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Difference Quotient Quiz · Q2

Same function as Q1 — but stop one step earlier. Reading the question's exact wording matters.

Question 2 (1 pt) Find and simplify the general difference quotient of $f(x) = 2x^2 + 1$.

A. $0$
B. $4xh + h^2$
C. $4x + 2h$
D. $2h$

1 · The intuition

Intuition

The phrase "general difference quotient" means the secant-slope formula before you take the limit. Q1 asked for the tangent slope (the limit), Q2 asks for the secant slope (the limit's input). Same first four steps of the 5-step dance; the limit never happens.

Two ways to know you've got the right answer:

The answer should still have an $h$ in it (a secant slope is a function of both $x$ and $h$). If your answer has no $h$, you took the limit.
The answer should be simpler than the raw difference quotient (the raw form is $\frac{4xh + 2h^2}{h}$; the simplified form is what's left after the one allowed $h$ cancellation).

2 · The formula

Formula

The general difference quotient (a.k.a. secant slope, a.k.a. average rate of change) is:

$$\frac{f(x+h) - f(x)}{h}$$

Three names, one object. The word "general" just means "valid for any $x$" (as opposed to a specific pair of points like $x = 1$ and $x = 3$). The word "quotient" is the giveaway: numerator divided by denominator.

3 · Worked solution

Worked solution

Given $f(x) = 2x^2 + 1$. Same expansion as Q1, but we stop after step 4.

Step 1 — Write the difference quotient.

$$\frac{f(x+h) - f(x)}{h}$$

Step 2 — Compute $f(x+h)$.

$$f(x+h) \;=\; 2(x+h)^2 + 1 \;=\; 2x^2 + 4xh + 2h^2 + 1$$

Step 3 — Subtract $f(x) = 2x^2 + 1$.

$$f(x+h) - f(x) \;=\; (2x^2 + 4xh + 2h^2 + 1) - (2x^2 + 1) \;=\; 4xh + 2h^2$$

Step 4 — Divide by $h$ and simplify.

$$\frac{4xh + 2h^2}{h} \;=\; \frac{h(4x + 2h)}{h} \;=\; 4x + 2h$$

No step 5. The question says "difference quotient" — that's the formula before the limit. We stop here.

Final answer:

$\dfrac{f(x+h) - f(x)}{h} = 4x + 2h$

4 · Why the other choices fail

Reading the distractors

A. $0$ — The constant $1$ from $f(x) = 2x^2 + 1$ has derivative $0$, but that's the limit of the difference quotient, not the difference quotient itself. Picking this means you took the limit and then read the question wrong.
B. $4xh + h^2$ — This is the numerator of the difference quotient before you simplify — and it's missing the leading 2! Correct numerator after the subtraction is $4xh + 2h^2$, not $4xh + h^2$. Easy to miss the 2 if you half-expand $(x+h)^2$ as $x^2 + 2xh + h^2$ and forget the outside 2.
C. $4x + 2h$ — This is the correct answer. ✓
D. $2h$ — This is the change in $f$ as $h \to 0$ from a linear function — for $f(x) = x$, the difference quotient is $\frac{(x+h) - x}{h} = \frac{h}{h} = 1$, and the limit is 1. For $2x^2 + 1$, the change is a lot more than $2h$.

5 · The "general difference quotient" vs "derivative" trap

Reading carefully

Three phrases that look similar but mean different things — sorted by how much work you've done:

Difference quotient	$\dfrac{f(x+h) - f(x)}{h}$	secant slope, still has $h$
Derivative	$\displaystyle\lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}$	tangent slope, $h$ is gone
"Simplify"	$\dfrac{4xh + 2h^2}{h} \to 4x + 2h$	the one allowed $h$ cancel

So: "difference quotient" = stop at step 4. "Derivative" or "tangent slope" or "limit" = go all the way to step 5. Watch the verb.

6 · Check your understanding

Practice

Find and simplify the general difference quotient of $g(x) = x^2 + 3x$.

Hint: this has two terms in $g(x)$, so when you compute $g(x+h)$ you'll have two expansions. Subtract carefully.

Show answer

Step 2: $g(x+h) = (x+h)^2 + 3(x+h) = x^2 + 2xh + h^2 + 3x + 3h$

Step 3: $g(x+h) - g(x) = (x^2 + 2xh + h^2 + 3x + 3h) - (x^2 + 3x) = 2xh + h^2 + 3h$

Step 4: $\dfrac{2xh + h^2 + 3h}{h} = \dfrac{h(2x + h + 3)}{h} = 2x + h + 3$

$2x + h + 3$

Sanity check: the difference quotient is a function of two variables ($x$ and $h$). Three terms, $h$ in two of them — that's what the secant slope should look like.

7 · What's next

Send Q3 whenever you have it. The 5-step pattern holds — just watch the wording (difference quotient vs. derivative vs. tangent slope) to know where to stop.

← Q1 (tangent slope) 📐 Calculus cheat sheet ← All Difference Quotient questions ↑ Top

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