Question

Hal makes an error when determining the difference quotient for the function

f(x)=x²+11x+2.

Hal's work:
A) (x+h)²+11(x+h)+2)-(x²+11x+2)/h
B) x²+h²+11x+11h+2-(x²+11x+2)/h
C) h²⁺₋11h/h= h+11
Which statement best describes the error?

Answer 1

Hal made an error by not properly expanding the binomial (x + h)² when calculating the difference quotient for the function f(x) = x² + 11x + 2. The correct expansion of the binomial should include the cross-term 2xh, which was omitted in Hal's work. Consequently, the simplified form of the difference quotient should contain the term 2x + 11, not just h + 11.

Step-by-step explanation:

The student, Hal, has made an error in calculating the difference quotient for the function f(x) = x² + 11x + 2. The correct approach to finding the difference quotient, which is f(x + h) - f(x) / h, starts with substituting x + h into the original function:

A) (x + h)² + 11(x + h) + 2 - (x² + 11x + 2) / h

This correctly represents the difference of the function values at x + h and x. The error occurred in step B where the squares were not properly expanded:

B) x² + 2xh + h² + 11x + 11h + 2 - (x² + 11x + 2) / h

Hal incorrectly expanded (x + h)² as x² + h² instead of x² + 2xh + h². The correct expansion includes the cross-terms 2xh. Hal's error is in not including these terms, which affects the simplification in step C. After canceling common terms and simplifying, the correct difference quotient should include 2x + 11 plus terms that cancel out when divided by h.