Calculus

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Standards For Mathematical Practice

Calculus

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1.0 Students demonstrate knowledge of both the formal definition and the graphical interpretation of limit of values of functions. This knowledge includes one-sided limits, infinite limits, and limits at infinity. Students know the definition of convergence and divergence of a function as the domain variable approaches either a number or infinity:  tools-icon Algebra I / Mathematics I
1.1  Students prove and use theorems evaluating the limits of sums, products, quotients, and composition of functions.  tools-icon Geometry / Mathematics II
1.2  Students use graphical calculators to verify and estimate limits.  tools-icon Algebra II / Mathematics III
1.3  Students prove and use special limits, such as the limits of (sin(x))/x and (1-cos(x))/x as x tends to 0.  tools-icon Statistics and Probability
2.0 Students demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function.  tools-icon

3.0 Students demonstrate an understanding and the application of the intermediate value theorem and the extreme value theorem.

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4.0 Students demonstrate an understanding of the formal definition of the derivative of a function at a point and the notion of differentiability:  tools-icon
4.1 Students demonstrate an understanding of the derivative of a function as the slope of the tangent line to the graph of the function.  tools-icon
4.2 Students demonstrate an understanding of the interpretation of the derivative as an instantaneous rate of change. Students can use derivatives to solve a variety of problems from physics, chemistry, economics, and so forth that involve the rate of change of a function.  tools-icon
4.3 Students understand the relation between differentiability and continuity.  tools-icon
4.4 Students derive derivative formulas and use them to find the derivatives of algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic functions.  tools-icon
5.0 Students know the chain rule and its proof and applications to the calculation of the derivative of a variety of composite functions.  tools-icon
6.0 Students find the derivatives of parametrically defined functions and use implicit differentiation in a wide variety of problems in physics, chemistry, economics, and so forth.  tools-icon
7.0 Students compute derivatives of higher orders.  tools-icon
8.0 Students know and can apply Rolle’s Theorem, the mean value theorem, and L’Hôpital’s rule.  tools-icon
9.0 Students use differentiation to sketch, by hand, graphs of functions. They can identify maxima, minima, inflection points, and intervals in which the function is increasing and decreasing.  tools-icon
10.0 Students know Newton’s method for approximating the zeros of a function.  tools-icon
11.0 Students use differentiation to solve optimization (maximum-minimum problems) in a variety of pure and applied contexts.  tools-icon
12.0 Students use differentiation to solve related rate problems in a variety of pure and applied contexts.  tools-icon
13.0 Students know the definition of the definite integral by using Riemann sums. They use this definition to approximate integrals.  tools-icon
14.0 Students apply the definition of the integral to model problems in physics, economics, and so forth, obtaining results in terms of integrals.  tools-icon
15.0 Students demonstrate knowledge and proof of the fundamental theorem of calculus and use it to interpret integrals as antiderivatives.  tools-icon
16.0 Students use definite integrals in problems involving area, velocity, acceleration, volume of a solid, area of a surface of revolution, length of a curve, and work.  tools-icon
17.0 Students compute, by hand, the integrals of a wide variety of functions by using techniques of integration, such as substi­tution, integration by parts, and trigonometric substitution. They can also combine these techniques when appropriate.  tools-icon
18.0 Students know the definitions and properties of inverse trigonometric functions and the expression of these functions as indefinite integrals.  tools-icon
19.0 Students compute, by hand, the integrals of rational functions by combining the techniques in standard 17.0 with the algebraic techniques of partial fractions and completing the square.  tools-icon
20.0 Students compute the integrals of trigonometric functions by using the techniques noted above.  tools-icon
21.0 Students understand the algorithms involved in Simpson’s rule and Newton’s method. They use calculators or computers or both to approximate integrals numerically.  tools-icon
22.0 Students understand improper integrals as limits of definite integrals.  tools-icon
23.0 Students demonstrate an understanding of the definitions of convergence and divergence of sequences and series of real numbers. By using such tests as the comparison test, ratio test, and alternate series test, they can determine whether a series converges.  tools-icon
24.0 Students understand and can compute the radius (interval) of the convergence of power series.  tools-icon
25.0 Students differentiate and integrate the terms of a power series in order to form new series from known ones.  tools-icon
26.0 Students calculate Taylor polynomials and Taylor series of basic functions, including the remainder term.  tools-icon
27.0 Students know the techniques of solution of selected elementary differential equations and their applications to a wide variety of situations, including growth-and-decay problems.  tools-icon
PDF-iconCalculus State Standards PDF