AP Calculus Premium: With 12 Practice Tests

This book is intended for students who are preparing to take either of the two Advanced Placement examinations in Mathematics offered by the College Board, and for their teachers. It is based on the curriculum framework published by the College Board, effective Fall 2019, and covers the topics listed there for both Calculus AB and Calculus BC.

THE COURSES

Calculus AB and BC are both full-year courses in the calculus of functions of a single variable. Both courses emphasize:

(1) student understanding of concepts and applications of calculus over manipulation and memorization;

(2) developing the student’s ability to express functions, concepts, problems, and conclusions analytically, graphically, numerically, and verbally, and to understand how these are related; and

(3) using a graphing calculator as a tool for mathematical investigations and for problem-solving.

Both courses are intended for those students who have already studied college-preparatory mathematics: algebra, geometry, trigonometry, analytic geometry, and elementary functions (linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise).

TOPIC OUTLINE FOR THE AB AND BC CALCULUS EXAMS

The AP Calculus course topics can be arranged into four content areas: 1. Limits, 2. Derivatives, 3. Integrals and the Fundamental Theorem, and 4. Series. The AB exam tests content areas 1, 2, and 3. The BC exam tests all four content areas. There are BC-only topics in content areas 2 and 3, as well. Roughly 40 percent of the points available for the BC exam are BC-only topics.

Content Area 1: Limits

Limits are used in many calculus concepts to go from the discrete to the continuous case. Students must understand the idea of limits so that a deeper understanding of definitions and theorems can be achieved. Students must be presented with different representations of functions when calculating limits. Working with tables, graphs, and algebraically defined functions with and without the calculator are essential skills that students need to master.

I.Understanding the behavior of a function

A.Limit—writing and interpreting

1.Limit definition—existence versus nonexistence

2.Writing limits using correct symbolic notation

a.One-sided limits

b.Limits at infinity

c.Infinite limits

d.Nonexisting limits

B.Estimating limits

1.Graphical and numerical representations of functions may be used to estimate limits

C.Calculating limits

1.Use theorems of limits to calculate limits of sums, differences, products, quotients, and compositions of functions

2.Use algebraic manipulation, trigonometric substitution, and Squeeze Theorem

3.L’Hospital’s Rule may be used to evaluate limits of indeterminate forms 00and ∞∞

D.Function behavior

1.Limits can be used to explain asymptotic (vertical and horizontal) behavior of functions

2.Relative rates of growth of functions can be compared using limits

II.Continuity of functions

A.Intervals of continuity and points of discontinuity

1.Definition of continuity

2.Some functions are continuous at all points in their domain

a.Polynomials

b.Rational functions

c.Power functions

d.Exponential functions

e.Logarithmic functions

f.Trigonometric functions

3.Types of discontinuities

a.Removable

b.Jump

c.Vertical asymptotes

B.Continuity allows the application of important calculus theorems

1.Continuity is a condition for the application of many calculus theorems

a.Intermediate Value Theorem

b.Extreme Value Theorem

c.Mean Value Theorem

Content Area 2: Derivatives

The derivative describes the rate of change. The concept of the limit helps us develop the derivative as an instantaneous rate of change. Many applications rely on the use of the derivative to help determine where a function attains a maximum or a minimum value. Again, students must be presented with different representations of functions and use different definitions of the derivative when calculating and estimating derivatives. Working with tables, graphs, and algebraically defined functions with and without the calculator are essential skills that students need to master.

I.The derivative as the limit of a difference quotient

A.Identifying the derivative

1.Difference quotients give the average rate of change on an interval; common forms include f(a+h)−f(a)h andf(x)−f(a)x−a

2.Instantaneous rate of change at a point is the limit of the difference quotient, provided it exists; f′(a)=limh→0f(a+h)−f(a)hor f′(a)=limx→af(x)−f(a)x−a

3.The derivative of the function f is given by f′(x)=limh→0f(x+h)−f(x)h

4.Various notations for the derivative of a function y = f(x) include dydx,f′(x), and y′

5.The derivative can be given using any of the representations in the rule of four: graphically, numerically, analytically, and verbally

B.Estimating the derivative

1.Tables and graphs allow the estimation of the derivative at a point

C.Calculating the derivative

1.Apply the rules for differentiating families of functions

a.Polynomial, rational, power, exponential, logarithmic, trigonometric, and inverse trigonometric

2.Differentiation rules can be used to find the derivatives of sums, differences, products, and quotients of functions

3.Chain Rule

a.Composite functions can be differentiated with the Chain Rule

b.Implicit differentiation

c.The derivative of an inverse function

4.Parametric, vector, and polar functions can be differentiated using the methods described above (BC only)

D.Higher-order derivatives

1.Differentiating the first derivative produces the second derivative, differentiating the second derivative produces the third derivative, and so on, provided these derivatives exist

2.Notations for higher-order derivatives: second derivatives d2ydx2,f″(x),and y″; third derivatives d3ydx3,f‴(x),and y‴; higher than third derivatives dnydxnor f(n)(x), where n is the number of the derivative

II.Using the derivative of a function to determine the behavior of the function

A.Analyzing the properties of a function

B.Connecting differentiability and continuity

III.Interpreting and applying the derivative

A.The meaning of a derivative

1.Instantaneous rate of change with respect to the independent variable

2.The units for the derivative of a function are the units of the function over the units for the independent variable

B.Using the slope of the tangent line

1.The slope of the line tangent to a graph at a point is the derivative at that point

2.The tangent line provides a local linear approximation of function values near the point of tangency

C.Solving problems

1.Related rates

a.Find the rate of change of one quantity by knowing the rate(s) of change of related quantities

2.Optimization

a.Finding the maximum or minimum value of a function on an interval

3.Rectilinear motion

a.Using the derivative to determine velocity, speed, and acceleration for particles moving along a line

4.Planar motion (BC only)

a.Using the derivative to determine velocity, speed, and acceleration for particles moving along a curve defined by parametric or vector functions

D.Differential equations

1.Verify a function is a solution to a given differential equation using derivatives

2.Estimate solutions to differential equations

a.Slope fields allow the visualization of a solution curve to a differential equation; students may be asked to draw a solution curve through a given point on a slope field

b.Euler’s method provides a numerical method to approximate points on the solution curve for a differential equation (BC only)

IV.The Mean Value Theorem (MVT)

A.If a function is continuous on the closed interval [a,b] and is differentiable on the open interval (a,b), then MVT guarantees the existence of a point in the open interval (a,b) where the instantaneous rate of change is equal to the average rate of change on the interval [a,b]