Equation of the tangent line at the indicated point

In summary, the equation of the tangent line is a mathematical expression that describes the slope and intercept of a line tangent to a given curve at a specific point. It can be calculated using the derivative of the curve at the given point and is important in calculus for approximating curve behavior and finding slope. The difference between the tangent line and the secant line is that the tangent line touches the curve at one point while the secant line intersects at two points. The equation of the tangent line can also be used to find the slope at other points by plugging in different x-values.
  • #1
carlarae
5
0

Homework Statement


Find an equation of the tangent line at the indicated point on the graph of the function.
y=f(x)=x^3/4 , (x,y)=(6,54)


Homework Equations





The Attempt at a Solution



I did the derivative which I get 3x^2/4 and then I plugged in the 6 and get 162. Is that the whole answer? right answer? it's asking for an equation and 162 doesn't look like an equation to me.
 
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  • #2
carlarae said:

Homework Statement


Find an equation of the tangent line at the indicated point on the graph of the function.
y=f(x)=x^3/4 , (x,y)=(6,54)


Homework Equations





The Attempt at a Solution



I did the derivative which I get 3x^2/4 and then I plugged in the 6 and get 162.
This is wrong. Show us how you got that number.
carlarae said:
Is that the whole answer? right answer?
No and no. The question asks for the equation of the tangent line to the curve at the point (6, 54). If you know the slope m of a line and a point (x0, y0) on it, the equation of the line is y - y0 = m(x - x0).
carlarae said:
it's asking for an equation and 162 doesn't look like an equation to me.
 
  • #3
So is the derivative of x^3/4 not 3x^2/4? That will make a big difference for me to take another crack at this.
 
  • #4
Welcome to PF, carlarae! :smile:

carlarae said:
So is the derivative of x^3/4 not 3x^2/4? That will make a big difference for me to take another crack at this.

Hmm, if I fill in x=6 in 3x^2/4 I get a different result...

But yes, the derivative of [itex]x^3 \over 4[/itex] is [itex]3x^2 \over 4[/itex].
 
  • #5
carlarae said:
So is the derivative of x^3/4 not 3x^2/4? That will make a big difference for me to take another crack at this.
Sorry I wasn't more specific. As I like Serena points out, your derivative is fine, but the value you got isn't.
 
  • #6
Is the function
[tex]\frac{x^3}{4}[/tex]
or
[tex]x^{\frac{3}{4}}[/tex]?
what you wrote was ambiguous. If the function is the first, then the derivative is
[tex]\frac{3}{4}x^2[/tex]
if the second, then the derivative is
[tex]\frac{3}{4}x^{-1/4}= \frac{3}{4x^{1/4}}[/tex]
 
  • #7
HallsofIvy said:
Is the function
[tex]\frac{x^3}{4}[/tex]
or
[tex]x^{\frac{3}{4}}[/tex]?
what you wrote was ambiguous.
It's the first. What she wrote actually isn't ambiguous, if you allow for exponentiation being higher in precedence than multiplication or division.
 

FAQ: Equation of the tangent line at the indicated point

What is the "equation of the tangent line"?

The equation of the tangent line is a mathematical expression that describes the slope and intercept of a line tangent to a given curve at a specific point.

How is the equation of the tangent line at a point calculated?

The equation of the tangent line can be calculated using the derivative of the curve at the given point. The derivative represents the slope of the tangent line at that point.

What is the importance of the tangent line in calculus?

The tangent line is important in calculus because it allows us to approximate the behavior of a curve at a specific point. It also helps us find the slope of a curve, which is essential in solving many real-world problems.

What is the difference between the tangent line and the secant line?

The tangent line is a line that touches the curve at one point, while the secant line is a line that intersects the curve at two points. The secant line can be thought of as the average slope between two points, while the tangent line represents the instantaneous slope at a single point.

Can the equation of the tangent line be used to find the slope of a curve at other points?

Yes, the equation of the tangent line can be used to find the slope of a curve at any point. By plugging in different x-values into the equation, we can calculate the slope of the tangent line at those points.

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