Limits
- Use graphical and numerical evidence to estimate limits and identify
situations where limits fail to exist.
- Apply rules to calculate limits.
- Use the limit concept to determine where a function is continuous.
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Derivatives
- Use the limit definition to calculate a derivative, or to determine
when a derivative fails to exist.
- Calculate derivatives (of first and higher orders) with pencil and
paper, without calculator or computer algebra software, using:
- Linearity of the derivative;
- Rules for products and quotients and the Chain Rule;
- Rules for constants, powers, trigonometric and inverse trignometric
functions, and for logarithms and exponentials.
- Use the derivative to find tangent lines to curves.
- Calculate derivatives of functions defined implicitly.
- Interpret the derivative as a rate of change.
- Solve problems involving rates of change of variables subject to a
functional relationship.
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Applications of the Derivative
- Find critical points, and use them to locate maxima and minima.
- Use critical points and signs of first and second derivatives to
sketch graphs of functions:
- Use the first derivative to find intervals where a function is
increasing or decreasing.
- Use the second derivative to determine concavity and find
inflection points.
- Apply the first and second derivative tests to classify critical
points.
- Use Differential Calculus to solve optimization problems.
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The Integral
- Find antiderivatives of functions; apply antiderivatives to solve
separable first-order differential equations.
- Use the definition to calculate a definite integral as a limit of
approximating sums.
- Apply the Fundamental Theorem of Calculus to evaluate definite
integrals and to differentiate functions defined as integrals.
- Calculate elementary integrals with pencil and paper, without
calculator or computer algebra software, using:
- Linearity of the integral;
- Rules for powers (including exponent -1) and exponentials, the
six trigonometric functions and the inverse sine, tangent and
secant;
- Simple substitution.
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Transcendental Functions
- Use the relation between the derivative of a one to one function and
the derivative of its inverse.
- Calculate with exponentials and logarithms to any base.
- Calculate derivatives of logarithmic, exponential and inverse
trigonometric functions. Use logarithmic differentiation.
- Use models describing exponential growth and decay.
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