Let f(x) = −x² + 7x + 2. When (f(x + h) − f(x))/h is evaluated and simplified it has the form Ax + Bh + C. Find A, B and C.

Step-by-step solution

1. 1

   Substitute $x+h$ everywhere $x$ appears.

   $f(x+h)=-(x+h)^{2}+7(x+h)+2.$

   Expanding $(x+h)^{2}=x^{2}+2xh+h^{2}$ and distributing the leading minus sign carefully:

   $f(x+h)=-x^{2}-2xh-h^{2}+7x+7h+2.$

   That minus sign in front of the square is the single most common place to slip.

2. 2

   Subtract $f(x)$ and watch the $h$-free terms disappear.

   $f(x+h)-f(x)=\bigl(-x^{2}-2xh-h^{2}+7x+7h+2\bigr)-\bigl(-x^{2}+7x+2\bigr).$

   The $-x^{2}$, $7x$ and $2$ terms cancel in pairs, leaving

   $-2xh-h^{2}+7h.$

   Every surviving term contains an $h$ — that has to happen, otherwise dividing by $h$ would blow up as $h\to0$.

3. 3

   Factor out $h$ and cancel.

   $\frac{h(-2x-h+7)}{h}=-2x-h+7,\qquad h\ne 0.$

4. 4

   Match the required form. Writing the result as $(-2)x+(-1)h+7$ and comparing with $Ax+Bh+C$:

   $A=-2,\qquad B=-1,\qquad C=7.$

5. 5

   Check against the derivative and a numeric case. Letting $h\to0$ gives $-2x+7$, which is exactly $f'(x)$ for $f(x)=-x^{2}+7x+2$ — the difference quotient must collapse to the derivative. Numerically at $x=3$, $h=0.1$: $f(3.1)=-9.61+21.7+2=14.09$ and $f(3)=-9+21+2=14$, so the quotient is $0.09/0.1=0.9$, while $-2(3)-0.1+7=0.9$ ✓.