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mathematics
precalculus
Thomas Calculus Early Transcendentals 13th Edition Joel R Hass, Christopher E Heil, Maurice D Weir - Solutions
Estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10-4 by (a) The Trapezoidal Rule and (b) Simpson’s Rule. 2 sin (x + 1) dx 0
Express the integrand as a sum of partial fractions and evaluate the integrals. S 0 dx (x + 1)(x² + 1)
The integrals are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. 4t³ - 1² + 16t 1² + 4 dt
A rectangular swimming pool is 30 ft wide and 50 ft long. The accompanying table shows the depth h(x) of the water at 5-ft intervals from one end of the pool to the other. Estimate the volume of water in the pool using the Trapezoidal Rule with n = 10 applied to the integral So V = -50 0 30.h(x) dx.
What probability density function describes the normal distribution? What are some examples typically modeled by a normal distribution? How do we usually calculate probabilities for a normal distribution?
Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. √x + 2√x − 1dx - -dx 2x√x – 1
The infinite region bounded by the coordinate axes and the curve y = -ln x in the first quadrant is revolved about the x-axis to generate a solid. Find the volume of the solid.
Estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10-4 by (a) The Trapezoidal Rule and (b) Simpson’s Rule. L cos (x + 7) dx
Express the integrand as a sum of partial fractions and evaluate the integrals. y² + 2y + 1 ,2 The (y² + 1)² dy
The integrals are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. - TT/2 V1 - cos 0 de
Express the integrand as a sum of partial fractions and evaluate the integrals. 1 √√3 31² + 1 + 4 1³ + t dt
In a normal distribution, what percentage of the population lies within 1 standard deviation of the mean? Within 2 standard deviations?
The accompanying table shows time-to-speed data for a sports car accelerating from rest to 130 mph. How far had the car traveled by the time it reached this speed? (Use trapezoids to estimate the area under the velocity curve, but be careful: The time intervals vary in length.) Speed change Zero to
Show that if the exponentially decreasing functionis a probability density function, then A = c. f(x) 0 So Aex if x < 0 if x ≥ 0
Find the centroid of the region in the first quadrant that is bounded below by the x-axis, above by the curve y = ln x, and on the right by the line x = e.
Suppose ƒ is a probability density function for the random variable X with mean m. Show that its variance satisfies Var (X) = هرا 20 X2f(X ) dX = 2. -
Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. (sect + cott)² dt
The graph of the equation x2/3 + y2/3 = 1 is an astroid. Find the area of the surface generated by revolving the curve about the x-axis. y 1 0 - 1 x2/3 + y2/3: = 1 X
Express the integrand as a sum of partial fractions and evaluate the integrals. 8x² + 8x + 2 18.12 (4x² + 1)² - dx
Find the centroid of the region in the plane enclosed by the curves y = ± (1 - x2)-1/2 and the lines x = 0 and x = 1.
Evaluate the integrals by using a substitution prior to integration by parts. 38+9 ds
Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. dy Ve²y - 1 S
Compute the mean and median for a random variable with the probability density functions f(x) x over [0, 4] 8x
The design of a new airplane requires a gasoline tank of constant cross-sectional area in each wing. A scale drawing of a cross-section is shown here. The tank must hold 5000 lb of gasoline, which has a density of 42 lb/ft3. Estimate the length of the tank by Simpson’s Rule. yo = 1.5 ft, 4
Find the length of the curve y = ln x from x = 1 to x = e.
Express the integrand as a sum of partial fractions and evaluate the integrals. √(5² 2s + 2 (s² + 1)(s - 1)³ ds
The sine-integral function,is one of the many functions in engineering whose formulas cannot be simplified. There is no elementary formula for the antiderivative of (sin t)/t. The values of Si(x), however, are readily estimated by numerical integration. Although the notation does not show it
Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. 6 dy √y (1 + y)
Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. Sivi 2 dx xV1 - 4 ln² x
Evaluate the integrals by using a substitution prior to integration by parts. SIVT xV1 x dx -
A diesel generator runs continuously, consuming oil at a gradually increasing rate until it must be temporarily shut down to have the filters replaced. Use the Trapezoidal Rule to estimate the amount of oil consumed by the generator during that week. Day Sun Mon Tue Wed Thu Fri Sat Sun Oil
Evaluate the integrals by using a substitution prior to integration by parts. TT/3 S™ x tan² x dx
Compute the mean and median for a random variable with the probability density functions fr ܐܨܓ 9 over [0, 3]
For what value or values of a does ax L ( 12 4 4 1 - 12/7) dx + 2x
Express the integrand as a sum of partial fractions and evaluate the integrals. 13² S481 s(s² + 9)² 2 ds
Compute the mean and median for a random variable with the probability density functions f(x) = = 2 x ≥ 1 x3 0 x < 1
Express the integrand as a sum of partial fractions and evaluate the integrals. S - x + 2 1 x3 dx
Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. Ja dx (x - 2)√x² - 4x + 3
Compute the mean and median for a random variable with the probability density functions f(x) = X 1≤x≤e 0 Otherwise
Evaluate the integrals by using a substitution prior to integration by parts. 1₁ In (x + x²) dx
Express the integrand as a sum of partial fractions and evaluate the integrals. 1 4 x² + x X xp -
Determine which are probability density functions and justify your answer. f(x) = x 1 over [0,1 + √3]
Expand the quotients by partial fractions. 2x + 2 x² - 2x + 1
The instructions for the integrals have two parts, one for the Trapezoidal Rule and one for Simpson’s Rule.I. Using the Trapezoidal Rulea. Estimate the integral with n = 4 steps and find an upper bound for |ET|.b. Evaluate the integral directly and find |ET|.c. Use the formula (|ET|/(true value))
Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. TT/3 Jπ/4 dx cos² x tan x
The instructions for the integrals have two parts, one for the Trapezoidal Rule and one for Simpson’s Rule.I. Using the Trapezoidal Rulea. Estimate the integral with n = 4 steps and find an upper bound for |ET|.b. Evaluate the integral directly and find |ET|.c. Use the formula (|ET|/(true value))
Determine which are probability density functions and justify your answer. f(x) = x ≥ 1 0 x < 1
Expand the quotients by partial fractions. [+2 z²(z − 1)
Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. 1-x V1 - x² =dx
What substitutions are made to evaluate integrals of sin mx sin nx, sin mx cos nx, and cos mx cos nx? Give an example of each case.
The instructions for the integrals have two parts, one for the Trapezoidal Rule and one for Simpson’s Rule.I. Using the Trapezoidal Rulea. Estimate the integral with n = 4 steps and find an upper bound for |ET|.b. Evaluate the integral directly and find |ET|.c. Use the formula (|ET|/(true value))
Determine which are probability density functions and justify your answer. f(x) = 2 cos 2x over 4
Determine which are probability density functions and justify your answer. f(x) = 8 T(4 + x²) 0 x ≥ 0 x < 0
Expand the quotients by partial fractions. 29 で 22 - 2 ع Z
Expand the quotients by partial fractions. 1² + 8 1²- 5t + 6
Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. S dx x - Vx
What substitutions are sometimes used to transform integrals involving √a2 - x2, √a2 + x2, and √x2 - a2 into integrals that can be evaluated directly? Give an example of each case.
The instructions for the integrals have two parts, one for the Trapezoidal Rule and one for Simpson’s Rule.I. Using the Trapezoidal Rulea. Estimate the integral with n = 4 steps and find an upper bound for |ET|.b. Evaluate the integral directly and find |ET|.c. Use the formula (|ET|/(true value))
Determine which are probability density functions and justify your answer. f(x) = 1/1/20 X over (0, e]
Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. -cot z sin² z dz
Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. 2ln z³ - dz 16z 1²
The instructions for the integrals have two parts, one for the Trapezoidal Rule and one for Simpson’s Rule.I. Using the Trapezoidal Rulea. Estimate the integral with n = 4 steps and find an upper bound for |ET|.b. Evaluate the integral directly and find |ET|.c. Use the formula (|ET|/(true value))
What restrictions can you place on the variables involved in the three basic trigonometric substitutions to make sure the substitutions are reversible (have inverses)?
Expand the quotients by partial fractions. 14 +9 14 + 91²
What is the goal of the method of partial fractions?
When the degree of a polynomial ƒ(x) is less than the degree of a polynomial g(x), how do you write ƒ(x)/g(x) as a sum of partial fractions if g(x)a. Is a product of distinct linear factors?b. Consists of a repeated linear factor?c. Contains an irreducible quadratic factor? What do you do if the
How are integral tables typically used? What do you do if a particular integral you want to evaluate is not listed in the table?
The instructions for the integrals have two parts, one for the Trapezoidal Rule and one for Simpson’s Rule.I. Using the Trapezoidal Rulea. Estimate the integral with n = 4 steps and find an upper bound for |ET|.b. Evaluate the integral directly and find |ET|.c. Use the formula (|ET|/(true value))
Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. dz e² + e -²
Express the integrand as a sum of partial fractions and evaluate the integrals. Si dx 1-x² 2
Let ƒ(X) be the uniform distribution for the random variable X in Example 11. Express the following probabilities as integrals.a. The probability that the arrow points either between South and West or between North and West.b. The probability that the arrow makes an angle of at least 2 radians.
Determine which are probability density functions and justify your answer. f(x) = x over [4,8]
The instructions for the integrals have two parts, one for the Trapezoidal Rule and one for Simpson’s Rule.I. Using the Trapezoidal Rulea. Estimate the integral with n = 4 steps and find an upper bound for |ET|.b. Evaluate the integral directly and find |ET|.c. Use the formula (|ET|/(true value))
Expand the quotients by partial fractions. 5x 5r – 13 (x − 3)(x - 2)
Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. S. 3 0 16x 8x² + 2 dx
The instructions for the integrals have two parts, one for the Trapezoidal Rule and one for Simpson’s Rule.I. Using the Trapezoidal Rulea. Estimate the integral with n = 4 steps and find an upper bound for |ET|.b. Evaluate the integral directly and find |ET|.c. Use the formula (|ET|/(true value))
The instructions for the integrals have two parts, one for the Trapezoidal Rule and one for Simpson’s Rule.I. Using the Trapezoidal Rulea. Estimate the integral with n = 4 steps and find an upper bound for |ET|.b. Evaluate the integral directly and find |ET|.c. Use the formula (|ET|/(true value))
Determine which are probability density functions and justify your answer. f(x) = 1/2 (2 (2 - x) over [0,2]
Expand the quotients by partial fractions. 5 – 7 x² - 3x + 2
What is a reduction formula? How are reduction formulas used? Give an example.
Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. (secx - tan tan x)² dx
Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. x2 x2 + 1 Xp.
What is the formula for integration by parts? Where does it come from? Why might you want to use it?
Determine which are probability density functions and justify your answer. In (1+ In 2) In 2 f(x) = 2* over 0,-
Expand the quotients by partial fractions. x + 4 (x + 1)²
When applying the formula for integration by parts, how do you choose the u and dν? How can you apply integration by parts to an integral of the form ∫ƒ(x) dx?
If an integrand is a product of the form sinn x cosm x, where m and n are nonnegative integers, how do you evaluate the integral? Give a specific example of each case.
How would you compare the relative merits of Simpson’s Rule and the Trapezoidal Rule?
The integrals are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. 1 2 8 dx x² - 2x + 2
Express the integrand as a sum of partial fractions and evaluate the integrals. S; dx X² 2 x² + 2x
Just as x = cos u and y = sin u are identified with points (x, y) on the unit circle, the functions x = cosh u and y = sinh u are identified with points (x, y) on the right-hand branch of the unit hyperbola, x2 - y2 = 1.Another analogy between hyperbolic and circular functions is that the variable
Use the shell method to find the volumes of the solids generated by revolving the regions bounded by the curves and line about the x-axis.y = |x|, y = 1
Find the volume of the solid generated by revolving the shaded region about the given axis.About the y-axis y 1 0/ x = tan (7) X
Find the areas of the surfaces generated by revolving the curves about the indicated axes. If you have a grapher, you may want to graph these curves to see what they look like. x = y³/3, 0≤ y ≤ 1;_y-axis
Find the areas of the surfaces generated by revolving the curves about the indicated axes. If you have a grapher, you may want to graph these curves to see what they look like. X = (1/3)y³/2y¹/2, 1 ≤ y ≤ 3; y-axis
Do the following.a. Set up an integral for the length of the curve.b. Graph the curve to see what it looks like.c. Use your grapher’s or computer’s integral evaluator to find the curve’s length numerically.y = tan x, -π/3 ≤ x ≤ 0
Use the shell method to find the volumes of the solids generated by revolving the regions bounded by the curves and line about the x-axis.x = y2, x = -y, y = 2, y ≥ 0
Find the volume of the solid generated by revolving the shaded region about the given axis.About the x-axis 72 0 y = sin x cos x ㅠ 2 → X
Find the center of mass of a thin plate covering the region bounded below by the parabola y = x2 and above by the line y = x if the plate’s density at the point (x, y) is δ(x) = 12x.
a. Suppose that the conical container in Example 5 contains milk (weighing 64.5 lb/ft3) instead of olive oil. How much work will it take to pump the contents to the rim?b. How much work will it take to pump the oil in Example 5 to a level 3 ft above the cone’s rim?Example 5The conical tank in
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