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## Calculus Word Problem Solver

Let us see with the help of examples how to solve problems:

### Solved Examples

**Question 1:**A one-mile recetrack has two semicircular ends connected by straight lines. Express the area enclosed by the track as a function of its semicircular radius. Determine its domain.

**Solution:**

**Step 1:**

The enclosed area consist of a rectangle whose dimensions are 'x' and 2r and two semicircles of radius r, then its combine area is $\pi r^2$.

=> A = Area or rectangular part + Area of 2 semicircles

= 2r x + $\pi r^2$

=> A = 2r x + $\pi r^2$ ..............................(1)

**Step 2:**

The perimeter of recetrack is the length of the two straight sides added to the lengths of the 2 semicircular arcs.

=> 2x + 2$\pi$ r = 1

=> 2x = 1 - 2$\pi$ r

=> x = $\frac{1 - 2\pi r}{2}$

**Step 3:**

Put the value of 'x' in equation (1).

=> A(r) = 2r($\frac{1 - 2\pi r}{2}$) + $\pi r^2$

= r - 2$\pi r^2$ + $\pi r^2$

= r - $\pi r^2$

since 'r' cant be negative, r$\geqslant $ 0.

**Step 4:**

The perimeter of the track is fixed so the maximum value of radius (r) occurs when x = 0.

=> 2x + 2$\pi$r = 1

=> 2$\pi$r = 1, ( because, assume x = 0)

=> r = $\frac{1}{2\pi}$

**Step 5:**

The area function and its domain are

A(r) = r - $\pi r^2$

where

0$\leq$ r $\leq$ $\frac{1}{2\pi}$.

**Question 2:**Does f(x) = $x^{\frac{2}{3}}$ have a tangent line at x = 0.

**Solution:**

**Step 1:**

Given f(x) = $x^{\frac{2}{3}}$

According to definition of derivative:

f '(x) = $\lim_{h\to0}$$\frac{f(x + h) - f(x)}{h}$

**Step 2:**

Check x = 0 is the tangent line to the curve or not.

=> $\lim_{x\to0}$$\frac{f(x) - f(0)}{x - 0}$

= $\lim_{x\to0}$ $\frac{x^\frac{2}{3} - 0}{x - 0}$

= $\lim_{x\to0}$ $\frac{x^\frac{2}{3}}{x}$

= $\lim_{x\to0}$$\frac{1}{x^\frac{1}{3}}$

But the limit does not exist.

LHL = $\lim_{x\to0^+} $$\frac{1}{x^\frac{1}{3}}$ = + $\infty$

RHL = $\lim_{x\to0^-}$ $\frac{1}{x^\frac{1}{3}}$ = - $\infty$

=> Curve have a tangent line at x = 0.