Derivatives Limits of the form arise whenever we calculate a rate of change in any of the sciences or engineering, such as a rate of reaction in chemistry.

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Presentation transcript:

Derivatives Limits of the form arise whenever we calculate a rate of change in any of the sciences or engineering, such as a rate of reaction in chemistry or a marginal cost in economics. Since this type of limit occurs so widely, it is given a special name and notation.

Derivatives Definition: The derivative of a function f at a number a, denoted by f ‘ (a), is if this limit exists.

Derivatives An equivalent way of stating the definition of the derivative is.

Derivatives Example 1: Find the derivative of the function f(x) = x 2 – 8x + 9 at the number a.

Derivatives We can now say that the tangent line to y = f(x) at (a, f(a)) is the line through (a, f(a)) whose slope is equal to f’(a), the derivative of f at a. Look at figure 1 on page 151. If we use the point-slope form of the equation of a line, we can write an equation of the tangent line to the curve y = f(x) at the point (a, f(a)): y – f(a) = f’(a)(x – a)

Derivatives Example 2: Find an equation of the tangent line to the parabola y = x 2 – 8x + 9 at the point (3, -6).

Derivatives Example 3: Let f(x) = 2 x. Estimate the value of f’(0) by using definition 2 and taking successively smaller values of h.

Derivatives The derivative f’(a) is the instantaneous rate of change of y = f(x) with respect to x when x = a. In particular, if s = f(t) is the position function of a particle that moves along a straight line, then f’(a) is the rate of change of the displacement s with respect to the time t. In other words, f’(a) is the velocity of the particle at time t = a. The speed of the particle is the absolute value of the velocity.

Derivatives Example 4: The position of a particle is given by the equation of motion s = f(t) = 1/(1 + t), where t is measured in seconds and s in meters. Find the velocity and the speed after 2 seconds.

Derivatives Example 5: A manufacturer produces bolts of a fabric with a fixed width. The cost of producing x yards of this fabric is C = f(x) dollars. a) What is the meaning of the derivative f’(x)? What are its units? b) In practical terms, what does it mean to say that f’(1000) = 9?

Derivatives ASSIGNMENT P. 155 (2, 4, 10, 13, 28)