section35_material

section35_material - MATH110BUSINESSCALCULUS

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MATH 110 BUSINESS CALCULUS CHAPTER 3 INTRODUCTION TO THE DERIVATIVE Section 3.5 Derivatives: Numerical and Graphical Viewpoints The instantaneous rate of change of f(x) at x = a is defined as 0 () ( ) () l im h f ah f a fa h + = [We say “ f prime of a equals the limit as h approaches 0, of the ratio ( ) f ah fa h +− ”] is also called the derivative of f(x) at x = a. Differentiation is the process of finding the derivative. Leibniz d Notation: 00 ( ) l hh f xh f x f d f fx hx d x →→ Δ == = Δ Thus, ( ) can be written as in Leibniz Notation xa df dx = The vertical line indicates that the derivative is being evaluated at x = a.
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Example: Estimate the derivative from the table of average rates of change. h ARCh of f over [3, 3+h] 1 4.7 0.1 5.96 0.01 5.989 0.001 5.99964 0.0001 5.9999984 -0.0001 6.0000085 -0.001 6.00042 -0.01 6.027 -0.1 6.10 -1 6.5 Solution: From the table, we see that (3) 6.0 f Example: Consider the function as representing the value of an ounce of silver in Indian rupees as a function of time t in days. Find the average rate of change of R(t)
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This note was uploaded on 10/19/2011 for the course MATH 110 taught by Professor Staff during the Spring '11 term at S.F. State.

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section35_material - MATH110BUSINESSCALCULUS

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