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mathematics
calculus 6th edition

Questions and Answers of Calculus 6th edition

Find the area under the parabola y = x2 from 0 to 1.
Use the properties of integrals to evaluate S₁² (4+ Jo (4 + 3x²) dx.
A particle moves along a line so that its velocity at time i is v(t) = t2 – t –6 (measured in meters per second). (a) Find the displacement of the particle during the time period 1 ≤ t ≤
Evaluate dx ¹6 13 X
Calculate ∫ tan x dx.
Figure 4 shows the power consumption in the city of San Francisco for a day in September (P is measured in megawatts; t is measured in hours starting at midnight). Estimate the energy used on that
If it is known that 10 f f(x) dx = 17 and fő f(x) dx = 17 and f 10 f(x) dx = 12, find fº f(x) dx.
Evaluate ∫04 √2x + 1 dx using (6) 6 THE SUBSTITUTION RULE FOR DEFINITE INTEGRALS If g' is continuous on [a, b] and f is continuous on the range of u = g(x), then ff(g(x))g'(x) dx = = *g(b) b)
The graph of a function f is shown in Figure 6. (a) Find the values of f(1) and f(5). (b) What are the domain and range of f?Figure 6 y 0 x
Find the area under the cosine curve from 0 to b, where 0 ≤ b ≤ π/2.
Sketch the graph of the function y = 3 – 2x and determine its domain and range.
Is the function f(x) = x3 one-to-one?
(a) As dry air moves upward, it expands and cools. If the ground temperature is 20°C and the temperature at a height of 1 km is 10°C, express the temperature T (in °C) as a function of the height
Draw the graph of the function f(x) = x2 + 3 in each of the following viewing rectangles.(a) [–2,2] by [–2, 2] (b) [–4,4] by [–4, 4](c) [–10, 10] by [–5, 30] (d) [–50, 50] by [–100,
Given the graph of y = √x, use transformations to graph y = √√x − 2₁ y = √√√x - 2₁y = -√√x₂y = 2√x, and y = -X.
Express the hypotenuse h of a right triangle with area 25 m2 as a function of its perimeter P.
Sketch the graph and find the domain and range of each function. (a) f(x) = 2x – 1 (b) g(x) = x2
Use a graphing device to compare the exponential function f(x) = 2x and the power function g(x) = x2. Which function grows more quickly when x is large?
Table 1 lists the average carbon dioxide level in the atmosphere, measured in parts per million at Mauna Loa Observatory from 1980 to 2002. Use the data in Table 1 to find a model for the carbon
Is the function g(x) = x2 one-to-one?
Determine an appropriate viewing rectangle for the function f(x) = √8 – 2x2 and use it to graph f.
Solve the inequality |x – 3| + |x + 2| < 11.
If f(x) = 2x2 – 5x + 1 and h ≠ 0, evaluate f(a+h)-f(a) h
Sketch the graph of the function f(x) = x2 + 6x + 10.
Graph the function y = 1/2 e–x – 1 and state the domain and range.
Use the linear model given by Equation 2 to estimate the average CO2 level for 1987 and to predict the level for the year 2010. According to this model, when will the CO2 level exceed 400 parts per
If f(1) = 5, f(3) = 7, and f(8) = –10, find f–1(7), f–1(5). and f–1(-10).
If f0(x) = x/(x + 1) and fn+1 = f0 ∘ fn for n = 0, 1, 2, . . ., find a formula for fn(x).
Figure 9 shows graphs of the number of hours of daylight as functions of the time of the year at several latitudes. Given that Philadelphia is located at approximately 40°N latitude, find a function
Sketch the graphs of the following functions. (a) y = sin 2x (b) y = 1 – sin x
Use a graphing device to find the values of x for which ex > 1000,000.
A ball is dropped from the upper observation deck of the CN Tower, 450 m above the ground, and its height h above the ground is recorded at 1-second intervals in Table 2. Find a model to fit the data
Find the inverse function of f(x) = x3 + 2.
Graph the function f(x) = sin 50x in an appropriate viewing rectangle.
Graph the function f(x) = sin x + 1/100 cos 100x.
Sketch the graph of the function y = |x2 – 1|.
Classify the following functions as one of the types of functions that we have discussed. (a) f(x) = 5x (c) h(x) = 1 + x 1- - (b) g(x) = x³ (d) u(t)=1t+5tª
Sketch the graphs of f(x) = √–1 – x and its inverse function using the same coordinate axes.
Draw the graph of the function y = 1/1 – x.
Find the domain of each function.(a) f(x) = √x + 2 (b) g(x) = 1/x2 – x
A function f is defined by Evaluate f(0), f(1), and f(2) and sketch the graph. f(x) 1- 1.1.² x if x ≤ 1 if x > 1
Use the laws of logarithms to evaluate log2 80 – log2 5.
If f(x) = √x and g(x) = √2 – x, find each function and its domain.(a) f ∘ g (b) g ∘ f (c) f ∘ f (d) g ∘ g
Find x if In x = 5.
Graph the function y = x3 + cx for various values of the number c. How does the graph change when c is changed?
Sketch the graph of the absolute value function f(x) = |x|.
Find f ∘ g ∘ h if f(x) = x/(x + 1), g(x) = x10, and h(x) = x + 3.
Solve the equation e5-3x = 10.
Find a formula for the function f graphed in Figure 17.Figure 17. VA 0 X
Find the solution of the equation cos x = x correct to two decimal places.
Given F(x) = cos2(x + 9), find functions f, g, and h such that F = f ∘ g ∘ h.
Express ln a + 1/2 ln b as a single logarithm.
Evaluate log8 5 correct to six decimal places.
Determine whether each of the following functions is even, odd, or neither even nor odd. (a) f(x) = x5 + x (b) g(x) = 1 – x4 (c) h(x) = 2x – x2
Sketch the graph of the function y = ln(x – 2) – 1.
Evaluate (a) sin–1(1/2) and (b) tan(arcsin 1/3).
Simplify the expression cos(tan–1 x).
Find an equation of the tangent line to the parabola y = x2 at the point P(1, 1).
Evaluate where c is a nonzero constant. lim X-0 1 + cx 1 X
Guess the value of 1 - ₂x1-x [-x X lim
The graph of a function f is given in Figure 1. Use it to sketch the graph of the derivative f'.Figure 1 y. 1 0 y = f(x) X
Figure 2 shows the graph of a function f. At which numbers is f discontinuous? Why?Figure 2 y 0 + 2 3 4 5 X
Find an equation of the tangent line to the parabola y = x2 at the point P(1, 1).
Use the Limit Laws and the graphs of f and g in Figure 1 to evaluate the following limits, if they exist.Figure 1 (a) lim [f(x) + 5g(x)] X-2 (b) lim [f(x)g(x)] x-1 f(x) X-2 g(x) (c) lim
Use a graph to find a number δ such thatIn other words, find a number δ that corresponds to ε = 0.2 in the definition of a limit for the function f(x) = x3 – 5x + 6 with a = 1 and L = 2. if
Find the infinite limits, limits at infinity, and asymptotes for the function f whose graph is shown in Figure 5.Figure 5 0 2 2 20
Estimate the value of lim 1²+ 9-3 1²
The flash unit on a camera operates by storing charge on a capacitor and releasing it suddenly when the flash is set off. The data in the table describe the charge Q remaining on the capacitor
(a) If f(x) = x3 – x, find a formula for f'(x).(b) Illustrate by comparing the graphs of f and f'.
Where are each of the following functions discontinuous? (a) f(x) = (c) f(x) = x²-x-2 x-2 2 x² - X - x-2 1 2 if x # 2 if x = 2 (b) f(x) = K (d) f(x) = [x] if x # 0 if x = 0
Find an equation of the tangent line to the hyperbola y = 3/x at the point (3, 1).
Evaluate the following limits and justify each step. (a) lim (2x²-3x + 4) X-5 (b) lim X-2 x³ + 2x² - 1 5 - 3x
Prove that lim (4.x - 5) = 7. X-3
Find 1 lim and X→∞ X lim 1
Guess the value of sin x lim x-0 X
Suppose that a ball is dropped from the upper observation deck of the CN Tower in Toronto, 450 m above the ground. Find the velocity of the ball after 5 seconds.
If f(x) = √x, find the derivative of f. State the domain of f'.
Suppose that a ball is dropped from the upper observation deck of the CN Tower, 450 m above the ground.(a) What is the velocity of the ball after 5 seconds?(b) How fast is the ball traveling when it
Find x² - 1 X 1 lim x1 x X
Use Definition 4 to prove thatData from definition 4 lim √√√x = 0. X x→0+
Evaluateand indicate which properties of limits are used at each stage. 3x²-x-2 lim x →∞0 5x² + 4x + 1
Investigate lim sin x-0 TT
Find f' if f(x) = 1-x 2 + x
Find the derivative of the function f(x) = x2 – 8x + 9 at the number a.
Find limx→1 g(x) where g(x) x + 1 if x # 1 if x = 1 TT
Show that the function f(x) = 1 –√1 – x2 is continuous on the interval [–1, 1].
Find the horizontal and vertical asymptotes of the graph of the function f(x) = 2x² + 1 3x - 5
Find limx³ + x-0 cos 5x 10,000
Where is the function f(x) = |x| differentiable?
Find lim X→-2 x³ + 2x² - 1 5 - 3x
Find an equation of the tangent line to the parabola y = x2 – 8x + 9 at the point (3, –6).
Use Definition 6 to prove thatData from definition 6 1 lim 2 ₂.X 0-x
Evaluate lim h→0 (3 + h)²9 h
Compute lim (√x² + 1 x). X→∞
If f(x) = x3 – x, find and interpret f"(x).
Where is the function continuous? f(x) In x + tan¹x x² - 1
A manufacturer produces bolts of a fabric with a fixed width. The cost of producing x yards of this fabric is C = f(x) dollars. (a) What is the meaning of the derivative f'(x)? What are its
Find lim 1-0 √² +93 2
Find lim x3 and lim x³. x →∞0
Find F'(x) if F(x) = √x2 + 1.

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