Finding Slopes and Equations of Secant and Tangent Lines for a Given Curve

[Jan Hill]

Homework Statement

Given a point P (3, 10) and the equation of a curve as x^2 -5x-4, find the slope of the secant and the equation of the tangent line to the curve

Homework Equations

The Attempt at a Solution

I tried using y = f(x + h) -f(x) all divided by h and got (x + h)^2 - 5(x + h) - 4 -x^2 - 5x-4 all divided by h

I got x^2 + 2xh + h^2 - 5x-5h -4-x^2 -4 all divided by h

which equals 2xh + h^2 - 5x-5h - 4+5x + 4 all divided by h

which equals 2x + h all divided by h

Can we then claim that the slope of the secant is probably 2 and substitute this into an equation of the form
y - 10 = 2(x-3)
or y = 2(x-3) + 10 to get the equation of the tangent line

 

[Mark44]

Homework Statement

Given a point P (3, 10) and the equation of a curve as x^2 -5x-4, find the slope of the secant and the equation of the tangent line to the curve

Homework Equations

The Attempt at a Solution

I tried using y = f(x + h) -f(x) all divided by h and got (x + h)^2 - 5(x + h) - 4 -x^2 - 5x-4 all divided by h

I got x^2 + 2xh + h^2 - 5x-5h -4-x^2 -4 all divided by h

which equals 2xh + h^2 - 5x-5h - 4+5x + 4 all divided by h

which equals 2x + h all divided by h

Lets' start by writing this in a more mathematical form.
[f(x + h) -f(x)]/h = [(x + h)^2 - 5(x + h) - 4 - (x^2 - 5x-4)]/h

Can we then claim that the slope of the secant is probably 2 and substitute this into an equation of the form
y - 10 = 2(x-3)
or y = 2(x-3) + 10 to get the equation of the tangent line

 

[Jan Hill]

so simplified this becomes

the limit as h approaches 0 of 2x + h - 5

which becomes 2x - 5

but what is the slope of the secant?

 

[Mark44]

Yes, so f'(x) = 2x - 5. This is the slope of the tangent line at a point (x, f(x)).

A secant line is a line that intersects two points of a curve.

I think what the first part of this problem is asking you to do is to find the slope of the secant line between P(3, 10) and a point (x, f(x)).

The second part is asking you for the slope of the tangent line at (3, 10), I think.

Have you written the problem here exactly as it's worded?