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

If f\left(x\right)=x^2-4x, find

a) f\left(-5\right)

b) f\left(y^2+1\right)

c) f\left(x+\Delta x\right)

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)

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