Welcome to Westonci.ca, your ultimate destination for finding answers to a wide range of questions from experts. Get detailed and precise answers to your questions from a dedicated community of experts on our Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
To find the solution to the initial value problem, we need to follow these steps:
1. Integrate the given derivative [tex]\( f'(x) \)[/tex] to find the general form of the function [tex]\( f(x) \)[/tex].
2. Determine the constant of integration using the given initial condition.
Let's start by integrating the derivative:
Given:
[tex]\[ f'(x) = 8x - 5 \][/tex]
Step 1: Integrate the given derivative
To find [tex]\( f(x) \)[/tex], we need to integrate [tex]\( f'(x) \)[/tex]:
[tex]\[ f(x) = \int (8x - 5) \, dx \][/tex]
Performing the integration gives us:
[tex]\[ f(x) = \int 8x \, dx - \int 5 \, dx \][/tex]
Now calculate each integral separately:
[tex]\[ \int 8x \, dx = 8 \cdot \frac{x^2}{2} = 4x^2 \][/tex]
[tex]\[ \int 5 \, dx = 5x \][/tex]
Putting these results together:
[tex]\[ f(x) = 4x^2 - 5x + C \][/tex]
where [tex]\( C \)[/tex] is the constant of integration.
Step 2: Determine the constant [tex]\( C \)[/tex] using the initial condition
We are given the initial condition:
[tex]\[ f(0) = 9 \][/tex]
Substitute [tex]\( x = 0 \)[/tex] and [tex]\( f(0) = 9 \)[/tex] into the equation:
[tex]\[ 9 = 4(0)^2 - 5(0) + C \][/tex]
[tex]\[ 9 = C \][/tex]
So, the constant [tex]\( C \)[/tex] is 9.
Final Solution:
Substituting [tex]\( C \)[/tex] back into the general solution gives us the specific solution:
[tex]\[ f(x) = 4x^2 - 5x + 9 \][/tex]
Hence, the solution to the initial value problem is:
[tex]\[ f(x) = 4x^2 - 5x + 9 \][/tex]
1. Integrate the given derivative [tex]\( f'(x) \)[/tex] to find the general form of the function [tex]\( f(x) \)[/tex].
2. Determine the constant of integration using the given initial condition.
Let's start by integrating the derivative:
Given:
[tex]\[ f'(x) = 8x - 5 \][/tex]
Step 1: Integrate the given derivative
To find [tex]\( f(x) \)[/tex], we need to integrate [tex]\( f'(x) \)[/tex]:
[tex]\[ f(x) = \int (8x - 5) \, dx \][/tex]
Performing the integration gives us:
[tex]\[ f(x) = \int 8x \, dx - \int 5 \, dx \][/tex]
Now calculate each integral separately:
[tex]\[ \int 8x \, dx = 8 \cdot \frac{x^2}{2} = 4x^2 \][/tex]
[tex]\[ \int 5 \, dx = 5x \][/tex]
Putting these results together:
[tex]\[ f(x) = 4x^2 - 5x + C \][/tex]
where [tex]\( C \)[/tex] is the constant of integration.
Step 2: Determine the constant [tex]\( C \)[/tex] using the initial condition
We are given the initial condition:
[tex]\[ f(0) = 9 \][/tex]
Substitute [tex]\( x = 0 \)[/tex] and [tex]\( f(0) = 9 \)[/tex] into the equation:
[tex]\[ 9 = 4(0)^2 - 5(0) + C \][/tex]
[tex]\[ 9 = C \][/tex]
So, the constant [tex]\( C \)[/tex] is 9.
Final Solution:
Substituting [tex]\( C \)[/tex] back into the general solution gives us the specific solution:
[tex]\[ f(x) = 4x^2 - 5x + 9 \][/tex]
Hence, the solution to the initial value problem is:
[tex]\[ f(x) = 4x^2 - 5x + 9 \][/tex]
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.