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Business Mathematics II
Project 1: Differentiation
CLASS ACTIVITY
Derivatives Let f be a function, which is defined on an open interval containing a real number x. Then the derivative of f at x is given by
ᡘጔ䙦ᡶ䙧 㐄 lim 〵፲⡨
- Let ᡘ䙦ᡶ䙧 㐄 け け⡸⡩.
(a) Use a difference quotient with ᡠ 㐄 0.0001 to estimate ᡘ䖓䙦4䙧.
x ᡘ䙦ᡶ ㎗ ᡠ䙧 ᡘ䙦ᡶ ㎘ ᡠ䙧
(b) Use Differentiating.xlsm to estimate f ′(^4 ).
- The graph of g is given below.
(a) On what interval(s) is the derivative of g positive? (b) On what interval(s) is the derivative of g negative? (c) For what value(s) of x is the derivative of g equal to zero?
FUNCTION
0
10
20
30
40
-4 -3 -2 -1 0 1 2 3 4 x
g ( x )
- Let ᡠ䖓䙦ᡶ䙧 㐄 4 · ᡶ ㎘ 9.
(a) Is h increasing or decreasing at ᡶ 㐄 2? (b) Is h increasing or decreasing at ᡶ 㐄 8?
Formulas:
If ᡘ䙦ᡶ䙧 㐄 ᡕ, where c is any constant, then ᡘ䖓䙦ᡶ䙧 㐄 0. If ᡘ䙦ᡶ䙧 㐄 ᡥ ᙨ ᡶ, where m is any constant, then ᡘ䖓䙦ᡶ䙧 㐄 ᡥ. If ᡙ䙦ᡶ䙧 㐄 ᡓ · ᡘ䙦ᡶ䙧, where a is any constant, then ᡙ䖓䙦ᡶ䙧 㐄 ᡓ · ᡘ䖓䙦ᡶ䙧. If ᡠ䙦ᡶ䙧 㐄 ᡘ䙦ᡶ䙧 ㎙ ᡙ䙦ᡶ䙧, then ᡠ䖓䙦ᡶ䙧 㐄 ᡘ䖓䙦ᡶ䙧 ㎙ ᡙ䖓䙦ᡶ䙧.
- Find the derivative of ᡘ䙦ᡶ䙧 㐄 6.
- Find the derivative of ᡘ䙦ᡶ䙧 㐄 6 ᙨ ᡶ.
- Suppose that ᡙ䙦ᡶ䙧 㐄 10 · ᡘ䙦ᡶ䙧 and that ᡘ䖓䙦ᡶ䙧 㐄 8. Find the derivative of ᡙ䙦ᡶ䙧.
- Suppose that ᡙ䙦ᡶ䙧 㐄 10 · ᡘ䙦ᡶ䙧 ㎘ 5 and that ᡘ䖓䙦ᡶ䙧 㐄 8. Find the derivative of ᡙ䙦ᡶ䙧.
Tangent Lines If ᡘ䖓䙦ᡓ䙧 exists, then the equation of the line that is tangent to the graph of f at the point 䙦ᡓ, ᡘ䙦ᡓ䙧䙧 is given by ᡷ 㐄 ᡘ䖓䙦ᡓ䙧 ᙨ 䙦ᡶ ㎘ ᡓ䙧 ㎗ ᡘ䙦ᡓ䙧.
- Suppose that ᡘ䙦8䙧 㐄 1 and that ᡘ䖓䙦8䙧 㐄 4. Find the equation of the line that is tangent to the graph of f at the point 䙦8, ᡘ䙦8䙧䙧.
- The graph of f and the tangent line at the point 䙦0, ᡘ䙦0䙧䙧 is given below.
Find the value of ᡘ䖓䙦0䙧.
FUNCTION AND TANGENT LINE
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4
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-8 -6 -4 -2 0 2 4 6 8 x
Function Tangent Line
