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mathematics
precalculus
Thomas Calculus Early Transcendentals 13th Edition Joel R Hass, Christopher E Heil, Maurice D Weir - Solutions
For what x > 0 does x(xx) = (xx)x? Give reasons for your answer.
Find the areas between the curves y = 2(log2 x)/x and y = 2(log4 x)/x and the x-axis from x = 1 to x = e. What is the ratio of the larger area to the smaller?
Use a definite integral to find the area of the region between the given curve and the x-axis on the interval [0, b].y = 3x2
The height H(ft) of a palm tree after growing for t years is given bya. Find the tree’s height when t = 0, t = 4, and t = 8.b. Find the tree’s average height for 0 ≤ t ≤ 8. H = Vt + 1 + 5t1/3 for 0 ≤ t ≤ 8.
The temperature T(°F) of a room at time t minutes is given bya. Find the room’s temperature when t = 0, t = 16, and t = 25.b. Find the room’s average temperature for 0 ≤ t ≤ 25. T = 853V/25 - t for for 0≤ t ≤ 25.
Show that if k is a positive constant, then the area between the x-axis and one arch of the curve y = sin kx is 2/k.
Use the Max-Min Inequality to show that if ƒ is integrable then f(x) ≥ 0 on [a, b] •[ f(x) f(x) dx ≥ 0.
Let ƒ be a function that is differentiable on [a, b]. We defined the average rate of change of ƒ over [a, b] to beand the instantaneous rate of change of ƒ at x to be ƒ′(x). In this chapter we defined the average value of a function. For the new definition of average to be consistent with the
Show that if ƒ is integrable then f(x) ≤ 0 on [a, b] cb 2 f(x) dx ≤ 0. =
Find the areas of the regions enclosed by the lines and curves.x = 2y2, x = 0, and y = 3
Find the areas of the regions enclosed by the lines and curves.x = y2 and x = y + 2
Show that the value of ∫01sin (x2) dx cannot possibly be 2.
Find the areas of the regions enclosed by the lines and curves.y2 - 4x = 4 and 4x - y = 16
Find dy/dx. y S 6 3 + 14 -dt
Specific heat Cv is the amount of heat required to raise the temperature of one mole (gram molecule) of a gas with constant volume by 1°C. The specific heat of oxygen depends on its temperature T and satisfies the formulaFind the average value of Cv for 20° ≤ T ≤ 675°C and the temperature at
Find dy/dx. = y 2 1 Lª dt 1² + 1 sec x
Find the areas of the regions enclosed by the lines and curves.x - y2 = 0 and x + 2y2 = 3
Find dy/dx. y ecost dt In x²
Find dy/dx. y = •X 2 √2 + cos³ t dt
Suppose that ƒ(x) is an antiderivative ofExpressin terms of F and give a reason for your answer. f(x) = V1 + x4.
Find dy/dx. y 7x² = = √₂ 2 √2 + cos³t dt
Find dy/dx. neVx y = =[₁² 1 In (t² + 1) dt
Find the areas of the regions enclosed by the lines and curves.x + y2 = 0 and x + 3y2 = 2
Find dy/dx. y = T/4 tan¹x eVt dt
Find the average value of ƒ(x) = mx + ba. Over [-1, 1] b. Over [-k, k]
Find the average value ofa. y = √3x over [0, 3] b. y = √ax over [0, a]
Find dy/dx. y = 0 sin ¹x dt VI - 21²
Is it true that the average value of an integrable function over an interval of length 2 is half the function’s integral over the interval? Give reasons for your answer.
To meet the demand for parking, your town has allocated the area shown here. As the town engineer, you have been asked by the town council to find out if the lot can be built for $10,000. The cost to clear the land will be $0.10 a square foot, and the lot will cost $2.00 a square foot to pave. Can
a. Verify that ∫ ln x dx = x ln x - x + C.b. Find the average value of ln x over [1, e].
Find the average value of ƒ(x) = 1/x on [1, 2].
Find dy/dx ifExplain the main steps in your calculation. y = V1 + 1² dt. X
Find dy/dx ifExplain the main steps in your calculation. y = cos x (1/(1 - 1²)) dt.
Is it true that every function y = ƒ(x) that is differentiable on [a, b] is itself the derivative of some function on [a, b]? Give reasons for your answer.
Skydivers A and B are in a helicopter hovering at 6400 ft. Skydiver A jumps and descends for 4 sec before opening her parachute. The helicopter then climbs to 7000 ft and hovers there. Forty-five seconds after A leaves the aircraft, B jumps and descends for 13 sec before opening his parachute. Both
What is the Net Change Theorem? What does it say about the integral of velocity? The integral of marginal cost?
Use a definite integral to find the area of the region between the given curve and the x-axis on the interval [0, b].y = 2x
Use the results of Equations (2) and (4) to evaluate the integral. 0 3b x² dx
a. Estimate the value ofby graphingover a suitably large interval of x-values.b. Now confirm your estimate by finding the limit with l’Hôpital’s Rule. As the first step, multiply ƒ(x) by the fractionand simplify the new numerator. lim (x - √x² + + x x)
Use the Substitution Formula in Theorem 7 to evaluate the integral.a.b. So 0 rV1 – r² dr
Use the Substitution Formula in Theorem 7 to evaluate the integral.a.b. 0 3 Vy + 1 dy
Find lim (√x² + 1 - Vx).
Use the Substitution Formula in Theorem 7 to evaluate the integral.a.b. 0 π/4 tan x sec² x dx
Express the limits as definite integrals.where P is a partition of [0, 2]. n lim Sc2 Δ.xk, |P|→0 k=1
Evaluate the indefinite integral by using the given substitutions to reduce the integrals to standard form. [² 2(2x + 4)5 dx, u = 2x + 4
Use the Substitution Formula in Theorem 7 to evaluate the integral.a.b. 0 TT 3 cos² x sin x dx
Express the limits as definite integrals.where P is a partition of [-1, 0] n lim Σ2ck3 Δ.xx. |P||→0 k=1
State the Mean Value Theorem’s three corollaries.
At what value(s) of x does cos x = 2x?
Express the limits as definite integrals.where P is a partition of [-7, 5] lim Σ(ck2 |P|→0 k=1 (ck2 – 34) Δ.xk,
At what value(s) of x does cos x = -x?
Evaluate the indefinite integral by using the given substitutions to reduce the integrals to standard form. [TVTX 7√7x 1 dx, u = 7x - 1 -
At what value(s) of x does e-x2 = x2 - x + 1?
At what value(s) of x does ln (1 - x2) = x - 1?
Evaluate the indefinite integral by using the given substitutions to reduce the integrals to standard form. √ 2x(x² + 5)-4 dx, 2x(x² + 5)-4 dx, u = x² + 5
Use the Substitution Formula in Theorem 7 to evaluate the integral.a.b. 1 Lea 1³ (1+14)³ dt
Express the limits as definite integrals.where P is a partition of [1, 4] n 1 Ck lim ||P||0 k=1 AXk,
Evaluate the indefinite integral by using the given substitutions to reduce the integrals to standard form. 4x3 (x4 + 1)² dx. dx, u = x² + 1
Express the limits as definite integrals.where P is a partition of [2, 3] n 1 lim |P|−0 1 - Sk k=1 Δεκε
Evaluate the indefinite integral by using the given substitutions to reduce the integrals to standard form. [ (3x + 2)(3x² + 4x)4 dx, u = 3x² + 4x
Suppose that ƒ and h are integrable and thatUse the rules in Table 5.6 to finda.b.c.d.e.f. Г'я f(x) dx = -1, 1,50 7 f(x) dx = 5, f 7 h(x) dx = 4.
Suppose that ƒ and g are integrable and thatUse the rules in Table 5.6 to finda.b.c.d.e.f. 2 fro 1 f(x) dx = -4, 5 Sis f(x) dx = 6, 5 [₁²80 1 g(x) dx = 8.
Use the Substitution Formula in Theorem 7 to evaluate the integral.a.b. ¹P E/1 (I + zl) ¹ LA Jo
Use the Substitution Formula in Theorem 7 to evaluate the integral.a.b. THEOREM 7-Substitution in Definite Integrals If g' is continuous on the interval [a, b] and f is continuous on the range of g(x) = u, then rg(b) [ f(g(x)) g'(x) dx = - f(u) du. g(a) Proof Let F denote any antiderivative of f.
Use the Substitution Formula in Theorem 7 to evaluate the integral.a.b. THEOREM 7-Substitution in Definite Integrals If g' is continuous on the interval [a, b] and f is continuous on the range of g(x) = u, then rg(b) [ f(g(x)) g'(x) dx = - f(u) du. g(a) Proof Let F denote any antiderivative of f.
Express the limits as definite integrals.where P is a partition of [0, 1] n lim Σ V4 - ck? Δ.κ. \P-0k=1
Express the limits as definite integrals.where P is a partition of [-π/4, 0] η limΣ (sec c) Δ.xk, |P| →0 k=1
Evaluate the indefinite integral by using the given substitutions to reduce the integrals to standard form. [s sin 3x dx, u = 3x
Evaluate the indefinite integral by using the given substitutions to reduce the integrals to standard form. + √x)1/3 ^^+1) √ Vx -dx, u = 1 + √x
Use the Substitution Formula in Theorem 7 to evaluate the integral.a.b. THEOREM 7-Substitution in Definite Integrals If g' is continuous on the interval [a, b] and f is continuous on the range of g(x) = u, then rg(b) [ f(g(x)) g'(x) dx = - f(u) du. g(a) Proof Let F denote any antiderivative of f.
Suppose that ∫12 ƒ(x) dx = 5. Finda.b.c.d. .2 S f(u) du
Evaluate the indefinite integral by using the given substitutions to reduce the integrals to standard form. [x xsin (2x²) dx, u = 2x²
Use the Substitution Formula in Theorem 7 to evaluate the integral.a.b. 0 1 X³ √x+ + 9 =dx
Express the limits as definite integrals.where P is a partition of [0, π/4] n lim Σ(tan ck) Δ.xk, ||P| |-0 k=1
Use the Substitution Formula in Theorem 7 to evaluate the integral.a.b. 1 SIVA 0 V4 + 5t dt
Evaluate the indefinite integral by using the given substitutions to reduce the integrals to standard form. J sec 2t tan 2t dt, u = 2t
Use the Substitution Formula in Theorem 7 to evaluate the integral.a.b. THEOREM 7-Substitution in Definite Integrals If g' is continuous on the interval [a, b] and f is continuous on the range of g(x) = u, then rg(b) [ f(g(x)) g'(x) dx = - f(u) du. g(a) Proof Let F denote any antiderivative of f.
Use the Substitution Formula in Theorem 7 to evaluate the integral.a.b. 0 TT/6 (1 - cos 3t) sin 3t dt
Evaluate the indefinite integral by using the given substitutions to reduce the integrals to standard form. 9r² dr V1-p³ 13 u = 1 - p³
Evaluate the indefinite integral by using the given substitutions to reduce the integrals to standard form. 2 [(1 - cos / )* sin 12 dt. u = 1 COS 2
Suppose that ƒ is integrable and that ∫03 ƒ(z) dz = 3 and ∫04 ƒ(z) dz = 7. Finda.b. 4 S 3 f(z) dz
Suppose that h is integrable and that ∫ 1-1 h(r) dr = 0 and ∫3-1 h(r) dr = 6. Finda.b. 3 J 1 h(r) dr
Evaluate the indefinite integral by using the given substitutions to reduce the integrals to standard form. [₁2 12(y² + 4y² + 1)²(y³ + 2y) dy, u = y² + 4y² + 1
Use the Substitution Formula in Theorem 7 to evaluate the integral.a.b. THEOREM 7-Substitution in Definite Integrals If g' is continuous on the interval [a, b] and f is continuous on the range of g(x) = u, then rg(b) [ f(g(x)) g'(x) dx = - f(u) du. g(a) Proof Let F denote any antiderivative of f.
Evaluate the indefinite integral by using the given substitutions to reduce the integrals to standard form. [v. √xsin²(x³/2 - 1) dx, u = x³/2 - 1
Graph the integrands and use known area formulas to evaluate the integrals. 4 -2 X + 3 dx
Use the Substitution Formula in Theorem 7 to evaluate the integral. 0 Vt5 + 2t (5t4 + 2) dt
Evaluate the indefinite integral by using the given substitutions to reduce the integrals to standard form. [ 12 cos² (1) dx. u - - X
Use the Substitution Formula in Theorem 7 to evaluate the integral. TT/6 S 0 cos 3 20 sin 20 de
Use the Substitution Formula in Theorem 7 to evaluate the integral. 4 1,21 dy 2 2√y(1 + √y)²
Evaluate the indefinite integral by using the given substitutions to reduce the integrals to standard form.a. Using u = 5x + 8b. Using S dx V5x + 8
Evaluate the indefinite integral by using the given substitutions to reduce the integrals to standard form.a. Using u = cot 2θ b. Using u = csc 2θ Jese² csc² 20 cot 20 de
Graph the integrands and use known area formulas to evaluate the integrals. 3 -3 V9 - x² dx
Graph the integrands and use known area formulas to evaluate the integrals. 3/2 J 1/2 (-2x + 4) dx
Use the Substitution Formula in Theorem 7 to evaluate the integral. 3π/2 0 [th* cors (9) sec² (2) do 6 TT
Graph the integrands and use known area formulas to evaluate the integrals. -4 V16 - x² dx 2
Use the Substitution Formula in Theorem 7 to evaluate the integral. TT 5(5 4 cos t)¹/4 sin t dt
Graph the integrands and use known area formulas to evaluate the integrals. -2 xp |x|
Use the Substitution Formula in Theorem 7 to evaluate the integral. So (4y - y² + 4y³ + 1)-2/3 (12y² − 2y + 4) dy 0
Use the Substitution Formula in Theorem 7 to evaluate the integral. 0 π/4 (1 - sin 2t)³/2 cos 2t dt
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