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- The second derivative tells whether the curve is concave up or concave down at that point. If the second derivative is positive at a point, the graph is bending upwards at that point. Similarly, if the second derivative is negative, the graph is concave down.
mathstat.slu.edu/~may/ExcelCalculus/sec-4-5-SecondDerivativeConcavity.htmlThe Second Derivative and Concavity - Saint Louis University
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Dec 21, 2020 · The key to studying \(f'\) is to consider its derivative, namely \(f''\), which is the second derivative of \(f\). When \(f''>0\), \(f'\) is increasing. When \(f''<0\), \(f'\) is decreasing. \(f'\) has relative maxima and minima where \(f''=0\) or is undefined.
Jul 18, 2021 · The first derivative gives you the slope of a line which is tangent to the graph at a point x x. The second derivative is a measure of how that slope changes as we vary x x. For example, consider the function f(x) =x2 f (x) = x 2. At x = −1 x = − 1, the slope of the tangent line is −2 − 2.
Describe how the second derivative of a function relates to its concavity and how to apply the second derivative test. Describe the relationship between inflection points and concavity and how to find the inflection points of a function.
Apr 24, 2022 · The second derivative tells us if a function is concave up or concave down. If \( f''(x) \) is positive on an interval, the graph of \( y=f(x) \) is concave up on that interval. We can say that \(f\) is increasing (or decreasing) at an increasing rate. If \( f''(x) \) is negative on an interval, the graph of \( y=f(x) \) is concave down on that ...
Sep 28, 2023 · The second derivative measures the instantaneous rate of change of the first derivative. The sign of the second derivative tells us whether the slope of the tangent line to \(f\) is increasing or decreasing. A differentiable function is concave up whenever its first derivative is increasing (or equivalently whenever its second derivative is ...
On the graph of a function, the second derivative corresponds to the curvature or concavity of the graph. The graph of a function with a positive second derivative is upwardly concave, while the graph of a function with a negative second derivative curves in the opposite way.
The second derivative tells us if a function is concave up or concave down. The first derivative: If f ′(x) f ′ (x) is positive on an interval, the graph of f (x) f (x) is increasing on that interval. If f ′(x) f ′ (x) is negative on an interval, the graph of f (x) f (x) is decreasing on that interval.
