Test 1: Calculus
Instruction: Create a pdf file with all the solutions, upload file on Google Drive and share the link in the form shared at the bottom of the page
1) If $f(x)$ is derivable in the interval $[a-h, a+h]$, prove that
(i) \(f(a+h)-f(a-h)=h\left[f^{\prime}\left(a+\theta_{1} h\right)+f^{\prime}\left(a-\theta_{1} h\right)\right]\), where \(0<\theta_{1}<1\)
(ii) \(f(a+h)-2 f(a)+f(a-h)=h\left[f^{\prime}\left(a+\theta_{2} h\right)-f^{\prime}\left(a-\theta_{2} h\right)\right]\), where \(0<\theta_{2}<1\)