Q.Find a point on the curve $y = (x - 3)^2$, where the tangent is parallel to the chord joining the points $(3, 0)$ and $(4, 1)$.

Concept: Mean Value Theorem — the tangent is parallel to the chord exactly where the derivative equals the chord's slope.

Step 1: Chord slope: $m = \dfrac{1-0}{4-3} = 1$.

Step 2: Tangent slope: $\dfrac{dy}{dx} = 2(x-3)$.

Step 3: Equate: $2(x-3)=1 \Rightarrow x = \dfrac72$. …

Setting Up

A tangent parallel to a chord means their slopes are equal. The chord's slope is the average rate of change of $y$ between the two given points; the tangent's slope at any $x$ is the derivative there. Finding where they match is exactly the geometric content of the Mean Value Theorem — for the smooth curve $y=(x-3)^2$ on $[3,4]$, MVT guarantees at least one such interior point.

Step 1 — Slope of the chord

The chord joins $(3,0)$ and $(4,1)$, both on the curve (check: $(3-3)^2=0$, $(4-3)^2=1$):

Step 2 — Slope of the tangent

Differentiating $y=(x-3)^2$:

$\frac{dy}{dx}=2(x-3).$ …

Method: Finding a Point Where the Tangent Is Parallel to a Given Chord

Use this method whenever a question gives two points on a curve (or a curve and an interval's endpoints) and asks you to find the point where the tangent line is parallel to the chord joining them. This is the geometric heart of the Mean Value Theorem, applied without necessarily stating the theorem by name.

Steps

Step 1: Confirm both given points genuinely lie on the curve

Substitute each point's $x$-coordinate into the curve's equation and check the resulting $y$-value matches. This matters because the whole method depends on the chord actually joining two points of the curve.

Step 2: Compute the slope of the chord

Step 3: Differentiate the curve's equation …

Common Mistakes

Mistake 1: Computing the chord slope with points in the wrong order

Why it's wrong: The chord slope is $\dfrac{y_2-y_1}{x_2-x_1}$; swapping the order for just the numerator or denominator (e.g. computing $\dfrac{0-1}{4-3}$) flips the sign and sends the whole solution wrong. Correct approach: keep $(3,0)$ and $(4,1)$ in the same order in both numerator and denominator: $\dfrac{1-0}{4-3}=1$.

Mistake 2: Sign error differentiating $(x-3)^2$

Why it's wrong: Using the chain rule, $\dfrac{dy}{dx}=2(x-3)$; a rushed differentiation sometimes drops the inner $-3$ or writes $2x-3$ instead, which changes the equation solved for $x$. Correct approach: apply the chain rule carefully — derivative of $(x-3)^2$ is $2(x-3)\cdot 1$, not $2x$ or $2x-3$. …