Math Problem Statement

h(t) =

t3 t6 + 1 2

Solution

It seems like you are trying to express a function h(t)h(t), but the formatting of the equation isn't clear. From what I see, it looks like you might want to express a fraction involving powers of tt. Could you confirm if this is what you're trying to write?

Is the function something like this?

h(t)=t3t6+1h(t) = \frac{t^3}{t^6 + 1}

If this is correct, I can help you analyze and solve the equation, or provide further details like finding limits, derivatives, or behavior as tt \to \infty. Let me know if you need any specific assistance with this equation.

Would you like me to proceed with solving or analyzing this?

5 Related Questions:

  1. What is the derivative of h(t)=t3t6+1h(t) = \frac{t^3}{t^6 + 1}?
  2. What happens to h(t)h(t) as tt \to \infty?
  3. How can we find the critical points of h(t)h(t)?
  4. What is the behavior of h(t)h(t) for large and small values of tt?
  5. How would you integrate h(t)h(t)?

Tip:

To simplify the analysis of rational functions, look at the degrees of the polynomials in the numerator and denominator to quickly understand behavior as tt becomes large or small.

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Math Problem Analysis

Mathematical Concepts

Rational Functions
Limits
Derivatives

Formulas

h(t) = \frac{t^3}{t^6 + 1}
Derivative of a rational function: \frac{d}{dt} \left(\frac{f(t)}{g(t)}\right) = \frac{f'(t)g(t) - f(t)g'(t)}{g(t)^2}
Limits: \lim_{t \to \infty} h(t)

Theorems

Limit Theorem
Quotient Rule for Differentiation

Suitable Grade Level

Grades 11-12 or First-year Calculus