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Given $f(x)=x^{3}-12 x .$ At which points will its tangent line (a) be horizontal; (b) have slope 15; (c) have slope 36?

(a) (-2,16),(2,-16)(b) (-3,9),(3,-9)(c) (-4,-16),(4,16)

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 1

Slope of a Curve

Derivatives

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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the function of X cube for X physical X cube minus 12 eggs. And we want to know at that point this slope to this line, the tangent to this line has is horizontal. So to do that, we need to calculate deliver the dysfunction, which is three X square minus 12. And if the tangent like the tourism horizontal, that means the derivative zero. So we put this equal to zero, and we solved that. Heh. Eggs is equal to do so that at X equal to two, the tangent to this care dysfunction is horizontal. What about 15? At what point? The slope of this care of is 15. So now we put this called to 15, and that tells us three x school got 27 so that x x square is nine an X s plus and minus three. Actually, at two points, it has a slope of 15. And finally, at what point? It has a slope up 36. So it put this go to 36 so you will get X Square equal to, And I'm sorry that the first part also it's just not just to it plus minus two at point plus minus two. You get a slope zero at point plus minus three. You get a slope, go to 15. And as we are following here at point plus, uh, XXVI equal to 16. That means at two points plus minus four, the slope is going to be 46 okay?

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