbol [ ] denotes the greatest integer function defined in Example 10, evaluate(b) If n is an integer, evaluate(c) For what values of does limx†’a [x] exist?
If f(x) = [x] + [-x], show that limx→2 f(x) exists but is not equal to f(2).
If p is a polynomial, show that limx →a P(x) = p(a).
If f(x) - 8 / x - 1 = 10, find f(x).
Show by means of an example that limx→a [f(x) g(x)] may exist even though neither limx→a f(x) nor limx→a g(x) exists.
Is there a number a such thatexists? If so, find the value of a and the value of the limit.
Use the given graph of to find a number such that if |x - 1|
A machinist is required to manufacture a circular metal disk with area 1000 cm2. (a) What radius produces such a disk? (b) If the machinist is allowed an error tolerance of ± 5 cm2 in the area of
(a) Find a number δ such that if |x -2| < δ, then |4x - 8 < ε, where ε = 0.01. (b) Repeat part (a) with ε = 0.01.
Prove the statement using the Îµ, Î´ definition of a limit and illustrate with a diagram like Figure 9.
Prove the statement using the Îµ, Î´ definition of a limit.
Use the given graph of f(x) = ˆšx to find a number such that if |x - 4|
Verify that another possible choice of for showing that limx→3 x2 = 9 in Example 4 is δ = min {2, ε/8}.
(a) For the limit limx →1 (x3 + x + 1) = 3, use a graph to find a value of δ that corresponds to ε = 0.4. (b) By using a computer algebra system to solve the cubic equation x3 + x + 1 = 3 + ε,
Use
If the function f is defined byProve that limx†’0 f(x) does not exist.
Prove that ln x = - ˆž.
Use a graph to find a number such that if |x - π/4 | < δ then |tan x - 1| < 0.2
For the limitillustrate Definition 2 by finding values of that correspond to Îµ = 0.2 and Îµ = 0.1.
Given that limx→π/2 tan2 x = ∞, illustrate Definition 6 by finding values of that correspond to (a) M = 1000 (b) M = 10,000.
Suppose and are continuous functions such that and limx→2 [3f(x) + f(x) g(x)] = 36. Find f(2).
Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a.F(x) = (x + 2x3)4, a = - 1
Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval. F(x) = 2x + 3 / x - 2, (2, ∞)
Explain why the function is discontinuous at the given number a. Sketch the graph of the function.(a) f(x) = 1/x + 2, a = - 2(b)
How would you "remove the discontinuity" of f? In other words, how would you define f(2) in order to make continuous at 2?F(x) = x2 - x - / x - 2
Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. (a) F(x) 2x2 - x - 1/ x2 + 1 (b) Q(x) = 3√x - 2 / x3 - 2
(a) From the graph of , state the numbers at which f is discontinuous and explain why.(b) For each of the numbers stated in part (a), determine whether f is continuous from the right, or from the
Locate the discontinuities of the function and illustrate by graphing. y = 1 / 1 + e1/x
Use continuity to evaluate the limit.
Show that f is continuous on (- ˆž, ˆž).
Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph of f.
For what value of the constant is the function f continuous on (- ˆž, ˆž)?
Which of the following functions f has a removable discontinuity at a? If the discontinuity is removable, find a function g that agrees with f for x ≠ a and is continuous at a.(a) f(x) = x4 - 1/ x
If f(x) = x2 + 10 sin x, show that there is a number such that f (c) = 1000.
Sketch the graph of a function that is continuous except for the stated discontinuity.(a) Discontinuous, but continuous from the right, at 2.(b) Removable discontinuity at 3, jump discontinuity at 5.
Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. (a) x4 + x - 3 = 0 (1, 2) (b) ex = 3 - 2x, (0, 1)
(a) Prove that the equation has at least one real root. (b) Use your calculator to find an interval of length 0.01 that contains a root. cos x = x3
(a) Prove that the equation has at least one real root. (b) Use your graphing device to find the root correct to three decimal places. 100e-x/100 = 0.01 x2
Prove that f is continuous at if and only if

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