Using Limits to Find Tangent Lines, Rates of Change, and Instantaneous Velocity

Introduction

The lecture walks through the core idea of calculus: turning the slope of a secant line into the slope of a tangent line by using limits. This process gives us the instantaneous rate of change of a function, which is the foundation for concepts like instantaneous velocity.

Tangent Lines and Limits

• Goal: Find the tangent line to a curve at a specific point P.
• Ingredients: a fixed point (x_0) (the x‑coordinate of P) and the slope at that point.
• Idea: Start with a second point (Q) at (x_0 + h). As (h) becomes infinitesimally small, (Q) approaches (P). The limit as (h\to0) turns the secant slope into the tangent slope.

The Difference Quotient

The slope of the secant line is expressed as the difference quotient:

[f(x_0 + h) – f(x_0)] / h

When we let (h) approach zero, the expression becomes the derivative (the tangent slope).

From Secant to Tangent – A Step‑by‑Step Example

1. Function: (f(x)=x^2).
2. Point: (x_0 = 1).
3. Compute (f(1+h)= (1+h)^2 = 1 + 2h + h^2) and (f(1)=1).
4. Form the difference quotient:

[1 + 2h + h^2 – 1] / h = (2h + h^2)/h = 2 + h

1. Take the limit as (h\to0): the tangent slope is 2.
2. Use point‑slope form (y - y_1 = m(x - x_1)) to write the tangent line: (y = 2x - 1).

General Formula for Any Point

Instead of plugging a specific (x_0) each time, keep (x) symbolic:

[f(x + h) – f(x)] / h  →  limit_{h→0}

After simplifying and canceling (h), the limit yields a formula for the derivative (f'(x)). Plug any (x) value later to get the slope at that point.

Average vs. Instantaneous Velocity

• Average velocity over an interval ([t_0, t_0+h]) is the secant slope:

[s(t_0 + h) – s(t_0)] / h

• Instantaneous velocity is the limit of the average velocity as (h\to0); it is exactly the derivative of the position function.

Motion Example

Position: (s(t)=500 - 16t^2). 1. Compute (s(5+h)=500 - 16(5+h)^2) and (s(5)=500 - 16·25 = 100). 2. Form the difference quotient and simplify:

[500 - 16(5+h)^2 – 100] / h = [-160h - 16h^2] / h = -160 - 16h

1. Limit as (h\to0) gives instantaneous velocity -160 (units per second) at (t=5).
2. The same algebra works for any (t); the general velocity formula is (v(t) = -32t).

Key Algebraic Tips

• Never separate a function’s argument: (f(x_0 + h)) is one whole expression, not (f(x_0) + h).
• Keep track of signs when distributing negatives.
• Factor out (h) to cancel it before taking the limit.
• Use a common denominator when dealing with complex fractions.

When Limits Fail

• Discontinuities (gaps, jumps) give no tangent because the limit does not exist.
• Sharp corners (cusps) also lack a unique tangent; the left‑hand and right‑hand limits differ.

Summary of Concepts

• The derivative is the limit of the difference quotient.
• Tangent slope = instantaneous rate of change.
• Average rate = secant slope; instantaneous rate = limit of the secant as the interval shrinks.
• Mastery of algebraic manipulation is essential for successful limit calculations.

The material above captures the entire lecture without needing to watch the video.

Limits let us shrink the distance between two points on a curve until they coincide, turning a secant slope into a tangent slope. This single idea provides the derivative, which is the instantaneous rate of change—whether it’s the slope of a curve, the velocity of a moving object, or any other real‑world rate.