Evaluate the difference quotient and simplify the result

Question

h(x) = x² + x + 4

h(x + Δx) − h(x) / Δx

Δx ≠ 0

Answer 1

The difference quotient is just like a formula. But I will LET h = Δx, otherwise the x's and the Δx's get confusing.

So, with my notation, the diffence quotient becomes: [ f(x+h)-f(x) ]/h

Step 1: Determine f(x+h): For your function f(x) = x²+x+4, so f(x+h) = (x+h)² + (x+h) + 4

multiply this out and simplify and get x² + 2xh + h² + x + h + 4

Step 2: Determine f(x+h) - f(x) Find the equation for the numerator -- already have f(x+h), now subtract the original function f(x) from the newly found f(x+h)

x² + 2xh + h² + x + h + 4 – (x² + x + 4)

x² + 2xh + h² + x + h + 4 – x² – x – 4 notice have x² – x² so those terms goes away, also have x – x, so those terms go away, also have 4 - 4, so those constants go away. All that is left when this is simplified is

2xh + h² + h this is the numerator : f(x+h) - f(x)

Step 3: Divide the numerator by h.

Final answer 2x + h + 1 or, using Δx's, this answer is:

2x + Δx + 1

Answer 2

Nick:x²+2*x*Δx + Δx² + x + Δx +4

h(x+Δx) = (x+Δx)² + (x+Δx) +4 = x²+2*x*Δx + Δx² + x + Δx +4

h(x) = x²+x+4

so h(x+Δx)-h(x) = x²+2*x*Δx + Δx² + x + Δx +4-x²-x-4 = 2*x*Δx + Δx² + Δx = Δx*(2*x+Δx+1)

This should give you enough to solve the problem. Let me know if you need further help!