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문제 7.2.43
Evaluate the integrals in Exercises 39–56.
43. ∫(from 0 to π)(sin t)/(2 - cos t) dt
문제 7.3.29
In Exercises 27–32, find dy/dx.
e^(2x)=sin(x+3y)
문제 7.1.33
Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.
f(x) = x² − 2x, x ≤ 1
문제 7.4.47
47. Carbon-14 The oldest known frozen human mummy, discovered in the Schnalstal glacier of the Italian Alps in 1991 and called Otzi, was found wearing straw shoes and a leather coat with goat fur, and holding a copper ax and stone dagger. It was estimated that Otzi died 5000 years before he was discovered in the melting glacier. How much of the original carbon-14 remained in Otzi at the time of his discovery?
문제 7.3.63
"In Exercises 59–86, find the derivative of y with respect to the given independent variable.
63. y = x^π"
문제 7.4.21
Solve the differential equation in Exercises 9–22.
21. (1/x)(dy/dx) = ye^(x²) + 2√y e^(x²)
문제 7.6.43
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
43. y=√(arcsin x)
문제 7.1.55
Show that increasing functions and decreasing functions are one-to-one. That is, show that for any x₁ and x₂ in I, x₂ ≠ x₁ implies f(x₂) ≠ f(x₁).
문제 7.5.39
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
39. lim (x → ∞) (ln 2x - ln(x + 1))
문제 7.3.79
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
79. y = θ sin(log₇ θ)
문제 7.7.23
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
23. y = (x²+1)sech(ln x)
(Hint: Before differentiating, express in terms of exponentials and simplify.)
문제 7.3.67
"In Exercises 59–86, find the derivative of y with respect to the given independent variable.
67. y = 7^(sec θ) ln 7"
문제 7.5.17
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
17. lim (θ → π/2) (2θ - π) / cos(2π - θ)
문제 7.2.23
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
23. y = ln(x)/(1+ln(x))
문제 7.2.27
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
27. y = θ(sin(lnθ) + cos(lnθ))
문제 7.5.53
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
53. lim (x → 1⁺) x^(1/(1 - x))
문제 7.2.14
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
14. y = ln(2θ+2)
문제 7.7.73
Evaluate the integrals in Exercises 31–78.
73. ∫dx/√(-2x-x²)
문제 7.2.59
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
59. y = √(t/(t+1))
문제 7.1.51
Let f(x) = x³ − 3x² − 1, x ≥ 2. Find the value of df⁻¹/dx at the point x = −1 = f(3).
문제 7.6.73
Evaluate the integrals in Exercises 53–76.
73. ∫(from 0 to ln√3) e^x dx/(1+e^(2x))
문제 7.3.137
137. Find a curve through the origin in the xy-plane whose length from x = 0 to x = 1 is L = ∫ from 0 to 1 of sqrt(1 + (1/4)e^x) dx.
문제 7.3.126
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
126. eʸ = y^(ln x)
문제 7.3.13
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = (x^2 - 2x + 2)e^(x)
문제 7.4.44
44. Silver cooling in air The temperature of an ingot of silver is 60°C above room temperature right now. Twenty minutes ago, it was 70°C above room temperature. How far above room temperature will the silver be
b. 2 hours from now?
문제 7.4.7
In Exercises 5–8, show that each function is a solution of the given initial value problem.
7. Differential Equation: xy' + y = -sin(x), x>0
Initial condition: y(π/2) = 0
Solution candidate: y = cos(x)/x
문제 7.3.23
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(cost+lnt)
문제 7.6.59
Evaluate the integrals in Exercises 53–76.
59. ∫(from 0 to 1)4ds/√(4-s²)
문제 7.5.55
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
55. lim (x → ∞) (ln x)^(1/x)
문제 7.3.5
5. e^(2t)-3e^t = 0