Section 2.1The Tangent and Velocity Problems

Math 1a

February 1, 2008

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I Grab a bingo card and start playing!

I Syllabus available on course websiteI Homework for Monday 2/4:

I Practice 2.1: 1, 3, 5, 7, 9I Turn-in 2.1: 2, 4, 6, 8I Complete the ALEKS initial assessment (course code

QAQRC-EQJA6)

Outline

Bingo

Velocity

Tangents

Outline

Bingo

Velocity

Tangents

Hatsumon

ProblemMy speedometer is broken, but I have an odometer and a clock.How can I determine my speed?

|−4

|−3

|−2

|−1

|0

|1

|2

|3

|4

Outline

Bingo

Velocity

Tangents

A famous solvable problem

ProblemGiven a curve and a point on the curve, find the line tangent tothe curve at that point.

But what do we mean by tangent?In geometry, a line is tangent to a circle if it intersects the circle inonly one place.

•

A famous solvable problem

ProblemGiven a curve and a point on the curve, find the line tangent tothe curve at that point.

But what do we mean by tangent?

In geometry, a line is tangent to a circle if it intersects the circle inonly one place.

•

A famous solvable problem

ProblemGiven a curve and a point on the curve, find the line tangent tothe curve at that point.

But what do we mean by tangent?In geometry, a line is tangent to a circle if it intersects the circle inonly one place.

•

Towards a definition of tangent

This doesn’t work so well for general curves, though:

Is this a tangent line?

Is this a tangent line?•

We need to think of tangency as a “local” phenomenon.

Towards a definition of tangent

This doesn’t work so well for general curves, though:

Is this a tangent line?

Is this a tangent line?•

We need to think of tangency as a “local” phenomenon.

Towards a definition of tangent

This doesn’t work so well for general curves, though:

Is this a tangent line?

Is this a tangent line?•

We need to think of tangency as a “local” phenomenon.

Towards a definition of tangent

This doesn’t work so well for general curves, though:

Is this a tangent line?

Is this a tangent line?•

We need to think of tangency as a “local” phenomenon.

Tangent

A line L is tangent to a curve C at a point P if

I L and C both go through P, and

I L and C have the same “slope” at P.

Slope of L = “m” in y = mx + b

=rise

run

Slope of C at a ≈ f (x)− f (a)

x − awhere x ≈ a

Tangent as a limiting process

I To find the tangent line through a curve at a point, we drawsecant lines through the curve at that point and find the linethey approach as the second point of the secant nears the first.

I For instance, it appears the tangent line to y =√

x through(4, 2) has slope 0.25.

Tangent as a limiting process

I To find the tangent line through a curve at a point, we drawsecant lines through the curve at that point and find the linethey approach as the second point of the secant nears the first.

I For instance, it appears the tangent line to y =√

x through(4, 2) has slope 0.25.

Same thing!

The infinitesimal rate of change calculation is the same in bothcases: finding velocities or finding slopes of tangent lines.

General rates of change

I The rate of change of f (t) at time t1 = the slope of y = f (t)at the point (t1, f (t1)).

I The units areunits of f (t)

units of t.