New AP Calculus Study Review

What’s included in the AP Calculus AB Examination

Limits and Continuity: Determining types of discontinuities, comprehending asymptotes, applying the Squeeze Theorem, defining limits, estimating limits from graphs and tables, determining limits using algebraic properties and manipulation, and using the Intermediate Value Theorem

Differentiation: Definition and Fundamentals—Estimating derivatives at a point, defining the derivative of a function, establishing the connection between differentiability and continuity, utilizing the Power, Product, and Quotient Rules, and calculating the derivatives of constants, sums, differences, and constant multiples, trigonometric functions, ex, and ln x

Differentiation: Inverse, Composite, and Implicit Functions—Using the Chain Rule, Differentiating Inverse Functions, Implicit Differentiation, and Computing Higher Order Derivatives

 Contextual Applications of Differentiation: Using rates of change in motion and other context, applying related rates, approximating using linearization, and applying L'Hospital's Rule are some examples of how to interpret derivatives in context.

The Mean Value Theorem, the Extreme Value Theorem, finding global and local extrema, applying the First and Second Derivative Tests, identifying intervals of increase and decrease, comprehending concavity, creating graphs, addressing optimization issues, and utilizing implicit relations are some of the analytical applications of differentiation.

Integration and Accumulation of Change—Identifying Reimann sums, definite integrals, accumulations of change, comprehending the Fundamental Theorem of Calculus, deciphering accumulation functions, locating anti-derivatives, and indefinite integrals; integrating through long division, substitutions, and square root

Differential Equations: Using separation of variables, sketching slope fields, confirming differential equation solutions, and modeling situations with differential equations

Uses of Integration: determining a function's average value, utilizing integrals to relate location, velocity, and acceleration, applying accumulation functions, determining the area between function curves, and calculating volumes from cross-sections and rotations

AP Calculus AB Grading Rubric

Score	Meaning	Percentage of Test Takers
5	Extremely qualified	20.4%
4	Well qualified	16.1%
3	Qualified	19.1%
2	Possibly qualified	22.6%
1	No recommendation	21.7%

 Important Concepts Covered

Critical Point

f'(x) = 0 or undefined (and endpoints on closed interval)

local minimum

f'(x) goes from - to +

local maximum

f'(x) goes from + to -

point of inflection

- concavity changes

- f''(x) goes from - to + or + to -

d/dx(xⁿ)

nxⁿ⁻¹

 

 

d/dx(sinx)

cosx

 

 

d/dx(cosx)

-sinx

 

 

d/dx(tanx)

sec²x

 

 

d/dx(cotx)

-csc²x

 

 

d/dx(secx)

secxtanx

 

 

d/dx(cscx)

-cscxcotx

 

 

d/dx(lnn)

(1/n)(dn/dx) [w/ n acting as "u"]

 

 

d/dx(eⁿ)

(eⁿ)(dn/dx) [w/ n acting as "u"]

 

 

d/dx [arcsin(u/a)]

(1/√a²-u²)(du/dx)

 

 

d/dx[arccosx]

-1/√1-x²

 

 

d/dx[arctan(u/a)]

(a/a²+u²)(du/dx)

 

 

d/dx[arccotx]

-1/1 + x²

 

 

d/dx[arcsec(u/a)]

(a/|u|√u²-a²)(du/dx)

 

 

d/dx[arccscx]

-1/|x|√x²-1

 

 

d/dx(aⁿ)

(aⁿ)(lna)(dn/dx) [w/ n acting as "u"]

 

 

d/dx(log₀x)

1/xln0 [w/ 0 acting as "a"]

 

 

Chain rule: d/dx[f(u)]

f'(u)(du/dx)

 

 

Product rule: d/dx(uv)

u'v + uv'

 

 

Quotient rule: d/dx(u/v)

(u'v - uv')/v²

 

 

Fundamental Theorem of Calculus

∫f(x)dx from a to b = F(a) - F(b)

[where F'(x) = f(x)]

 

 

Intermediate Value Theorem

If the function f(x) is continuous on [a,b], and y is a number between f(a) and f(b), then there exists at least one number x = c in the open interval (a,b) such that f(c) = y

 

 

Mean Value Theorem

If the function f(x) is continuous on [a,b], and the first derivative exists on the interval (a,b), then there exists at least one number x = c in the open interval (a,b) such that f'(c) = [f(b) - f(a)]/b-a

 

 

Rolle's Theorem

If the function f(x) is continuous n [a,b], and the first derivative exists on the interval (a), and f(a) = f(b), then there is at least 1 number x = c in (a,b) such that f'(c) = 0

 

 

Average Value of a Function

f(c) = (1/b-a)∫f(x)dx on a to b

 

 

Disk Method

V = π∫[R(x)]² on x=a to x=b

 

 

Washer Method

V = π∫([R(x)]²-[r(x)]²)dx

 

 

General Volume equation (not rotated)

V = ∫Area(x)dx on a to b

 

 

velocity

d/dt(position)

 

 

acceleration

d/dt(velocity)

 

 

displacement

∫vdt from 0 to t

 

 

distance

∫|v|dt from initial time to final time

 

 

average velocity

∆x/∆t

 

 

L'Hopital's Rule

If f(x)/g(b) = 0/0 or ±∞/±∞, then the limit as x approaches a of f(x)/g(x) = the limit as x approaches a of f'(x)/g'(x)

 

 

sin2x

2sinxcosx

 

 

cos2x

cos²x - sin²x or 1-2sin²x

 

 

cos²x

1/2(1 + cos2x)

 

 

sin²x

1/2(1-cos2x)

 

 

sin²x + cos²x

1

 

 

1 + tan²x

sec²x

 

 

cot²x + 1

csc²x

 

 

∫tanxdx

ln|secx| + C or -ln|cosx| + C

 

 

∫secxdx

ln|secx + tanx| + C

 

Find lim x →∞ f(x), calculator allowed

Use TABLE [ASK], find y values for large values of x, i.e. 999999999999

cos(pi/3)------------------1/2

cos(pi/2)----------------------0

 

cos(pi)----------------------------1