Make it easier to study later on by summarizing your notes. You will be taught and tested on the concepts highlighted by these questions. Creating FLASHCARDS with an answer and an example on each card will support you as concepts build up in this course. Also, just go though and answer each question to study in general.
- What is a function? Draw a graph that represents a function and one that does not represent a function. Explain why.
- What is domain? When you want to find the domain of a function what should you think about?
- What is an even or odd function? How do you verify if a function is even or odd?
- How do you find the points of intersection of the graphs of two functions?
- How do you find x and y intercepts?
- What would you do if you have to find roots of functions, zeroes of functions or solutions of functions?
- What is the point slope form of the equation of the line?
- How do the slopes of parallel and perpendicular lines compare?
- What is the vertical line test and how is it used?
- How do you evaluate f+g, f-g, f*g, f/g, and f( g(x))?
- How do we sketch the graphs of functions using transformations?
- What is the definition of the limit? Give an example on how to read/estimate the limit of a function from a graph or a table.
- How can a limit fail to exist?
- Note some examples of how different techniques are used to evaluate limits - Special Trig limits, rationalizing technique, factoring technique.
- How do you find limits of piece-wise defined functions?
- Create a table of basic trig function values ( 0o, 30o 45o….) Remember these.
- What does it mean for a function to be continuous? Conditions?
- What is a removable/ non-removable discontinuity?
- How can you check the continuity of a function at a point? Find the value of c that makes a piecewise defined function continuous.
- How do you evaluate one sided limits? Give an example.
- How do you find the equations of vertical asymptotes? Using limits?
- How do you find horizontal asymptotes of a function? What limits do you evaluate?
- What is a difference quotient?
- How can the secant lines be used to approximate the slope of the tangent line graphically?
- How do you find the derivative using the limiting process?
- What is the relationship between continuity and differentiability?
- How do you find the derivative at a given point (x,y)?
- How do you find the equation of the tangent line?
- What is a normal line and how do you find its equation?
- How can you find the x-values of horizontal tangents of a function?
- How can you find the x values of vertical tangent of a function?
- Note down the derivative rules.
- How can you show if a piecewise defined function is differentiable where it splits?
- What are the derivatives of trigonometric functions? Know the basic trig values from the table.
Applications of derivatives
- What do you think about when drawing the graph of f ‘ given the graph of f ?
- What does the graph of f’ tell about f? Give an example on how to read the graph.
- What does the graph of f’’ tell about f? Give an example on how to read the graph.
- How do you find critical values of f(x)?
- How to you find the intervals where f(x) is increasing or decreasing?
- What does the first derivative test say about finding relative extrema?
- What does concave up and concave down mean graphically? How do you find concavity on a given interval?
- How do you find points of inflection?
- What does the second derivative test say? What do you do when it fails?
- What is the Intermediate values theorem? Explain it graphically.
- What is the Extreme Value Theorem? How is it used to find absolute extrema?
- What is the Rolle’s Theorem? Can you explain it graphically too.
- What does the Mean Value Theorem say? Explain it graphically too.
- How do we find velocity and acceleration if the position function is given?
- How do we find the average rate of change of a function? Average velocity?
- What do we mean by instantaneous rate of change? What does it have to do with the derivative? Instantaneous velocity?
- How are speed and velocity related?
- How can you determine when an object is moving forward or backwards or standing still?
- How do you find the velocity or acceleration of an object given the graph of the position function?
- Learn how to analyze the position, velocity and acceleration graphs.
- What can we say about the speed of an object if velocity and acceleration have the same signs/ opposite signs?
- Can you determine the total distance traveled by a particle from the position graph?
- What is an antiderivative? (indefinite Integral)
- Does a function have a unique antiderivative? Explain.
- Learn the basic integration rules. See page 250.
- How can you check the result of a indefinite integral?
- How are areas under graphs approximated by sums?
- What is an upper sum? Lower sum? Riemann Sum?
- What is the difference between a definite integral and an indefinite integral?
- Why do we say that the definite integral represents “signed area” and NOT geometric area?
- What does the first fundamental theorem of calculus say in your own words? (And how do you use it?)
- What does the Mean Value Theorem for Integrals say?
- How can you find the average value of a function over an interval?
- What does the Second Fundamental Theorem say?
- How do you use the u-substitution method to evaluate integrals?
- How does knowing whether a function is even or odd help us in integrating it?
- What is the trapezoidal sum? Simpson’s rule (BC)?
Logarithmic, Exponential, and other transcendental functions – Derivatives and Integrals
- How is the natural logarithmic function defined using the integral?
- What are the properties of the natural log function?
- What is the definition of the number e?
- How do you find the derivative of the natural log function?
- What is logarithmic differentiation?
- What is the log rule for integration?
- How do you compute the derivative of the inverse function?
- How do you integrate and differentiate exponential functions and logarithmic function with other bases?
- Describe the difference between a general solution and a particular solution?
- What is a slope field and how do you interpret it?
- How is the Euler’s method used to approximate the solution of a differential equation? (BC)
- Describe how to recognize and solve differential equations that can be solved by separation of variables?
- How do you model exponential growth and decay with differential equations?
- What situations are modeled by the logistic differential equation? (BC)
Applications of Integration
- How do you use integrals to find the area bounded by two curves?
- How do you use integrals to find the volumes of solids?
- Describe how Disk Method, Washer Method, and Shell Method works?
- How can you find the length of a curve? (BC)
More methods for Integration
- How do you find the antiderivative using integration by parts?
- What is partial fraction decomposition and how is it used to find antiderivatives?
- What are improper integrals?
- Define the terms converges and diverges when working with improper integrals?
- When and how do you use the L’hopital’s rule?
Sequences and Series
- What is the difference between a sequence and a series?
- What does converges or diverges mean?
- How do you check if a sequence converges or diverges?
- What are the tests by which you can check the convergence or divergence of a series? (know the conditions of each test (page 644))
- What is the alternating series remainder?
- How can you find the radius and interval of convergence using ratio and root tests?
- What is a taylor polynomial? Taylor series/ Maclauren series?
- How can you compute the Langrange error bound or the remainder for Pn taylor polynomial?
- What is a power series? How do you express a function by a power series?