Math Problem Statement

Find g(x) if g'(x) = (60x2 − 80x) · 74x3 − 8x2 and g(0) = 1. g(x) =

Solution

To find g(x)g(x), given g(x)=(60x280x)(74x38x2)g'(x) = (60x^2 - 80x) \cdot (74x^3 - 8x^2) and g(0)=1g(0) = 1, we need to integrate g(x)g'(x) with respect to xx and then apply the initial condition g(0)=1g(0) = 1.

Step 1: Simplify g(x)g'(x)

Rewrite g(x)g'(x): g(x)=(60x280x)(74x38x2)g'(x) = (60x^2 - 80x)(74x^3 - 8x^2)

Expand g(x)g'(x) by multiplying the terms: g(x)=(60x274x3)(60x28x2)(80x74x3)+(80x8x2)g'(x) = (60x^2 \cdot 74x^3) - (60x^2 \cdot 8x^2) - (80x \cdot 74x^3) + (80x \cdot 8x^2)

This gives: g(x)=4440x5480x45920x4+640x3g'(x) = 4440x^5 - 480x^4 - 5920x^4 + 640x^3 Combine like terms: g(x)=4440x56400x4+640x3g'(x) = 4440x^5 - 6400x^4 + 640x^3

Step 2: Integrate g(x)g'(x) to find g(x)g(x)

Now, we integrate g(x)g'(x) with respect to xx: g(x)=(4440x56400x4+640x3)dxg(x) = \int (4440x^5 - 6400x^4 + 640x^3) \, dx

Integrating each term separately:

  1. 4440x5dx=44406x6=740x6\int 4440x^5 \, dx = \frac{4440}{6}x^6 = 740x^6
  2. 6400x4dx=64005x5=1280x5\int -6400x^4 \, dx = -\frac{6400}{5}x^5 = -1280x^5
  3. 640x3dx=6404x4=160x4\int 640x^3 \, dx = \frac{640}{4}x^4 = 160x^4

Thus, g(x)=740x61280x5+160x4+Cg(x) = 740x^6 - 1280x^5 + 160x^4 + C

Step 3: Apply the initial condition g(0)=1g(0) = 1

Substitute x=0x = 0 and g(0)=1g(0) = 1: g(0)=74006128005+16004+C=1g(0) = 740 \cdot 0^6 - 1280 \cdot 0^5 + 160 \cdot 0^4 + C = 1 This simplifies to C=1C = 1.

Final Answer

Therefore, g(x)=740x61280x5+160x4+1g(x) = 740x^6 - 1280x^5 + 160x^4 + 1

Would you like more details on any part of the solution?

Here are five related questions to consider:

  1. How would the solution change if g(0)g(0) had a different value?
  2. How can you check the correctness of g(x)g(x) by differentiating it?
  3. What techniques can be used to simplify products of polynomials before integrating?
  4. How would the solution differ if there were additional terms in g(x)g'(x)?
  5. How does the degree of g(x)g(x) relate to the degree of g(x)g'(x)?

Tip: Always remember to apply initial conditions after integrating to determine the constant of integration accurately.

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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