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75. f(y) = 3y³ +5 Y ; F(1) = 3, y>0

75. f(y) = 3y³ +5 Y ; F(1) = 3, y>0

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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(Question 75 and 79) Show work, thank you! Answers bellow 75 = y^3+5 In y+2, y>0 79 = g(x) = 7/8 x^8 -x^2/2 + 13/8
### Particular Antiderivatives For the following functions \( f \), find the antiderivative \( F \) that satisfies the given condition. 69. \( f(x) = x^5 - 2x^2 + 1 \); \( F(0) = 1 \) 70. \( f(x) = 4\sqrt{x} + 6 \); \( F(1) = 8 \) 71. \( f(x) = 8x^3 + \sin x \); \( F(0) = 2 \) 72. \( f(t) = \sec^2 t \); \( F\left(\frac{\pi}{4}\right) = 1 \), \(-\frac{\pi}{2} < t < \frac{\pi}{2}\) 73. \( f(v) = \sec v \tan v \); \( F(0) = 2 \), \(-\frac{\pi}{2} < v < \frac{\pi}{2}\) 74. \( f(u) = 2e^u + 3 \); \( F(0) = 8 \) 75. \( f(y) = \frac{3y^3 + 5}{y} \); \( F(1) = 3 \), \( y > 0 \) 76. \( f(\theta) = 2 \sin \theta - 4 \cos \theta \); \( F\left(\frac{\pi}{4}\right) = 2 \) Note: No graphs or diagrams are included in this text. The task involves finding a function \( F \) whose derivative is \( f \) and that satisfies the initial condition provided for each problem.
Transcribed Image Text:### Particular Antiderivatives For the following functions \( f \), find the antiderivative \( F \) that satisfies the given condition. 69. \( f(x) = x^5 - 2x^2 + 1 \); \( F(0) = 1 \) 70. \( f(x) = 4\sqrt{x} + 6 \); \( F(1) = 8 \) 71. \( f(x) = 8x^3 + \sin x \); \( F(0) = 2 \) 72. \( f(t) = \sec^2 t \); \( F\left(\frac{\pi}{4}\right) = 1 \), \(-\frac{\pi}{2} < t < \frac{\pi}{2}\) 73. \( f(v) = \sec v \tan v \); \( F(0) = 2 \), \(-\frac{\pi}{2} < v < \frac{\pi}{2}\) 74. \( f(u) = 2e^u + 3 \); \( F(0) = 8 \) 75. \( f(y) = \frac{3y^3 + 5}{y} \); \( F(1) = 3 \), \( y > 0 \) 76. \( f(\theta) = 2 \sin \theta - 4 \cos \theta \); \( F\left(\frac{\pi}{4}\right) = 2 \) Note: No graphs or diagrams are included in this text. The task involves finding a function \( F \) whose derivative is \( f \) and that satisfies the initial condition provided for each problem.
--- **77-86. Solving Initial Value Problems** Find the solution of the following initial value problems: **77.** \( f'(x) = 2x - 3; \quad f(0) = 4 \) **78.** \( g'(x) = 7x^6 - 4x^3 + 12; \quad g(1) = 24 \) **79.** \( g'(x) = 7x \left( x^6 - \frac{1}{7} \right); \quad g(1) = 2 \) **80.** \( h'(t) = 1 + 6 \sin t; \quad h\left( \frac{\pi}{3} \right) = -3 \) **81.** \( f'(u) = 4(\cos u - \sin u); \quad f\left( \frac{\pi}{2} \right) = 0 \) **82.** \( p'(t) = 10e^t + 70; \quad p(0) = 100 \) **83.** \( y'(t) = \frac{3}{t} + 6; \quad y(1) = 8, \quad t > 0 \) **84.** \( u'(x) = \frac{xe^{2x} + 4e^x}{xe^x}; \quad u(1) = 0, \quad x > 0 \) ---
Transcribed Image Text:--- **77-86. Solving Initial Value Problems** Find the solution of the following initial value problems: **77.** \( f'(x) = 2x - 3; \quad f(0) = 4 \) **78.** \( g'(x) = 7x^6 - 4x^3 + 12; \quad g(1) = 24 \) **79.** \( g'(x) = 7x \left( x^6 - \frac{1}{7} \right); \quad g(1) = 2 \) **80.** \( h'(t) = 1 + 6 \sin t; \quad h\left( \frac{\pi}{3} \right) = -3 \) **81.** \( f'(u) = 4(\cos u - \sin u); \quad f\left( \frac{\pi}{2} \right) = 0 \) **82.** \( p'(t) = 10e^t + 70; \quad p(0) = 100 \) **83.** \( y'(t) = \frac{3}{t} + 6; \quad y(1) = 8, \quad t > 0 \) **84.** \( u'(x) = \frac{xe^{2x} + 4e^x}{xe^x}; \quad u(1) = 0, \quad x > 0 \) ---
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