# Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1 Problem 1

### d) $f\left(x+1\right)-f\left(x-1\right)$

SOLUTION:

Part a

$f\left(-5\right)=\left(-5\right)^2-4\left(-5\right)$

$f\left(-5\right)=25+20$

$f\left(-5\right)=45$

Part b

$f\left(y^2+1\right)=\left(y^2+1\right)^2-4\left(y^2+1\right)$

$f\left(y^2+1\right)=y^4+2y^2+1-4y^2-4$

$f\left(y^2+1\right)=y^4-2y^2-3$

Part c

$f\left(x+\Delta x\right)=\left(x+\Delta x\right)^2-4\left(x+\Delta x\right)$

$f\left(x+\Delta x\right)=\left(x+\Delta x\right)\left[\left(x+\Delta x\right)-4\right]$

$f\left(x+\Delta x\right)=\left(x+\Delta x\right)\left(x+\Delta x-4\right)$

Part d

$f\left(x+1\right)-f\left(x-1\right)=\left[\left(x+1\right)^2-4\left(x+1\right)\right]-\left[\left(x-1\right)^2-4\left(x-1\right)\right]$

$f\left(x+1\right)-f\left(x-1\right)=\left[x^2+2x+1-4x-4\right]-\left[x^2-2x+1-4x+4\right]$

$f\left(x+1\right)-f\left(x-1\right)=x^2-x^2+2x-4x+2x+4x+1-4-1-4$

$f\left(x+1\right)-f\left(x-1\right)=4x-8$

$f\left(x+1\right)-f\left(x-1\right)=4\left(x-2\right)$