# Question on the slope of the tangent

• kiss89
In summary, to find the points on the graph of y=(1/3)x^3-5x-(4/x) where the slope of the tangent is horizontal, we need to find the derivative of the function and set it equal to zero. This is because the derivative of a function represents the slope of the tangent line at any given point on the graph. By setting the derivative equal to zero, we can find the x-values where the slope of the tangent is horizontal. Alternatively, we could also use the "quotient difference" formula to find the slope of the tangent at any given point.
kiss89
Find the points on the graph of y= (1/3)x^3-5x- (4/x) at which the slope of the tangent is horizontal.

what i know:
- we have to use m=[f(a+h)-f(a)]/h
- if we change the equation we can get 3x^4 - 15x^2 -12
- the slope of the tangent is zero.

THANX

hint: how does the derivative of a function relate to the slope of the tangent of its graph?

Find the derivative, and set it equal to the slope of a horizontal line. What is that?

What everyone is saying is "DO IT"! By the way are you really required to use the "quotient difference" formula? It's not too difficult but tedious and most problems like this allow the use of derivative formulas.

Gib Z said:
Find the derivative, and set it equal to the slope of a horizontal line. What is that?

Do you mean, "what is the slope of a horizontal line?"

Draw a graph that shows a horizontal line. Pick two points $(x_1,y_1)$ and $(x_2,y_2)$ on the line. Do you know how to calculate the slope of a line from two points?

yes that's what i meant, but i knew the answer..set the derivative to zero is what i meant

## 1. What does the slope of the tangent represent?

The slope of the tangent at a point on a curve represents the instantaneous rate of change, or the rate at which the curve is changing at that specific point.

## 2. How is the slope of the tangent calculated?

The slope of the tangent can be calculated using the derivative of the function at the given point. It is the limit of the change in the y-coordinate over the change in the x-coordinate as the change in x approaches 0.

## 3. Can the slope of the tangent be negative?

Yes, the slope of the tangent can be negative. This indicates that the curve is decreasing at that point.

## 4. What is the significance of the slope of the tangent in real-world applications?

The slope of the tangent is used in many real-world applications, such as calculating the speed of an object at a specific time or predicting how the stock market will change based on current trends.

## 5. How does the slope of the tangent relate to the concavity of a curve?

The slope of the tangent can determine the concavity of a curve. If the slope of the tangent is increasing, the curve is concave up. If the slope of the tangent is decreasing, the curve is concave down.

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