You are given $f(x)$, where $f(x)$ is a function such as $3x^4-6x^2-4x+2$. You are asked to find the length of the curve between two points $(a,f(a)) \text{ and } (b,f(b))$. How would I do this for:
- Polynomials $3x^2, 4x^3-5x^2-x+2$
- Exponentials $e^x, 5^x$
- Rationals $\dfrac{x+1}{x},\dfrac{3x^2-5x+2}{x^2+2x+2}$
- Logarithmic $\ln x, \log x^3$
- Trigonometric $4\sin x, 5\cos x$
Since you have tagged calculus:
The length of a curve for a differentiable function from $a$ to $b$:
$$ L=\int_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^2} dx $$