Q.Case Study - 3 : A volleyball player serves the ball which takes a parabolic path given by the equation $h(t) = -\frac{7}{2}t^2 + \frac{13}{2}t + 1$, where $h(t)$ is the height of ball at any time $t$ (in seconds), $(t \geq 0)$. Based on the above information, answer the following questions :

Concept: Continuity of polynomial functions and optimization using calculus.

(i) Continuity of $h(t)$

A polynomial function is continuous everywhere on $\mathbb{R}$ because it is defined by a finite sum of power functions, each of which is continuous. The function $h(t) = -\frac{7}{2}t^2 + \frac{13}{2}t + 1$ is a quadratic polynomial, so it is continuous for all $t \in \mathbb{R}$, and in particular for $t \geq 0$. …

The trajectory of the volleyball is modeled by a quadratic function. Before we find when the ball reaches its peak, we need to understand what continuity means and why polynomials behave so nicely.

(i) Continuity of $h(t)$

A function is continuous at a point if it has no breaks, jumps, or holes there. More formally, $f$ is continuous at $c$ if $\lim_{x \to c} f(x) = f(c)$.

The function $h(t) = -\frac{7}{2}t^2 + \frac{13}{2}t + 1$ is a polynomial. Polynomials are built from powers of $t$ and constants using only addition, subtraction, and multiplication—operations that preserve continuity. Since each term $t^n$ is continuous for all real $t$, and sums and scalar multiples of continuous functions remain continuous, every polynomial is continuous on its entire domain.

Yes, $h(t)$ is a continuous function for all $t \in \mathbb{R}$, and in particular for $t \geq 0$. There are no breaks or jumps in the ball's trajectory.

(ii) Time at which height is maximum

The parabola $h(t) = -\frac{7}{2}t^2 + \frac{13}{2}t + 1$ opens downward (since the coefficient of $t^2$ is negative), so it has a maximum at its vertex.

For a quadratic $at^2 + bt + c$, the vertex occurs at $t = -\frac{b}{2a}$. This comes from completing the square or from calculus: the derivative $h'(t) = 2at + b$ vanishes when $t = -\frac{b}{2a}$. …