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 $x^2$.

Would you like to start with a specific question from the assignment or go through them one by one? Feel free to request details for any part!