Are any tangents to the graph f(x)=x^2-3x horizontal?

In summary, the conversation discusses finding tangents to the graph f(x)=x^2-3x and using a shortcut formula to do so, instead of using limits. It is also mentioned that the derivative must equal 0 for the tangent line to be horizontal.
  • #1
fiziksfun
78
0
"At what points, if any, are the tangents to the graph f(x)=x^2-3x horizontal?"

I know how to figure this out using a graphing calculator and using the 'short-cut' (nx^n-1), but I can't do it using limits. Help!

I'd like to use the equation lim x->a = f(x) - f(a) / x - a

Thanks!
 
Physics news on Phys.org
  • #2
why would you do it like that? one yields the other. just use the "shortcut"
 
  • #3
As ice109 said, once you have worked out the general formula from the definition, you can use that formula and not have to go back to the definition every time! That's the whole point of the formula (which you call a "shortcut").

However, since you ask, it's not particularly difficult: f(x)= x^2- 3x and f(a)= a^2- 3a.
f(x)- f(a)= x^2- 3x- a^2+ 3a= (x^2- a^2)- (3x-3a)= (x-a)(x+a)- 3(x-a). Now, when you divide that by x-a, what do you get? What is the limit of that as x goes to a?

I assume you know that "tangent line horizontal" means that the derivative must be 0.
 

What is a tangent to a graph?

A tangent to a graph is a straight line that touches the curve of the graph at only one point, and has the same slope as the curve at that point.

Can a tangent to the graph f(x)=x^2-3x be horizontal?

Yes, a tangent to the graph f(x)=x^2-3x can be horizontal if it touches the curve at a point where the slope of the curve is 0. This would occur at the vertex of the parabola where the slope is 0.

How do you determine if a tangent to a graph is horizontal?

To determine if a tangent to a graph is horizontal, you need to find the derivative of the function and set it equal to 0. If the derivative is 0 at a point, then the tangent to the graph at that point is horizontal.

Are there any horizontal tangents to the graph f(x)=x^2-3x?

Yes, there are two horizontal tangents to the graph f(x)=x^2-3x. One at the vertex where x=3/2 and another at the point where x=-3/2.

What is the significance of a horizontal tangent to a graph?

A horizontal tangent to a graph indicates that the slope of the curve at that point is 0, which means the function is not changing. This can represent a maximum or minimum point on the graph, or a point of inflection where the curve changes direction.

Similar threads

I Inverse Functions: x=f(y) and X=f^-1(y)
    • Calculus
Replies
4
Views
1K
B Why is Big-O about how rapidly the Taylor graph approaches that of f(x)?
    • Calculus
Replies
19
Views
2K
B Linearization of a function error viewed with differentials
    • Calculus
Replies
14
Views
1K
I Why prove Taylor's theorem with p(x)=q(x)−((b−a)^n/(b−x)^n)q(a)?
    • Calculus
Replies
9
Views
851
I The Basic Area Problem (introduction to the topic of integrals)
    • Calculus
Replies
24
Views
2K
B Second Derivative Question -- Help Understanding the Importance Please
    • Calculus
Replies
13
Views
2K
B Tangents to a curve as functions of different variables
    • Calculus
Replies
5
Views
1K
MHB What values of x make the graph of f(x) have a horizontal tangent?
    • Calculus
Replies
2
Views
1K
I Derivative of F = f(x)/f(x+dx)
    • Calculus
Replies
15
Views
1K
MHB Application of Differentiation Problem - Need help
    • Calculus
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top