State the condition on the derivative f' that can be used to show that a function f is increasing.

b Define the function arctan.

c Explain how one, starting from the definition of arctan, may derive an expression for the derivative of this function, and carry out that calculation.

Expert Solution

a) The derivative of a function may be used to determine whether the function is increasing or decreasing on any intervals in its domain. If f′(x) > 0 at each point in an interval I, then the function is said to be increasing on I.

b) Every mathematical function has an inverse. Even simple operations such as addition and multiplication have inverses - subtraction and division. The arctan is the inverse of the tangent function and is used to compute the angle measure from the tangent ratio of a right triangle, designated by the formula:

tan = opposite / adjacent

The term 'arc' is used because when measuring an angle in radians, the arc length of a portion of a circle bisected by an angle (with the vertex at the center of the circle) equals the angle measure. The radian is the standard unit of measurement for an angle and is equal to approximately 57 degrees. It is based on the radius of a circle.

We can use arctan to determine an angle measure when the sides opposite and side adjacent angles are known.The arctan has practical application in architecture, buildings, landscaping, physics and engineering.

C) I am attaching a picture fir better understanding.

Colby Messinger answered 2 years ago

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