Find​ C(x) if Upper C prime left parenthesis x right parenthesis equals 5 x squared minus 7 x plus 4 and Upper C left parenthesis 6 right parenthesis equals 260.

To find the function \( C(x) \), we start with the information provided:

1. The derivative of \( C(x) \) is:
   \[
   C'(x) = 5x^2 - 7x + 4
   \]

2. We are also given that \( C(6) = 260 \).

---

### Step 1: Integrate \( C'(x) \) to find \( C(x) \)
The antiderivative of \( 5x^2 - 7x + 4 \) is calculated as follows:
\[
C(x) = \int (5x^2 - 7x + 4) \, dx
\]

- For \( 5x^2 \):
  \[
  \int 5x^2 \, dx = \frac{5x^3}{3}
  \]

- For \( -7x \):
  \[
  \int -7x \, dx = -\frac{7x^2}{2}
  \]

- For \( 4 \):
  \[
  \int 4 \, dx = 4x
  \]

Thus, the general form of \( C(x) \) is:
\[
C(x) = \frac{5x^3}{3} - \frac{7x^2}{2} + 4x + C_0
\]
where \( C_0 \) is the constant of integration.

---

### Step 2: Use the initial condition \( C(6) = 260 \)
Substitute \( x = 6 \) and \( C(x) = 260 \) into the equation:
\[
260 = \frac{5(6)^3}{3} - \frac{7(6)^2}{2} + 4(6) + C_0
\]

Simplify each term:
- \( \frac{5(6)^3}{3} = \frac{5(216)}{3} = 360 \)
- \( \frac{7(6)^2}{2} = \frac{7(36)}{2} = 126 \)
- \( 4(6) = 24 \)

Thus:
\[
260 = 360 - 126 + 24 + C_0
\]

Combine the constants:
\[
260 = 258 + C_0
\]

Solve for \( C_0 \):
\[
C_0 = 2
\]

---

### Final Answer:
The function \( C(x) \) is:
\[
C(x) = \frac{5x^3}{3} - \frac{7x^2}{2} + 4x + 2
\]

---

Let me know if you'd like further details! Here are some related questions for practice:

1. How do you calculate definite integrals from a derivative?
2. What are the steps to solve an initial value problem?
3. How do you verify that a function satisfies a given differential equation?
4. What role does the constant of integration play in antiderivatives?
5. How can you find derivatives of cubic functions?

**Tip:** Always check your solution by differentiating the final expression to ensure it matches the original derivative.

Find C(x) if C'(x) = 5x^2 - 7x + 4 and C(6) = 260.

Learn how to determine C(x) from its derivative C'(x) = 5x² - 7x + 4, given C(6) = 260. This problem involves integration, applying the Fundamental Theorem of Calculus, and solving for constants of integration. Suitable for high school or college calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation