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study help
mathematics
calculus 6th edition
Calculus 6th Edition James Stewart - Solutions
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 ≤ 4. (b) Find the distance traveled during this time period.
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 day.Figure 4 P 800 600 400 200 0 3 ليا 6 9 12 15 18 21 1 Pacific Gas & Electric
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) f(u) du Ig(a)
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 h (in kilometers), assuming that a linear model is appropriate. (b) Draw the graph of the function
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, 1000]
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 dioxide level. Year 1980 1982 1984 1986 1988 1990 TABLE I CO₂ level (in
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 million?Equation 2 C = 1.55192t - 2734.55
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 that models the length of daylight at Philadelphia.Figure 9 20 18 16 14 12 Hours 10 6 4 2 20°N -0-
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 and use the model to predict the time at which the ball hits the ground. Time (seconds) 0 1234 in
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 |x-1|< 8 then |(3 – 5x + 6) – 2|
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 (measured in microcoulombs) at time t (measured in seconds after the flash goes off ). Use the data to
(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 hits the ground?
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 units? (b) In practical terms, what does it mean to say that f'(1000) = 9? (c) Which do you think is
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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![6 THE SUBSTITUTION RULE FOR DEFINITE INTEGRALS If g' is continuous on [a, b] and f is continuous on the range](https://dsd5zvtm8ll6.cloudfront.net/images/question_images/1698/8/2/0/0416541efc96f4e01698820042767.jpg)
![(a) lim [f(x) + 5g(x)] X-2 (b) lim [f(x)g(x)] x-1 f(x) X-2 g(x) (c) lim](https://dsd5zvtm8ll6.cloudfront.net/images/question_images/1698/5/0/4/729653d2019681481698504729577.jpg)
![(a) f(x) = (c) f(x) = x-x-2 x-2 2 x - X - x-2 1 2 if x # 2 if x = 2 (b) f(x) = K (d) f(x) = [x] if x # 0 if x](https://dsd5zvtm8ll6.cloudfront.net/images/question_images/1698/5/1/9/450653d599a4087a1698519451870.jpg)
