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study help
mathematics
precalculus
Questions and Answers of
Precalculus
Find the indefinite integral by making a change of variables.∫ sec5 7x tan 7x dx
Find the indefinite integral.∫ 6x3 √3x4 + 2 dx
Find the indefinite integral.∫ sin3 x cos x dx
Evaluate the definite integral. Use a graphing utility to verify your result.∫1-1 x(x2 + 1)3 dx
Evaluate the definite integral. Use a graphing utility to verify your result. 1 So J+ √1 + x dx
Find the indefinite integral.∫ x sin 3x2 dx
Evaluate the definite integral. Use a graphing utility to verify your result. So x²(x³ - 2)³ dx
Find the indefinite integral.∫ cosθ /√1 - sinθ dθ
Consider the region bounded by the graphs of f (x) = 8x/(x + 1), x =0, x = 4, and y = 0, as shown in the figure. To print an enlarged copy of the graph, go to MathGraphs.com.(a) Redraw the figure,
Find the indefinite integral.∫ sin x /√cos x dx
Evaluate the definite integral. Use a graphing utility to verify your result. -6 S. 3. √3 3√√√x²-8 X =dx
Find the indefinite integral.∫ x√8 - x dx
Evaluate the definite integral. Use a graphing utility to verify your result.∫40 1/√2x + 1 dx
Find the indefinite integral.∫ √1+ √x dx
Evaluate the definite integral. Use a graphing utility to verify your result.∫91 1 /√x(1 + √x)2 dx
Evaluate the definite integral. Use a graphing utility to verify your result.∫20 x / √1 + 2x2 dx
Evaluate the definite integral. Use a graphing utility to verify your result.∫54 x /√2x - 6 dx
Evaluate the definite integral. Use a graphing utility to verify your result. 2π (y + 1)√1- y dy
Evaluate the definite integral. Use a graphing utility to verify your result. 0 2π x² √√x + 1dx -1
Find the area of the region. Use a graphing utility to verify your result. n/2 0 2 -1 -2 y (cos x + sin 2x) dx 2 X
Evaluate the integral using the properties of even and odd functions as an aid.∫2-2 (x3 - 2x) dx
Evaluate the integral using the properties of even and odd functions as an aid.∫π-π (cos x + x2) dx
(a) Verify that sin u - u cos u + C = ∫ u sin u du.(b) Use part (a) to show that ∫π20 sin √x dx = 2π.
The function f (x) = kxn (1 - x)m, 0 ≤ x ≤ 1 where n > 0, m > 0, and k is a constant, can be used to represent various probability distributions. If k is chosen such thatthen the
The function f (x) = kxn (1 - x)m, 0 ≤ x ≤ 1 where n > 0, m > 0, and k is a constant, can be used to represent various probability distributions. If k is chosen such thatthen the
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.∫10-10 (ax3 + bx2 + cx + d) dx = 2 ∫10-10 (bx2 + d) dx
Consider the functions f and g, where f (x) = 6 sin x cos2 x and g(t) = ∫t0 f (x) dx.(a) Use a graphing utility to graph f and g in the same viewing window.(b) Explain why g is nonnegative.(c)
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.∫ 3x2(x3 + 5)-2 dx = -(x3 + 5)-1 + C
The sales S (in thousands of units) of a seasonal product are given by the modelwhere t is the time in months, with t = 1 corresponding to January. Find the average sales for each time period.(a) The
Use l’Hôpital’s rule to find the limit. (In x)² lim x-0+ ln (sin x)
a. Identify the function’s local extreme values in the given domain, and say where they occur.b. Which of the extreme values, if any, are absolute?c. Support your findings with a graphing
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. + z²x = (
Find the function with the given derivative whose graph passes through the point P.ƒ′(x) = 2x - 1, P(0, 0)
a. Find the open intervals on which the function is increasing and decreasing.b. Identify the function’s local and absolute extreme values, if any, saying where they occur.h(x) = x1/3(x2 - 4)
The positions of two particles on the s-axis are s1 = sin t and s2 = sin (t + π/3), with s1 and s2 in meters and t in seconds.a. At what time(s) in the interval 0 ≤ t ≤ 2π do the particles
Use l’Hôpital’s rule to find the limit. lim x²ex X78
Use l’Hôpital’s rule to find the limit. lim x->0+ 3x + 1 X sin X
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. x² = y X - 3 2
Find the function with the given derivative whose graph passes through the point P. g'(x) x² +2x, P(-1, 1)
a. Find the open intervals on which the function is increasing and decreasing.b. Identify the function’s local and absolute extreme values, if any, saying where they occur.k(x) = x2/3(x2 - 4)
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. y = √|₁x| = = [V=x, -X, X < 0 L√x₂ x ≥ 0
Give the velocity y = ds/dt and initial position of an object moving along a coordinate line. Find the object’s position at time t. υ || 2 COS 2t s(73) = 1
Use l’Hôpital’s rule to find the limit. x - sin x lim x0 x tan x
Determine all critical points for each function.ƒ(x) = x(4 - x)3
Give the acceleration a = d2s/dt2, initial velocity, and initial position of an object moving on a coordinate line. Find the object’s position at time t.a = et, v(0) = 20, s(0) = 5
Use l’Hôpital’s rule to find the limit. (ex - 1)² lim x0 x sin x
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. y = √√x - 4| Vx
a. Identify the function’s local extreme values in the given domain, and say where they occur.b. Which of the extreme values, if any, are absolute?c. Support your findings with a graphing
Determine all critical points for each function. y = x² + 2 XIN
Determine all critical points for each function.g(x) = (x - 1)2(x - 3)2
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = xe1/x
Give the acceleration a = d2s/dt2, initial velocity, and initial position of an object moving on a coordinate line. Find the object’s position at time t.a = 9.8, v(0) = -3, s(0) = 0
Use l’Hôpital’s rule to find the limit. lim 0-0 0-sin cos 0 tan 0 - 0
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. y et X
Determine all critical points for each function. f(x) = x² x - 2
Give the acceleration a = d2s/dt2, initial velocity, and initial position of an object moving on a coordinate line. Find the object’s position at time t.a = -4 sin 2t, v(0) = 2, s(0) = -3
Use l’Hôpital’s rule to find the limit. sin 3x lim x-0 - 3x + x² sin x sin 2x
a. Identify the function’s local extreme values in the given domain, and say where they occur.b. Which of the extreme values, if any, are absolute?c. Support your findings with a graphing
Give the acceleration a = d2s/dt2, initial velocity, and initial position of an object moving on a coordinate line. Find the object’s position at time t. a 9 TT COS 3t TT v(0) = 0, s(0) = -1
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = ln (3 - x2)
Determine all critical points for each function.y = x2 - 32√x
Determine all critical points for each function. g(x) = √2x - x²
It took 14 sec for a mercury thermometer to rise from -19°C to 100°C when it was taken from a freezer and placed in boiling water. Show that somewhere along the way the mercury was rising at the
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = x (ln x)2
The intensity of illumination at any point from a light source is proportional to the square of the reciprocal of the distance between the point and the light source. Two lights, one having an
Use l’Hôpital’s rule to find the limit. lim x1+x 1 1 1 In x
Find the function with the given derivative whose graph passes through the point P. P(0.3) f'(x) = ²x, P 0,
a. Find the open intervals on which the function is increasing and decreasing.b. Identify the function’s local and absolute extreme values, if any, saying where they occur.ƒ(x) = e2x + e-x
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. y = √√√x³ + 1 -3
Use l’Hôpital’s rule to find the limit. lim (cscx-cot x + cos x) x-0+
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. y 8x x² + 4
The range R of a projectile fired from the origin over horizontal ground is the distance from the origin to the point of impact. If the projectile is fired with an initial velocity v0 at an angle a
Use l’Hôpital’s rule to find the limit. cos - 1 A lim 0 0e01
The strength S of a rectangular wooden beam is proportional to its width times the square of its depth.a. Find the dimensions of the strongest beam that can be cut from a 12-in.-diameter cylindrical
Find the function’s absolute maximum and minimum values and say where they are assumed.ƒ(x) = x4/3, -1 ≤ x ≤ 8
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. || y = 5 x4 + 5
a. Find the open intervals on which the function is increasing and decreasing.b. Identify the function’s local and absolute extreme values, if any, saying where they occur.ƒ(x) = e√x
Use l’Hôpital’s rule to find the limit. lim h→0 eh - (1 + h) h²
Find the function with the given derivative whose graph passes through the point P.r′(t) = sec t tan t - 1, P(0, 0)
Find the function’s absolute maximum and minimum values and say where they are assumed.ƒ(x) = x5/3, -1 ≤ x ≤ 8
a. Find the open intervals on which the function is increasing and decreasing.b. Identify the function’s local and absolute extreme values, if any, saying where they occur.ƒ(x) = x ln x
Use l’Hôpital’s rule to find the limit. lim et + 1² et - t
Give the velocity y = ds/dt and initial position of an object moving along a coordinate line. Find the object’s position at time t.v = 9.8t + 5, s(0) = 10
Find the function’s absolute maximum and minimum values and say where they are assumed.g(θ) = θ3/5, -32 ≤ θ ≤ 1
The stiffness S of a rectangular beam is proportional to its width times the cube of its depth.a. Find the dimensions of the stiffest beam that can be cut from a 12-in.-diameter cylindrical log.b.
a. Find the open intervals on which the function is increasing and decreasing.b. Identify the function’s local and absolute extreme values, if any, saying where they occur.ƒ(x) = x2 ln x
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = |x2 - 1|
Give the velocity y = ds/dt and initial position of an object moving along a coordinate line. Find the object’s position at time t.v = 32t - 2, s(0.5) = 4
Find the function’s absolute maximum and minimum values and say where they are assumed.h(θ) = 3θ2/3, -27 ≤ θ ≤ 8
Determine all critical points for each function.y = x2 - 6x + 7
Give the velocity y = ds/dt and initial position of an object moving along a coordinate line. Find the object’s position at time t.v = sin πt, s(0) = 0
Determine all critical points for each function.ƒ(x) = 6x2 - x3
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = |x2 - 2x|
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = (2 - x2)3/2
Use l’Hôpital’s rule to find the limit. lim (In 2x In (x + 1))
Find all possible functions with the given derivative.a. y′ = sin 2t b. y′ = cos t/2c. y′ = sin 2t + cos t/2
a. Find the open intervals on which the function is increasing and decreasing.b. Identify the function’s local and absolute extreme values, if any, saying where they occur.ƒ(x) = x1/3(x + 8)
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. y = x√8 - x²
The height above ground of an object moving vertically is given by s = -16t2 + 96t + 112, with s in feet and t in seconds. Finda. The object’s velocity when t = 0;b. Its maximum height and when it
Find all possible functions with the given derivative.a.b.c. y' 1 2√x
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