Calculus 1

Review Material | Transcendental Functions | Limits | Differentiation | Applications of Derivatives | Integration

The Mathematics Department has a standard of performance expected of calculus students. The following pages describe in detail what you need to be able to do and what you need to understand to receive an A in Calculus 1. This provides an excellent checklist you can use each day to make sure you have learned the material well enough. Before an exam you can use the list of expectations as a checklist to make sure you have reviewed everything.

- Real Numbers
- Be able to state the density property of rationals and irrationals.
- Absolute Value
- Be able to state the definition of absolute value. Be able to use its properties. Be able to use the distance interpretation of absolute value for solving inequalities.
- Analytic Geometry
- Know the distance and midpoint formulas in the plane and the equation of a circle.
- Lines
- Know and be able to find the slope of a line, the point-slope form, slope-intercept form, and the general linear equation form for the equation of a line. Be able to determine whether lines are parallel or perpendicular by using their slopes.
- Functions
- Be able to state the definition of a function. Know how to find the domain, range, and codomain. Be able to state the definition for 1-1. Know how to check for 1-1 using the equation for a function or using horizontal line test. Be able to state the definition for onto and know how to check for onto using the equation for a function or by looking at its graph.
- Classification of Functions
- Know the definitions of algebraic and transcendental functions, polynomials, rational functions, even/odd functions. Given an equation for a function be able to classify it.
- Trigonometry
- You must know and be able to do all of the following on exams. So do not rely on calculators or computers to do this in your daily homework. You need the practice!
- Memorize the radian measure of 30, 45, 60, 90, 180, 270, 360 degree angles.
- Be able to mentally evaluate all six trig functions for these angles.
- Memorize the lengths of sides of 45-45-90 and 30-60-90 triangles.
- Be able to define the trig functions using 1) right triangles and 2) unit circle.
- Memorize the basic trig identities.
- Be able to sketch the graphs of sin, cos, and tan.

- Inverse Functions
- State the defining relationship for an inverse function and identify its domain and range. Illustrate the reflective property of the graph of f
^{-1}by drawing its graph along with the graph of f. Given a defining expression for y = f(x), find the defining expression x = f^{-1}(y). State the definition of a 1-1 function. Determine whether a function is 1-1 by applying the definition of 1-1 to the defining expression of the function and by applying the horizontal line test to its graph. - Inverse Trig Functions
- State the property defining each of the six inverse trig functions. State this in words, for example, y = arcsin x means y is the angle whose sin is x. Memorize the range of each inverse trig function; this range is the restricted domain that makes the corresponding trig function 1-1. Evaluate inverse trig functions.
- Definition of e
- State the definition of e as a limit.
- Exponential Function Base e
- Sketch the graph of e
^{x}and describe its properties. Identify and use the basic properties of the exponential function, in particular its composition with ln x. Solve exponential equations. - Natural Logarithm
- Sketch the graph of ln x and describe its properties. Recognize and apply the basic algebraic properties of the logarithm.
- Bases Other Than e
- Identify and apply the basic properties of a
^{x}and log_{a}x.

- Definition
- Be able to state the epsilon-delta definition of limit.
- **Evaluation Techniques
- Be able to compute limits using direct substitution, algebraic properties, cancellation of factors, squeeze theorem, special trig limits. Be able to identify indeterminate forms, be able to choose an appropriate evaluation technique and use it to find the limit.
- One-Sided Limits
- Know the notation for one-sided limits and how to compute them. Be able to determine limits that equal infinity and know how to find vertical asymptotes.
- Continuity
- Be able to state the limit definition of continuity. Be able to identify those x's in the domain of a function where the function is continuous and discontinuous by 1) looking at the graph of the function and 2) analyzing the equation(s) for the function. Be able to identify removable and nonremovable discontinuities on graphs by 1) looking at the graph of the function and 2) analyzing the equation(s) for the function. Know the algebraic properties of continuity and how to determine continuity for composite functions
- Intermediate Value Theorem
- Be able to state the Intermediate Value Theorem (know its hypotheses and conclusion) and know how to apply it, for example, for finding zeros of functions.

- Definition
- State the definition of derivative (as a limit of a difference quotient).
- State the definition of left and right derivatives and be able to evaluate each of them as a limit or using a graph.
- Draw the picture that accompanies the definition of derivative (curve, secant line, tangent line) and explain how the slopes of secant lines are related to the slope of the tangent line in the above definitions.
- Know the various interpretations of the derivative (slope of tangent line, limit of slopes of secant lines, and rate of change of a variable with respect to another variable.
- Relationship to Continuity
- Know the theorem: "differentiable implies continuous." Remember that the converse ("continuous implies differentiable") is false, the absolute value function providing a counter-example.
- Non-Differentiability
- Be able to identify points of non-differentiability on graphs (cusps, vertical tangents, discontinuities).
- Mechanics
- Be able to describe how the derivative is related to the study of rectilinear motion (position function, velocity, acceleration, speed [absolute value of velocity]). Know the picture that defines instantaneous velocity and how to interpret it.
- Higher Derivatives
- Know the definition and notation for higher derivatives and be able to compute them.
- **Differentiation Rules
- Be able to use the differentiation rules: sums, constant multiples, power rule, product rule, quotient rule, chain rule.
- **Trigonometric Functions
- Memorize and be able to use the derivatives of the six trig functions together with the differentiation rules above.
- **Exponential and Log Functions
- Be able to differentiate e
^{x}, ln x, ln |x|, a^{x}, and log_{a}x together with the differentiation rules above. Apply the process of logarithmic differentiation. - Implicit Differentiation
- Know when to use implicit differentiation (when it is difficult or impossible to solve explicitly for y as a function of x) and how to perform implicit differentiation.
- **Derivatives of Inverse Functions
- State and apply the inverse function theorem for a real-valued function of one variable, namely, differentiate f
^{-1}. Be able to differentiate the inverse trig functions. - Related Rates
- Know the procedure for solving related rates problems and be able to apply it.
- Newton's method
- Be able to derive the formula for Newton's method from a diagram (which you should be able to produce). Be able to carry out one step in Newton's method. Know the ways Newton's method can fail.

- **Sketching graphs
- Be able to sketch the graph of a function from a list of properties involving first and second derivatives and vertical and horizontal asymptotes.
- Extrema of functions
- Know the definitions critical points, relative and absolute extrema and how to find these points on the graph of a function.
- Non-Differentiability
- Be able to identify points of non-differentiability on graphs (cusps, vertical tangents, discontinuities).
- Rolle's Theorem, Mean Value Theorem
- Know the hypotheses and conclusion of each theorem. Be able to draw a diagram illustrating the conclusion of each theorem. Given an equation or graph of a function, be able to determine if the hypotheses of these theorems are satisfied. Be able to prove simple applications of the mean value theorem as given in lecture.
- **First Derivative
- Know how to use the first derivative to find intervals where f is increasing or decreasing. Know how to apply the first derivative test. Be able to state the definitions of monotonic and strictly monotonic functions and how to determine whether a function has one of these properties.
- **Second derivative
- Know how to use the second derivative to find intervals of concavity and inflection points. Know how to use the second derivative test to check for relative maxima and minima.
- Limits at infinity
- Be able to evaluate limits of functions as x approaches infinity or negative infinity, and to sketch the graphs of functions having horizontal asymptotes.
- Optimization problems
- Be able to analyze, set up, and solve max-min problems for functions of one variable.
- Differentials
- Be able to state the linear approximation equation (including the limit for the error term). Be able to reproduce the diagram from which this equation comes. Know what the notation for x, y, dx, and dy means. Be able to use the differential to approximate differences (delta y) and relative error.

- Indefinite integrals
- Be able to use basic integration properties to find antiderivatives. This includes antiderivatives of polynomials, trig functions, and exponential functions. Remember to include an arbitrary constant.
- Definite integrals
- Be able to interpret and use sigma notation for sums and memorize basic summation formulas. Set up and evaluate a definite integral as the limit of a Riemann sum. Use integral properties to evaluate a definite integral. In particular, exploit properties of even and odd functions when integrating over an interval centered at the origin. Understand a definite integral as (positive and negative) area bounded by a curve.
- **Fundamental theorems of calculus
- Be able to use the fundamental theorem and second fundamental theorem of calculus. Apply and interpret the mean value theorem for integrals. Compute the average value of a function.
- **u-Substitution
- Recognize an integrand which is susceptible to u-substitution. Perform all the steps of a u-substitution including adjustment of limits of integration for definite integrals.
- Numerical integration
- Be able to set up and evaluate the expressions for the trapezoidal rule and Simpson's rule to approximate the value of a definite integral. Compute error bounds for the trapezoidal rule and Simpson's rule.
- Transcendental Functions
- Be able to define ln x as an integral. Be able to use ln x as an antiderivative for integration purposes, in particular, to integrate tan x, cot x, sec x, and csc x. Be able to use the inverse trig functions for integration.