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4.1 to 4.5 (Indeterminate Forms and l’Hospital’s Rule (∞/∞, 0/0, In…
4.1 to 4.5
Indeterminate Forms and l’Hospital’s Rule
∞/∞
0/0
In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this substitution does not give enough information to determine the original limit, it is said to take on an indeterminate form.
0*∞
∞-∞
1^∞
0^∞
Guidelines for Sketching a Curve
B. Intercepts
C. Symmetry
A. Domain
D. Asymptotes
E. Intervals of Increase or Decrease
F. Local Maximum and Minimum Values
G. Concavity and Points of Inflection
H. Sketch the Curve
4.1 Maximum and Minimum Values
The maximum and minimum
values of f are called extreme values of f.
f (c) is called a local minimum value of f because
f (c) ≤ f (x) for x near c [in the interval (b, d )
Extreme Value Theorem
Fermat’s Theorem.
4.2 The Mean Value Theorem
Rolle’s Theorem
Tangent line
Secant line
4.3 How Derivatives Affect the
Shape of a Graph
Local Extreme Values
The first derivative test
The second derivative test