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Problem 7.1.26
Evaluate the following derivatives.
d/dx (x^{x¹⁰})
Problem 7.1.11
7–28. Derivatives Evaluate the following derivatives.
d/dx ((ln 2x)⁻⁵)
Problem 7.1.12
7–28. Derivatives Evaluate the following derivatives.
d/dx (ln³(3x² + 2))
Problem 7.1.13
7–28. Derivatives Evaluate the following derivatives.
d/dx ((2x)⁴ˣ)
Problem 7.1.14
7–28. Derivatives Evaluate the following derivatives.
d/dx (x^{π})
Problem 7.1.16
7–28. Derivatives Evaluate the following derivatives.
d/dt ((sin t)^{√t})
Problem 7.1.19
7–28. Derivatives Evaluate the following derivatives.
d/dy (y^{sin y})
Problem 7.1.20
7–28. Derivatives Evaluate the following derivatives.
d/dt (t^{1/t})
Problem 7.1.21
7–28. Derivatives Evaluate the following derivatives.
d/dx (e^{-10x²})
Problem 7.1.3
Evaluate ∫ 4ˣ dx.
Problem 7.1.29
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫₀³ (2x - 1) / (x + 1) dx
Problem 7.1.30
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫ (x²) / (4x³ + 7) dx
Problem 7.1.40
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫₋₂² (e^{z/2}) / (e^{z/2} + 1) dz
Problem 7.1.44
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫₀^{π/2} 4^{sin x} cos x dx
Problem 7.1.67
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume x > 0 and y > 0.
d. 2ˣ = 2² ˡⁿ ˣ
Problem 7.1.70
Behavior at the origin Using calculus and accurate sketches, explain how the graphs of f(x) = xᵖ ln x differ as x → 0⁺ for p = 1/2, 1, and 2.
Problem 7.1.71a
Zero net area Consider the function f(x) = (1 − x)/x
a. Are there numbers 0 < a < 1 such that ∫₁₋ₐ¹⁺ᵃ f(x) dx = 0?
Problem 7.1.72
Derivative of ln|x| Differentiate ln x, for x > 0, and differentiate ln(−x), for x < 0, to conclude that d/dx (ln|x|) = 1/x
Problem 7.1.73b
Properties of exp(x) Use the inverse relations between ln x and exp(x), and the properties of ln x, to prove the following properties:
b. exp(x − y) = exp(x) / exp(y)
Problem 7.1.73c
Properties of exp(x) Use the inverse relations between ln x and exp(x), and the properties of ln x, to prove the following properties:
c. (exp(x))ᵖ = exp(px), p rational
Problem 7.1.5
Express 3ˣ, x^{π}, and x^{sin x} using the base e.
Problem 7.1.45
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫₁² (1 + ln x) x^x dx
Problem 7.1.51
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫ 3^{-2x} dx
Problem 7.1.54
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫₀^{π} 2^{sin x} · cos x dx
Problem 7.1.59
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫₀ˡⁿ ² (e^{3x} − e^{−3x}) / (e^{3x} + e^{−3x}) dx
Problem 7.1.63
63–66. Calculator limits Use a calculator to make a table similar to Table 7.1 to approximate the following limits. Confirm your result with l’Hôpital’s Rule.
limₕ→₀ (1 + 2h)^{1/h}
Problem 7.1.64
63–66. Calculator limits Use a calculator to make a table similar to Table 7.1 to approximate the following limits. Confirm your result with l’Hôpital’s Rule.
limₕ→₀ (1 + 3h)^{2/h}
Problem 7.1.67b
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume x > 0 and y > 0.
b. ln 0 = 1
Problem 7.1.67c
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume x > 0 and y > 0.
c. ln (x + y) = ln x + ln y
Problem 7.1.68
Logarithm properties Use the integral definition of the natural logarithm to prove that ln(x/y) = ln x - ln y.