AP CALCULUS PERIODIC REVIEW

1: Limits and Continuity A function y = f(x) is continuous at x = a if: i) f(a) is defined (it exists) ii) iii) Otherwise, f is discontinuous at x = a.

2: Intermediate Value Theorem A function y = f(x) that is continuous on a closed interval [a,b] takes on every value between f(a) and f(b). Note: If f is continuous on [a,b] and f(a) and f(b) differ in sign, then the equation f(x) = 0 has at least one solution in the open interval (a,b). a b f(a) f(b)

3: Limits of Rational Functions as x 

Note: The limit will be the ratio of the leading coefficient of f(x) to g(x).

4: Horizontal and Vertical Asymptotes

5: Average Rate vs. Instantaneous Rate of Change Average Rate of Change: If (a, f(a)) and (b, f(b)) are points on the graph of y=f(x), then the average rate of change of y with respect to x over the interval [a, b] is: a b f(a) f(b)

5: Average Rate vs. Instantaneous Rate of Change Instantaneous Rate of Change: If (x 0, y 0 ) is a point on the graph of y=f(x), then the instantaneous rate of change of y with respect to x at x 0 is f’(x 0 ). a b f(a) f(b)

6: Limit Definition of a Derivative

AKA Difference Quotient Geometrically, the derivative of a function at a point is the slope of the tangent line to the graph of the function at that point.

7: The Number e is actually a limit

8: Rolle’s Theorem If f is continuous on [a,b] and differentiable on (a, b) such that f(a) = f(b), then there is at least one number c in the open interval (a, b) such that f’(c) = 0. f’(c) = 0.

9: Mean Value Theorem If f is continuous on [a,b] and differentiable on (a, b), then there is at least one number c in the open interval (a, b) such that: a b f(a) f(b)

10: Extreme Value Theorem If f is continuous on a closed interval [a,b], then f(x) has both a maximum and a minimum on [a,b]. a b CONSIDER

11: Max / Min of Functions To find maximum and minimum values of a function y = f(x), locate 1. the point(s) where f’(x) changes sign. To find the candidates first find where f’(x) = 0 or is infinite or does not exist. 2. the end points, if any, on the domain of f(x).

12: Increasing and Decreasing Intervals If f’(x) > 0 for every x in (a, b), then f is increasing on [a, b]. If f’(x) < 0 for every x in (a, b), then f is decreasing on [a, b]. a b

13: Concavity and POI If f’’(x) > 0 for every x in (a, b), then f is concave up [a, b]. If f’’(x) < 0 for every x in (a, b), then f is concave down [a, b]. a b

To locate the points of inflection of y = f(x), find the points where f’’(x) = 0 or where f’’(x) fails to exist. These are the only candidates where f(x) may have a POI. Then test these points to make sure that f’’(x) 0 on the other (changes sign). a b

14: Differentiability and Continuity Differentiability implies continuity: If a function is differentiable at a point x = a, it is continuous at that point. The converse is false, that is, continuity does NOT imply differentiability.

15: Linear Approximation The linear approximation of f(x) near x = x o is given by 11.1

16: Comparing Rates of Change The exponential function y = e x grows rapidly as x  while the logarithmic function y = ln x grows very slowly as x . ln x x2x2 x3x3 3x3x

Exponential functions like y = 2 x or y = e x grow more rapidly as x  than any positive power of x. The function y = ln x grows slower as x  than any nonconstant polynomial.

Another way to look at this, as x  : 1. f(x) grows faster than g(x) if If f(x) grows faster than g(x) as x , then g(x) grows slower than f(x) as x .

Another way to look at this, as x  : 2. f(x) and g(x) grow at the same rate as x  if

17: Inverse Functions 1. If f and g are two functions such that f(g(x)) = x for every x in the domain of g, and, g(f(x)) = x, for every x in the domain of f, then, f and g are inverse functions of each other. ln x exex f(x) = e x g(x) = ln x f(g(x)) = e ln x = x

17: Inverse Functions 4. If f is differentiable at every point on an interval I, and f’(x)  0 on I, then g = f -1 (x) is differentiable at every point of the interior of the interval f(I) and f(x) = e x g(x) = ln x f(g(x)) = e ln x = x f’(g(x)) g’(x) = 1 f’(g(x)) g’(x) = 1/(f’(g(x)))

18: Properties of e x 1. The exponential function y = e x is the inverse function of y = ln x. 2. The domain of y = e x is the set of all real numbers and the range is the set of all positive numbers, y>0.

18: Properties of e x 4. y = e x is continuous, increasing, and concave up for all x. 3.

18: Properties of e x

19: Properties of ln x 1. The domain of y = ln x is the set of all positive numbers, x > x.

19: Properties of ln x 3. y = ln x is continuous and increasing everywhere on its domain.

19: Properties of ln x

20: Trapezoidal Rule If a function, f, is continuous on the closed interval [a, b] where [a, b] has been partitioned into n subintervals of equal length, each length (b – a) / n, then:

21: Properties of the Definite Integral If f(x) and g(x) are continuous on [a, b]:

21: Properties of the Definite Integral If f(x) and g(x) are continuous on [a, b]:

21: Properties of the Definite Integral If f(x) and g(x) are continuous on [a, b]:

21: Properties of the Definite Integral If f(x) is an even function, then

21: Properties of the Definite Integral If f(x) is an odd function, then

21: Properties of the Definite Integral If f(x)  0 on [a, b], then

21: Properties of the Definite Integral If g(x)  f(x) on [a, b], then

22: Definition of a Definite Integral as the Limit of a Sum Suppose that a function f(x) is continuous on the closed interval [a, b]. Divide the interval into n equal subintervals, of length

22: Definition of a Definite Integral as the Limit of a Sum Choose one number in each subinterval, i.e., x 1 in the first, x 2 in the second, …., x i in the i th,…., and x n in the n th. Then:

23: First Fundamental Theorem of Calculus

23: Second Fundamental Theorem of Calculus

24: PVA

The speed of an object is the absolute value of the velocity,  v(t) . It tells how fast it is going disregarding its direction. The velocity of an object tells how fast it is going and in which direction. Velocity is an instantaneous rate of change.

24: PVA The acceleration is the instantaneous rate of change of velocity—it is the derivative of the velocity—that is, a(t) = v’(t). Negative acceleration (deceleration) means that the velocity is decreasing.

24: PVA The average velocity of a particle over the time interval t 0 to another time t, is: Where s(t) is the position of the particle at time t.

25: Average Value The average value of f(x) on [a, b] is

26: Area Between Curves If f and g are continuous functions such that g(x)  f(x) on [a, b], then the area between the curves is

27: Volume of Solids of Revolution For volumes of solids rotated around the x (or y) axis, volume = a b

27: Volume of Solids of Revolution For washer method, volume = where f(x) is the large radius, and g(x) is the small radius. a b

27: Volume of Solids of Revolution For cylinder method, volume =

28: Volume of Solids with Known Cross Sections 1. For cross sections of area A(x), taken perpendicular to the x-axis, volume = 2. For cross sections of area A(x), taken perpendicular to the y-axis, volume =

28: Volume of Solids with Known Cross Sections Some examples of these volumes are shown in the next four slides:

29: Solving Differential Equations: Graphically and Numerically (Slope Fields) At every point (x, y) a differential equation of the form dy/dx = f(x, y), gives the slope of the member of the family of solutions that contains that point. At each point in the plane, a short segment is drawn whose slope is equal to the value of the derivative at that point. These segments are tangent to the solution’s graph at the point.

29: Solving Differential Equations: Graphically and Numerically (Slope Fields) x y O You may be given an initial condition: This tells you exactly which of the possible solutions is the answer.

30: Solving Differential Equations by Separating the Variables Example of a differential equation: 1. Rewrite the equation as an equivalent equation with all the x and the dx terms on one side and all the y and dy terms on the other. 2. Antidifferentiate both sides to obtain an equation without dx or dy, but with one constant of integration. 3. Use the initial condition (given) to evaluate this constant.

f f ’ f ’’