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- https://k12.libretexts.org/Bookshelves/Mathematics/Calculus/07%3A_Differentiation_-_Increasing_and_Decreasing_Values_and_Extrema
- https://k12.libretexts.org/Bookshelves/Mathematics/Calculus/07%3A_Differentiation_-_Increasing_and_Decreasing_Values_and_Extrema/7.03%3A_First_Derivative_TestThe first derivative test says that if f is a continuous function and that x=c is a critical value of f, then if f′ changes from positive to negative at x=c then f has a local maximum at x=c, if f′ ch...The first derivative test says that if f is a continuous function and that x=c is a critical value of f, then if f′ changes from positive to negative at x=c then f has a local maximum at x=c, if f′ changes from negative to positive at x=c then f has a local minimum at x=c, and if f′ does not change sign at x=c then f has neither a local maximum nor minimum at x=c.
- https://k12.libretexts.org/Bookshelves/Mathematics/Calculus/08%3A_Differentiation_-_Derivative_Applications/8.01%3A_Absolute_Extrema_and_OptimizationThis kind of problem is an optimization problem, and the solution that is the maximum or minimum value of the function in the region is the optimal solution. The number of tires that the company sells...This kind of problem is an optimization problem, and the solution that is the maximum or minimum value of the function in the region is the optimal solution. The number of tires that the company sells is a function of the price charged and can be modeled by the formula \( T(x)=−x^3+36.5x^2+50x+250 \nonumber\), where x is the priced charged for each tire in dollars.
- https://k12.libretexts.org/Bookshelves/Mathematics/Calculus/09%3A_Integral_-_Area_Computation/9.02%3A_AntiderivativeIt should be no surprise then that there would be a name for the function f(x), or family of functions, that can generate f′(x) when differentiated: f(x) and f′(x) are a pair of inverse functions, and...It should be no surprise then that there would be a name for the function f(x), or family of functions, that can generate f′(x) when differentiated: f(x) and f′(x) are a pair of inverse functions, and f(x) is called an antiderivative of f′(x). We refer to ∫f(x)dx as “the indefinite integral of f(x) with respect to x”. The function f(x) is called the integrand and the constant C is called the constant of integration.
- https://k12.libretexts.org/Bookshelves/Mathematics/Calculus/00%3A_Front_Matter
- https://k12.libretexts.org/Bookshelves/Mathematics/Calculus/08%3A_Differentiation_-_Derivative_Applications
- https://k12.libretexts.org/Bookshelves/Mathematics/Calculus/09%3A_Integral_-_Area_Computation/9.06%3A_Reimann_SumsIn general, Riemann Sums are of form \( \sum_{i=1}^n f ( x_i^∗) △x \nonumber\) where each \( x_i^∗ \nonumber\) is the value we use to find the length of the rectangle in the ith sub-interval. For exam...In general, Riemann Sums are of form \( \sum_{i=1}^n f ( x_i^∗) △x \nonumber\) where each \( x_i^∗ \nonumber\) is the value we use to find the length of the rectangle in the ith sub-interval. For example, evaluate the Riemann Sum for f(x)=x3 from x=0 to x=3 using n=6 sub-intervals, and take the sample points to be the midpoints of the sub-intervals.
- https://k12.libretexts.org/Bookshelves/Mathematics/Calculus
- https://k12.libretexts.org/Bookshelves/Mathematics/Calculus/01%3A_Limit/1.04%3A_Limits_of_Composite_FunctionsDid you know that you can track the "behavior" of a function? Well, you can, and I'm not talking about whether the function is making curfew or not. When we talk about the "behavior" of a function, we...Did you know that you can track the "behavior" of a function? Well, you can, and I'm not talking about whether the function is making curfew or not. When we talk about the "behavior" of a function, we look at how the function "shows up" when graphed. As we solve the function, we can get closer and closer and closer - the limits helps us to understand this movement of the function on the graph. Video: Find Limits of Composite Functions Graphically Practice: Limits of Composite Functions
- https://k12.libretexts.org/Bookshelves/Mathematics/Calculus/02%3A_Limit_-_Types_of_Limits/2.05%3A_Limits_Involving_Radical_FunctionsUsing direct substitution to find the limit results in the indeterminate form ∞ / ∞ . To transform the radical expression to a better form, use the fact that the value of x is going to larger and larg...Using direct substitution to find the limit results in the indeterminate form ∞ / ∞ . To transform the radical expression to a better form, use the fact that the value of x is going to larger and larger positive values. Using direct substitution to find the limit of the function results in the indeterminate form 0 / 0 . To transform the radical expression to a better form, do the following:
- https://k12.libretexts.org/Bookshelves/Mathematics/Calculus/09%3A_Integral_-_Area_Computation
