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I see you've uploaded an image of a calculus assignment. I'll assist with the questions on the assignment step by step. Here's a summary of the questions:

1. **Inequalities**: Solve two inequalities to find the range of values of \( x \).
2. **Greatest Integer Function**:  
   (a) Define functions based on the greatest integer less than or equal to \( x \) and determine the range.  
   (b) Evaluate a given limit involving \( f(t) \) and \( f'(1) \).  
   (c) Find a limit expression involving \( t \) raised to a power.
3. **Inverse of a function**: Find the inverse of \( \varphi(x) \).
4. **Continuity Proof**: Prove that a piecewise function \( f(x) \) is continuous at \( x = 0 \) using the \( \epsilon-\delta \) definition of continuity.
5. **Gradient Function**: Differentiate a given function involving trigonometric terms.
6. **Mean-Value Theorem**: Use the Mean-Value Theorem to show a logarithmic inequality.
7. **Differentiation**:  
   (a) Differentiate a square root function from first principles.  
   (b) Find the 7th derivative of a function using the Leibniz rule.
8. **Maclaurin Series**:  
   (a) Find the equation of a tangent to a curve.  
   (b) Derive the Maclaurin series for a function.  
   (c) Expand \( e^{\sin x} \) up to \( x^2 \).

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The original calculus assignment covers various problems including inequalities, greatest integer function, limits, continuity, differentiation, Mean-Value Theorem, and Maclaurin series.

Explore a comprehensive calculus assignment that includes problems on inequalities, the greatest integer function, limits, continuity proofs, differentiation, the Mean-Value Theorem, and Maclaurin series expansions. Suitable for university-level students looking to enhance their calculus knowledge.

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