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Level 560

Applications of the Derivative


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Extreme Value Theorem
If a function f(x) is continuous on a closed interval, the f(x) has both a maximum and a minimum value in the interval.
critical point
a point in the domain of the function at which the derivative is zero or undefined.
Rolle's Theorem
If f(x) is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists a point c∈(a,b), where f '(c) = 0.
Mean Value Theorem
if f(x) is continuous and differentiable, slope of tangent line equals slope of secant line at least once in the interval (a, b)
It is increasing
If f '(x) > 0 on (a,b), then f(x) has what characteristic?
It is decreasing
If f '(x) < 0 on (a,b), then f(x) has what characteristic?
It is constant
If f '(x) = 0 on (a,b), then f(x) has what characteristic?
To find a relative maximum on f(x) (by the 1st derivative test)...
Find a point where f '(x) is zero or undefined, AND where f '(x) changes from positive to negative.
To find a relative minimum on f(x) (by the 1st derivative test)...
Find a point where f '(x) is zero or undefined, AND where f '(x) changes from negative to positive.
It is concave upward
If f "(x) > 0 on (a,b), then f(x) has what characteristic?
It is concave downward
If f "(x) < 0 on (a,b), then f(x) has what characteristic?
It is linear
If f "(x) = 0 on (a,b), then f(x) has what characteristic?
Inflection Point
a point in the domain of the function at which a tangent line exists and the concavity of the function changes.
To find a point of inflection on f(x)...
Find a point where f "(x) is zero or undefined, where a tangent line exists, AND where f "(x) changes sign.
To find a relative maximum on f(x) (by the 2nd derivative test)...
Find a point where f '(x) is zero AND where f "(x) is negative.
To find a relative minimum on f(x) (by the 2nd derivative test)...
Find a point where f '(x) is zero AND where f "(x) is positive.
f '(x) * dx
If y = f(x) is a differentiable function and dx ≠ 0, then dy =
Tangent Line
Another name for a linear approximation to the curve is a
f '(a)*(x − a) + f(a)
The linear approximation of f(x) near x = a is