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mathematics
calculus

Questions and Answers of Calculus

Sketch a rough graph of the yield of a crop as a function of the amount of fertilizer used.
Suppose that the graph of f 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 + 2f (x) (d) y = ½ f (x)
The graph of f is given. Draw the graphs of the following functions.(a) y = f (x €“ 8) (b) y = €“f (x)(c) y = 2 €“ f (x) (d) y = ½ f (x) €“ 1(e) y
Use transformations to sketch the graph of the function.
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
Find an expression for the function whose graph consists of the line segment from the point (–2, 2) to the point (–1, 0) together with the top half of the circle with center the origin and radius
If f(x) = In x and g(x) = x2 – 9, find the functions f o g, g o f, f o f, g o g, and their domains.
Express the function F(x) =1/√x + √x as a composition of three functions.
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
A small-appliance manufacturer finds that it costs $9000 to produce 1000 toaster ovens a week and $12,000 to produce 1500 toaster ovens a week.(a) Express the cost as a function of the number of
If f(x) = 2x + In x, find f–1(2).
Find the inverse function of f(x) = x + 1 / 2x + 1.
Find the exact value of each expression.(a) e2In3 (b) log10 25 + log104 (c) tan (arc sin ½) (d) sin (cos –1(4/5))
Solve each equation for x.(a) ex = 5 (b) In x = 2(c) eex = 2 (d) tan-1x = 1
The half-life of palladium-100, 100Pd, is four days. (So half of any given quantity of 100Pd will disintegrate in four days) The initial mass of a sample is one gram.(a) Find the mass that remains
The population of a certain species in a limited environment with initial population 100 and carrying capacity 1000 is where t is measured in years.(a) Graph this function and estimate how long it
Graph members of the family of functions f(x) = In (x2 – c) for several values of c. How does the graph change when changes?
Graph the three functions y = xa, y = ax, and y = log ax on the same screen for two or three values of a >. For large values of x, which of these functions has the largest values and which has the
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.
The altitude perpendicular to the hypotenuse of a right triangle is 12 cm. Express the length of the hypotenuse as a function of the perimeter.
Solve the equation | 2x – 1 | – | x + 5 | = 3.
Solve the inequality | 2x – 1 | – | x – 3 | > 5.
Sketch the graph of the function f (x) = | x2 – 4 | x | + 3 |.
Sketch the graph of the function g (x) = | x2 – 4 | – | x2 – 4 |.
Draw the graph of the equation x + |x| = y + | y |.
Draw the graph of the equation x4 – 4x2 – x2y2 + 4y2 = 0.
Sketch the region in the plane consisting of all points (x, y) such that | x | + | y | < 1.
Sketch the region in the plane consisting of all points (x, y) such that x – y | + | x | – |y| < 2
Evaluate (log2 3) (log3 4) (log4 5) . . ., (log 31 32).
(a) Show that the function f (x) = In (x + √x2 + 1 is an odd function. (b) Find the inverse function of f.
Solve the inequality In (x2 – 2x – 2 < 0.
Use indirect reasoning to prove that log2 5is an irrational number.
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?
Is it true that f o (g + h) = f o g + f o h?
Prove that if n is a positive integer then 7n – 1 is divisible by 6.
Prove that 1 + 3 + 5 + . . . + (2n – 1) = n2.
If f0(x) = x2 and fn+1(x) = f 0(f n(x)) for n = 0, 1, 2. . . , find a formula for fn (x)..
(a) If f0(x) =1/2 – x and fn+1 = f0 o fn for n = 0, 1, 2., find an expression for fn(x) and use mathematical induction to prove it.(b) Graph f0, f1, f2, f3 on the same screen and describe the
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)
A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after t minutes. When the data in the table are graphed, the slope of the tangent
The point P(1, 1/2) lies on the curve y =x/(1 + x).(a) If Q is the point (x, x/(1 + x)), use your calculator to find the slope of the secant line PQ(correct to six decimal places) for the following
The point P (2, In2) lies on the curve y = In x.(a) If Q is the point (x, In x), use your calculator to find the slope of the secant line PQ (correct to six decimal places) for the following values
If a ball is thrown into the air with a velocity of 40 ft/s, its height in feet after seconds is given by y = 40t - 16t2.(a) Find the average velocity for the time period beginning when t = 2 and
If an arrow is shot upward on the moon with a velocity of 58 m/s, its height in meters after seconds is given by h = 58t - 0.83t2. (a) Find the average velocity over the given time intervals:(i) [1,
The displacement (in feet) of a certain particle moving in a straight line is given by s = t3/6, where is measured in seconds.(a) Find the average velocity over the following time periods:(i) [1, 3]
The position of a car is given by the values in the table.(a) Find the average velocity for the time period beginning when t = 2 and lasting(i) 3 seconds (ii) 2 seconds (iii) 1 second(b) Use the
The point P (1, 0) lies on the curve y = sin (10π /x). (a) If Q is the point (x, sin (10π/x), find the slope of the secant line PQ (correct to four decimal places) for x = 2, 1.5, 1.4,
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.
In this situation is it possible that limx→1 f(x) exists? Explain.Explain what it means to say that
Explain the meaning of each of the following.
For the function f whose graph is given, state the value of the given quantity, if it exists. If it does not exist, explain why.(a) limx→0 f(x) (b limx→3€“ f(x)(c)
Use the given graph of f to state the value of each quantity, if it exists. If it does not exist, explain why.(a) limx→1- f(x) (b) limx→1+ f(x)(c) limx→1 f(x) (d) limx→5
For the function whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why.(a) lim x→ €“2€“ g(x)(b) lim x→ €“2+
For the function whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why.(a) lim t→ 0€“ g(t)(b) lim t→ 0+ g(t)(c) lim t→0
For the function R whose graph is shown, state the following.(a) lim x→2 R(x) (b) lim x→ 5 R (x)(c) lim x→ €“3€“ R (x) (d) lim x→€“3+ R(x)(e) The
For the function f whose graph is shown, state the following.(a) lim x→ €“7 f(x)(b) lim x→ €“3 f(x)(c) lim x→ 0 f(x)(d) lim x→ €“6€“
A patient receives a 150-mg injection of a drug every 4 hours. The graph shows the amount f(t) of the drug in the bloodstream after hours. (Later we will be able to compute the dosage and time
Use the graph of the function f(x) = 1 / (1 + e1/x) to state the value of each limit, if it exists. If it does not exist, explain why. (a) limx→0– f (x) (b) limx→0+ f (x) (c)
Sketch the graph of the following function and use it to determine the values of for which lim x→ a f (x) exists:
Sketch the graph of an example of a function f that satisfies all of the given conditions.
Limx→0– f (x) = 1, limx→0+ f (x) = –1, limx→2– f (x) = 0, limx→2+ f (x) = 1, f (2) = 1, f (0) is undefined.
Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places).
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.
Determine the infinite limit.
Determine lim x→1– 1 / x3 – 1 and lim x→+ 1 / x3 – 1 (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
(a) Find the vertical asymptotes of the function y = x / x2 – x – 2(b) Confirm your answer to part (a) by graphing the function.
(a) Estimate the value of the limitx→0 (1 + x) 1/x to five decimal places. Does this number look familiar? (b) Illustrate part (a) by graphing the function y = (1 + x)1/x
The slope of the tangent line to the graph of the exponential function y = 2x at the point (0, 1) is limx→0 (2x – 1)/x. Estimate the slope to three decimal places.
(a) Evaluate the function f(x) = x2 €“ (2x/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.
(a) Evaluate h(x) = (tan x – x)/x3 for x = 1, 0.5, 0.1, 0.05, 0.01, and 0.005. (b) Guess the value of lim x→0 tan x – x / x3. (c) Evaluate h(x) for successively smaller values of until
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.
In the theory of relativity, the mass of a particle with velocity v is where mo is the rest mass of the particle and is the speed of light. What happens as v → c -?
Use a graph to estimate the equations of all the vertical asymptotes of the curve y = tan (2sin x) – π < x < π Then find the exact equations of these asymptotes.
(a) Use numerical and graphical evidence to guess the value of the limit(b) How close to 1 does have to be to ensure that the function in part (a) is within a distance 0.5 of its limit?
Given that lim x→ a f(x) = €“ 3 lim x→ a g(x) = 0 lim x→ a h(x) = €“ 8 Find the limits that exist, if the limit does not exist, explain why.
The graphs of f and g are given. Use them to evaluate each limit, if it exists. If the limit does not exist, explain why.
Evaluate the limit and justify each step by
(a) What is wrong with the following equation? x2 + x €“ 6 / x €“ 2 = x + 3(b) In view of part (a), explain why the equation is correct.
Evaluate the limit if it exist.
a) Estimate the value ofby graphing the function f(x) = x / (√1 + 3x €“ 1).(b) Make a table of values of for x close to 0 and guess the value of the limit.(c) Use the Limit Laws to
(a) Use a graph ofto estimate the value of lim x →0 f(x) to two decimal places.(b) Use a table of values of f(x)to estimate the limit to four decimal places.(c) Use the Limit Laws to find the
Use the Squeeze Theorem to show that limx→0 x2 cos20πx = 0. Illustrate by graphing the functions f(x) = – x2, g(x) x2 cos 20πx, and h(x) = x2 on the same screen.
Use the Squeeze Theorem to show thatIllustrate by graphing the functions f, g, and h (in the notation of the Squeeze Theorem) on the same screen.
If 1 < f(x) < x2 + 2x + 2 for all x, fin lim x→–1 f(x)
If 3x < f(x) < x3 + 2 for 0 < x < 2 evaluate lim x→–1 f(x)
Prove that limx→0 x4 cos 2/x = 0.
Prove that limx→0+ √xe sin (π/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.(i) limx→0+ sgn x(ii)
Let(a)Find limx→2- f(x) and limx→2+ f(x)(b) Does limx→2 f(x) exist?(c) Sketch the graph of f.
Let F(x) = x2 – 1 / |x – 1| (a) Find (i) limx→1+ F(x) (i) limx→1– F(x) (b) Does limx→1 F(x) exit? (c) Sketch the graph of F.
Let(a) Evaluate each of the following limits, if it exists.(i) limx→1+ h(x) (ii) limx→0 h(x) (iii) limx→1 h(x)(iv) limx→2- h(x) (v) limx→2+ h(x) (vi) limx→2 h(x)(b)
(a) If the symbol [ ] denotes the greatest integer function defined in Example 10, evaluate (i) lim x→–2+ [x] (ii) lim x→–2 [x] (iii) lim x→–2.4 [x] (b) If n is an
Let f(x) = x – [x]. (a) Sketch the graph of f. (b) If n is an integer, evaluate (i) lim x→ n– f[x] (ii) lim x→ n+ f[x] (c) For what values of a does lim x→ a f[x] exist?
If f(x) = [x] + [–x], show that lim x→ 2 f(x) exists but is not equal to f (2).
In the theory of relativity, the Lorentz contraction formula L = L0√1 – v2/c2 expresses the length L of an object as a function of its velocity v with respect to an observer, where L0 is the
If p is a polynomial, show that |lim x→ a p(x) = p(a).
If r is a rational function, use Exercise 53 to show that lim x→ a r(x) = r(a) for every number a in the domain of r.
IfProve that lim x→ o f(x) = 0.
Show by means of an example that lim x →a [f(x) + g(x)] may exist even though neither lim x →a f(x) nor lim x →a g(x) exists.

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