AP CalculusAB 1st Semester Exam Review Sheet (Baggett 2012)
Your Calculus Exam is on Tuesday, December 12, 2010, at 12:30 pm. C Block is in room A-3 and F-Block is in A-4. Remember to bring at least two pencils with erasers and your calculator with new or recently charged batteries.
The tests that you have had this term are Test Chapter 1, Test 2.1-2.3, 2.5-2.6, Test 3.1- 3.2, and Test on 3.6-3.7; 4.1-4.3, 4.5-4.6. There will be no precalculus questions on the exam – but you will use precalculus skills to complete many calculus problems. Major Quizzes were: Quiz 1.1, Quiz 2.2-2.3, limit derivative take-home quiz, Quiz 3.1-3.2, & quiz 3.5-3.6. There were also some hw quizzes and worksheets that were assigned – make sure you go through my answer keys – and have the corrections to your quizzes, tests, and worksheets – prior to the weekend – this will make your studying time more efficient.
The exam will be in two parts.
Part I is multiple choice and “short answer” and noncalculator. You need to know derivatives of polynomials, trig functions, inverse trig functions, y = ex, y = lnx, and be able to use these along with your rules for derivatives of a sum, difference, product, quotient, chain rule, inverse of a function, implicit differentiation, and logarithmic differentiation. Also be able to write an equation of a line in point-slope form. Precalculus facts that will help include unit circle values in radians, graphs of ,and the basic graphs of
You also need know the following theorems (know the hypothesis, conclusion, and a geometric sketch to illustrate the theorem when appropriate): Intermediate Value Theorem, Rolles Theorem, and Mean Value Theorem. Definitions to know include continuity at a point and over an interval; both definitions of the derivative (the limit definitions) and be able to use them; critical number; point of inflection (or inflection point). And, don’t forget the First Derivative Test and the Second Derivative Test. Make a list of these things tonight (or at least several days prior to the exam) and start reviewing them if you do not remember them.
Part II is calculator and show any necessary work. For this part, make sure you can do the following with a calculator:
1. Use calculator to find zeros and maximums and minimums – using a function or its derivative.
2. Solve an equation by graphing—using “calculate zero” or “calculate intersection”.
3. Use the table feature.
Review of concepts covered to date
AP Calculus AB Test 1stuff
Precalculus material is not on the exam except for when it applies to calculus—such as writing the equation of a tangent line.
Chapter 1 test covered the following:
Precalculus
Lines: (Used to write equations of tangent and normal lines.)
· Determine the slope of a line given two points;
· Given two points or a point and the slope, write the equation of the line in point-slope form: note: I gave you the point slope form as . Know how to write the equation of a vertical line & horizontal line.
· Be able to change from point-slope form to slope-intercept or general form
· Write an equation of a line given information about the line and a line parallel or perpendicular to it. For instance, the line that is parallel to and has the same y-intercept as the line 3x + 5y = 10.
Functions and Graphs: (this stuff will make the calculus questions easier in many cases. i.e., knowing the domain of the function
· Given a function or relation, name the domain and, in some cases, the range.
· Given a function in equation form, determine whether the function is odd, even, or neither without the aid of a graphing calculator.
· Graph the basic functions – including and
· Given the graph of a function, determine whether it is odd, even, or neither.
· Determine whether a relation has symmetry with respect to x-axis, y-axis, or origin given either an equation or the graph.
· An odd function has what kind of symmetry? An even function has what kind of symmetry?
· Graph a piecewise function.
· Write an equation for a piecewise function given its graph.
· Write a formula for a given function, such as: write a formula that expresses the area of a rectangle in terms of its width if the perimeter is known to be 50; or write a formula that expresses the weekly wages of a person as a function of the number of hours worked if the pay is $5.50/hour.
· Composition of functions
Calculus Begins:
Limits:
· Given the graph of a function, determine the value of one–sided and two-sided limits.
· Determine the limit of a function without graphing using direct substitution, factor and cancel, rationalizing the denominator or numerator, or the fact that .
Limits (continued)
*These you should be able to do analytically—without a graphing calculator.
Analytically determine the limit of a function as . You might have to factor and cancel, rationalize the numerator, rationalize the denominator, simplify a complex fraction, etc.
Determine whether a function goes to infinity or negative infinity if the limit does not exist.
Determine the limit of a function as . Remember you can sometimes use
“end behavior” here.
Know , .
Determine limits involving piece-wise functions.
Determine vertical asymptotes of a function and the behavior to the left and to the right of
the asymptotes using limits.
Determine horizontal asymptotes of a function using limits. [You need to explore
]
Limits involving ex, lnx, ax and ends of polynomial functions. That is, which one increases the fastest or
has the steepest slope on the ends.
Know the addition, subtraction, multiplication, division, and root rules for limits—be able to use them, not state them. [That is, , etc.]
Given a graph, determine limits as x approaches certain values from the left or right. The graph
will include open circles, asymptotes, discontinuous functions, etc. Don’t miss easy points on this section!!
Find the limit using a graph and/or table of values.
Given information about limits, draw a possible graph for the function.
Know the three-part definition for continuity at x = a that I gave you in your notes. Be able to use the definition to explain why a function is discontinuous at a point. Also know the differences among a jump, removable, and infinite discontinuity.
Use the limit definition for the (instantaneous) rate of change of a function to determine the rate of change at a given point. [Find the rate of change of at x = 7.]
Differentiation (Derivatives)
Ch 3 Sections
1. Know the 2 limit definitions for the derivative. Be able to use them to determine the derivative of a function at a value of x , such as at x = 3, and/or the general equation f ¢(x).
2. Be able to answer questions about f ¢(x) given a graph of f; answer questions about f ¢ given a graph of f.
3. Given a graph of f, sketch a graph of f ¢, and vice versa.
4. Given a table of values for a function or for its derivative, draw a graph of the derivative or of the function.
5. Find the average rate of change of a function over an interval; find the instantaneous rate of change of a function at a given value of the domain. Very beginning stuff.
6. Write the equations of the tangent and normal lines at a given value of the domain.
7. Given the position function, s(t ), for a particle moving along a line, determine
(a) the velocity and acceleration functions;
(b) the displacement over a given time period;
(c) the average velocity over a given time period;
(d) the intervals over which the particle is moving to the left, right, and changing direction;
(e) the intervals over which the particle is speeding up and slowing down,
etc.
(f) make sure you can draw the table that shows t, v, and a.
(g) draw the number line graph showing the position and movement of the particle.
8. Find derivatives involving the 6 trig functions.
9. Knowing the following is helpful:
****10. Find derivatives of functions involving power rule, product rule, quotient rule, and chain rule.
11. Determine rate of change, slopes of tangent & normal lines, equations of tangent and normal lines, and velocity & acceleration functions using functions that involve trig, chain rule, etc.
12. Use implicit differentiation to determine the derivative when the equation cannot be solved or is not easy to solve for y in terms of x. This involves trig functions as well.
13. Determine and simplify using implicit differentiation.
14. Problem like #25 in section 3.5.
15. Determine slopes of tangents at points on implicitly defined curves. Applications of these.
Sections 4.1-4.3; 4.5-4.61. Analytically determine absolute (global) maximum and minimum of a function over a closed interval. A theorem assures us that if the function is continuous on a closed interval, there must be an absolute maximum and absolute minimum value. Remember the max and mins are y values.
2. State Rolle’s Theorem and draw a picture to illustrate it
3. State the Mean Value Theorem (MVT) and draw a picture to illustrate it.
4. Given a function and a closed interval
(a) determine whether or not the MVT holds.
(b) if it does, find the “c” value that is guaranteed by the MVT.
5. State the Intermediate Value Theorem and find the “c” value(s).
6. Determine the critical numbers of a function. These are the values c in the domain of the function for which the derivative is zero or does not exist.
7. Determine the local maxima and minima, intervals where function is increasing, and intervals where function is decreasing using 1st Derivative Test. Make sure you understand the 1st Derivative Test.
8. Determine the points of inflection and where function is concave up and concave down using the 2nd derivative.
9. Given a function, find relative extrema, points of inflection, intervals where function is increasing, decreasing, concave up and concave down using both the 1st and 2nd Derivatives Tests. If the 2nd derivative is easy to find, use the Second Derivative Test. If it is difficult to find, use the First Derivative Test. Don’t forget—if the second derivative is 0 at a critical number, the 2nd Derivative Test fails—that is, no conclusion can be drawn. The critical number is not necessarily a point of inflection. If c is a point of inflection, then . The converse is not always true.
10. Given a graph of , answer questions about the graph of f.
11. Given a graph of , answer questions about the graph of f.
12. Know how to sketch a function – using the tools that you have been taught to this point (i.e. domain, asymptotes, 1st and 2nd derivatives, etc.). Sketches of the functions will help answer numerous questions.
AGAIN – be able to to find the derivatives of functions involving the following techniques/tools:
1. Use the Power Rule for all values of the power:
2. Use the Sum, Product, and Quotient Rules:
3. Use the Chain Rule:
4. Derivatives of the trig functions, including product, quotient, and chain rules:
5. Derivatives of the inverse trig functions, including product, quotient, and chain rules.
6. Derivatives of using the product, quotient, chain rules, etc.
7. Implicit differentiation:
9. Other good things to remember:
; ;
Do problems!!!!! Go through tests, quizzes, and worksheets. If you messed up on something on an earlier quiz or test – figure out why you messed up, and fix it.