Math Problem Statement
Find g(x) if g'(x) = (60x2 − 80x) · 74x3 − 8x2 and g(0) = 1. g(x) =
Solution
To find , given and , we need to integrate with respect to and then apply the initial condition .
Step 1: Simplify
Rewrite :
Expand by multiplying the terms:
This gives: Combine like terms:
Step 2: Integrate to find
Now, we integrate with respect to :
Integrating each term separately:
Thus,
Step 3: Apply the initial condition
Substitute and : This simplifies to .
Final Answer
Therefore,
Would you like more details on any part of the solution?
Here are five related questions to consider:
- How would the solution change if had a different value?
- How can you check the correctness of by differentiating it?
- What techniques can be used to simplify products of polynomials before integrating?
- How would the solution differ if there were additional terms in ?
- How does the degree of relate to the degree of ?
Tip: Always remember to apply initial conditions after integrating to determine the constant of integration accurately.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Differentiation
Integration
Polynomial Algebra
Formulas
g'(x) = (60x^2 − 80x) · (74x^3 − 8x^2)
g(x) = ∫(4440x^5 − 6400x^4 + 640x^3) dx
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 11-12
Related Recommendation
Find the Function g(x) Given Its Derivative g'(x) = 3x^2 - 5
Find Function f(x) Given its Derivative f'(x) = 2x^2 + 5x - 4 and Initial Condition f(0) = 2
Solve for f(x) Given f′(x) = 6x − 5 and f(4) = 0
Find the Function f(x) Given the Derivative f'(x) = 1 + 3√x and Initial Condition f(4) = 26
Find f(x) Given f'(x) = 4x^3 + 2x - 5x^9 and f(1) = 1 - Detailed Solution