Math Is Fun Forum

nycguitarguy

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 571

Difference Quotient of f

Michael Sullivan introduces the calculus idea of the difference quotient of f at the end of section 3.3 in chapter 3.

The slope of the secant line containing the two points
(x, f(x)) and (x + h, f(x + h)) on the graph of a function y = f(x) may be given as

m_sec = [f(x + h) - f(x)]/[(x + h) - x] which leads to [f(x + h) - f(x)]/h, where h cannot = 0.

A. Express the slope of the secant line for the function f(x) = 2x + 5 in terms of x and h. Be sure to simplify.

Note: m_sec = slope of the secant line.

sec_m = [2(x + h) + 5 - (2x + 5)]/h

My answer is 2.

You say?

B. Find m_sec for h = 0.5, 0 1, and 0.01 at x = 1. What value does m_sec approach as h approaches 0?

After working it out on paper, I get m_sec = 2. I also get 2 when h = 0.1 and h = 0.01. I conclude that m_sec tends to 2 as h tends to 0.

C. Is this what we call the limit in calculus?

D. Find the equation of the secant line at x = 1 with h = 0.01.

I need help with part D.

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The Rapture is the central event in biblical prophecy. The greatest truth about the Rapture is not its timing but it's reality. The Rapture will be the great disappearance.

Dr. David Jeremiah

Bob

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Registered: 2010-06-20

Posts: 10,214

Re: Difference Quotient of f

A, B, and C all correct.

D.sec_m = [2(x + h) + 5 - (2x + 5)]/h

Find the equation of the secant line at x = 1 with h = 0.01.

Sub in those values

sec_m = [2 times 1.01 + 5 - (2 + 5)]/0.01 = 0.02/0.01 = 2

So the line has gradient 2.

function is y = 2x + 5 so if x = 1, y = 7

So we have a line that has gradient 2 and goes through (1,7)

Because we're dealing with a straight line that is going to give y = 2x + 5 again.

Not sure why this question is even being asked.

I'd have chosen a function that isn't a line, eg y = x^2

Maybe that's the next question?

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycguitarguy

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 571

Re: Difference Quotient of f

A, B, and C all correct.

D.sec_m = [2(x + h) + 5 - (2x + 5)]/h

Find the equation of the secant line at x = 1 with h = 0.01.

Sub in those values

sec_m = [2 times 1.01 + 5 - (2 + 5)]/0.01 = 0.02/0.01 = 2

So the line has gradient 2.

function is y = 2x + 5 so if x = 1, y = 7

So we have a line that has gradient 2 and goes through (1,7)

Because we're dealing with a straight line that is going to give y = 2x + 5 again.

Not sure why this question is even being asked.

I'd have chosen a function that isn't a line, eg y = x^2

Maybe that's the next question?

Bob

Yes, part D is weird.

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The Rapture is the central event in biblical prophecy. The greatest truth about the Rapture is not its timing but it's reality. The Rapture will be the great disappearance.

Dr. David Jeremiah