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If f'(t) = 9t² + e' and f(0) = 4. Compute f(3).

If f'(t) = 9t² + e' and f(0) = 4. Compute f(3).

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 8:** If \( f'(t) = 9t^2 + e^t \) and \( f(0) = 4 \), compute \( f(3) \). **Explanation:** - \( f'(t) = 9t^2 + e^t \) is the derivative of the function \( f(t) \). - You are given an initial condition: \( f(0) = 4 \). - To find \( f(3) \), you need to integrate \( f'(t) \) to find \( f(t) \) and use the initial condition to solve for the constant of integration. Finally, substitute \( t = 3 \) into the solved function \( f(t) \) to find the value.
Transcribed Image Text:**Question 8:** If \( f'(t) = 9t^2 + e^t \) and \( f(0) = 4 \), compute \( f(3) \). **Explanation:** - \( f'(t) = 9t^2 + e^t \) is the derivative of the function \( f(t) \). - You are given an initial condition: \( f(0) = 4 \). - To find \( f(3) \), you need to integrate \( f'(t) \) to find \( f(t) \) and use the initial condition to solve for the constant of integration. Finally, substitute \( t = 3 \) into the solved function \( f(t) \) to find the value.
**Problem 7.** Find \( k \) so that \[ \int x^2 e^{x^3} \, dx = k e^{x^3} + C. \] **Explanation:** This problem is asking for the constant \( k \) that will satisfy the equation, where \( C \) is the constant of integration. The equation involves an integral of the product of a polynomial function \( x^2 \) and an exponential function \( e^{x^3} \). This often requires techniques such as substitution to evaluate the integral.
Transcribed Image Text:**Problem 7.** Find \( k \) so that \[ \int x^2 e^{x^3} \, dx = k e^{x^3} + C. \] **Explanation:** This problem is asking for the constant \( k \) that will satisfy the equation, where \( C \) is the constant of integration. The equation involves an integral of the product of a polynomial function \( x^2 \) and an exponential function \( e^{x^3} \). This often requires techniques such as substitution to evaluate the integral.
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