Since the second derivative is zero at x = 0, the test is inconclusive, and further analysis is needed to determine the nature of the extrema.Ĭonsider the function f(x) = x^4 - 6x^2. The first derivative of this function is f'(x) = 3x^2, and the second derivative is f''(x) = 6x. Since the second derivative is negative at x = 0, the function has a local maximum at x = 0.Ĭonsider the function f(x) = x^3. The first derivative of this function is f'(x) = -2x, and the second derivative is f''(x) = -2. Since the second derivative is positive at x = 0, the function has a local minimum at x = 0.Ĭonsider the function f(x) = -x^2. The first derivative of this function is f'(x) = 2x, and the second derivative is f''(x) = 2. It is important to note that the Second Derivative Test can only be applied to a function that is twice differentiable, and that it only gives information about the extrema at a single point, not on an interval.Ĭonsider the function f(x) = x^2. If the second derivative is zero at a critical point, the test is inconclusive and further analysis is needed. If the second derivative is negative at a critical point, the function has a local maximum at that point.Ħ. If the second derivative is positive at a critical point, the function has a local minimum at that point.ĥ. Evaluate the second derivative at each critical point.Ĥ. Find the second derivative of the function.ģ. Find the critical points of the function by setting the first derivative equal to zero and solving for x.Ģ. Here are the steps to use the Second Derivative Test:ġ. It is based on the fact that the sign of the second derivative of a function can indicate whether a critical point is a maximum, minimum, or neither. The best way to prepare for this challenging exam is to complete as many practice problems and exams as possible.The Second Derivative Test is a powerful tool in calculus that can be used to determine the nature of extrema (maxima and minima) of a function. Part B consists of four problems in which a calculator is not permitted.īoth the multiple choice and free response questions expect you to evaluate, analyze, conceptualize, and develop functions and representations both at face value in in real-world contexts. Part A has two free response questions that require a graphing calculator. Section II is also broken down into a Part A and Part B, but the calculator usage is reversed. Part A does not allow a graphing calculator and Part B allows a graphing calculator, with some questions requiring its use in order to get an answer. Section I is further broken down into Part A and Part B.
Section II is the free response section which you have 90 minutes to complete. Section I is the multiple choice section which you have 105 minutes to complete. The AP Calculus AB exam is divided into two sections, each worth 50% of your final score.
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