Math Calculus Review Part 4

Extreme Value Theorem If f is continuous on the interval [a, b], f has both a maximum and a minimum value in the interval. Note that brackets [ ] refer to a closed interval including the endpoints while parentheses ( ) refer to an interval not including the endpoints. Mean Value Theorem If f is continuous on the interval [a, b] and differentiable on the interval (a, b), there exists a point c f(b) − f(a) on (a, b) such that f0 (c) = . b−a In other words, somewhere on the interval the slope of the tangent line equals (at least once) the slope of the secant line connecting the two endpoints. Rolle’s Theorem Rolle’s Theorem is a special case of the Mean Value Theorem. If f is continuous on the interval [a, b], differentiable on the interval (a, b), and f(a) = f(b), then there exists a point c on (a, b) such that f0 (c) = 0. Integrals This chapter was designed for a test on integrals administered by Jonathan Chernick to his AP5 Calculus BC class on November 26, 2008. It is not covered in Math 12H/4H. Definite Integrals Definition definite integral area between a curve and the x-axis (area underneath the x-axis is negative) A finite number of rectangles can be used to estimate this area. A larger number of rectangles will give a more accurate estimate, and an infinite number of rectangles can give an exact answer. Zb a [f(x)dx] ≈ Ak = n X ak = a1 + a2 + · · · + an−1 + an k=1 Riemann Sums This area can be expressed as the infinite limit of Riemann sums. As n gets larger the width of the rectangles gets smaller and when n approaches infinity, the exact area is calculated. If f(x) is a continuous on the closed interval [a, b], the definite integral of f(x) between a and b is: !  Zb n X b−a [f(x)]dx = lim f(ck ) n→∞ n a k=1 where ck are sample points in the interval. Notation Rb When considering the expression a [f(x)]dx, the function f(x) is called the integrand and the interval [a, b] is the interval of integration. a and b are the lower and upper limits of integration, respectively. 5 AP is a registered trademark of the College Board, which was not involved in the production of, and does not endorse, this product. Rectangular Approximation Method Rectangular Approximation Method (RAM) is a method of estimating definite integrals by calculating the area of a certain number of rectangles. A larger number of rectangles will give a more accurate estimate. Left Rectangular Approximation Method (LRAM) Zb [f(x)dx] ≈ ∆x(f(a) + f(a + ∆x) + · · · + f(b − 2∆x) + f(b − ∆x)) a a where ∆x is the width of the rectangles ( b− n ) and n is the number of rectangles. Right Rectangular Approximation Method (RRAM) Zb [f(x)dx] ≈ ∆x(f(a + ∆x) + f(a + 2∆x) + · · · + f(b − ∆x) + f(b)) a a where ∆x is the width of the rectangles ( b− n ) and n is the number of rectangles. Midpoint Rectangular Approximation Method (MRAM) Zb [f(x)dx] ≈ ∆x(f(a + a ∆x ∆x ) + f(a + ∆x) + · · · + f(b − ∆x) + f(b − )) 2 2 a where ∆x is the width of the rectangles ( b− n ) and n is the number of rectangles. Trapezoidal Approximation Method Zb [f(x)dx] ≈ a   1 (∆x) (f(a) + 2f(a + ∆x) + · · · + 2f(b − ∆x) + f(b)) 2 a where ∆x is the width of the trapezoids ( b− n ) and n is the number of trapezoids. An integral approximated with this rule on a concave-up function will be an overestimate because the trapezoids include all of the area under the curve and extend over it. Using this method on a concave-down function yields an underestimate because area is unaccounted for under the curve, but none is counted above. Graphing Calculator These instructions are designed for a TI-84 Plus calculator, but they may used on other Texas Instruments graphing calculators, though slight modification may be necessary. Unless otherwise specified, the graphing calculator should be in radian mode. Definite Integral Rectangular Approximations In some cases it may be easier or required to calculate rectangular approximations of definite integrals using the graphing calculator, especially when using a large number of rectangles. The program RAM must be added to the calculator’s memory. Once installed, set the y1 of the calculator’s graph to the function being integrated and run the program with PRGM −→ RAM. Definite Integral Calculations In some cases it may be easier or required to calculate definite integrals using the graphing calculator, especially when the function is too complex. It can also be used to check one’s answer. Fundamental Theorem of Calculus Every continuous function has an antiderivative. Part I Rx If f is continuous on the closed interval [a, b] and F(x) = a [f(t)dt] on the closed interval [a, b], then F is differentiable on the open interval (a, b) and F0 (x) = f(x) for all x in the open interval (a, b). By definition F(x) is the antiderivative of f(x) in the open interval (a, b). Part II If f is continuous on the closed interval [a, b] and F is an antiderivative of f, then: Zb [f(x)dx] = F(b) − F(a) a It is therefore possible to calculate a definite integral using rules for antiderivatives (indefinite integrals). Corollary Integration and differentiation are inverses of each other. If f is continuous on the closed interval [a, b] then: Z x  d [f(t)dt] = f(x) dx a d du Z u  [f(t) dt] = f(u)du a Integral Rules Rules for calculating the integrals of general functions have been developed. As a result, it is possible to calculate the integrals of a wide variety of functions. In many cases the use of multiple rules are required. In the following rules, C represents the constant of integration. Constant Function The definite integral of a constant function is a rectangle with the height being the constant and the width being the interval of integration. Z [cdx] = cx + C Zb [cdx] = c(b − a) a where c is a constant. Addition/Subtraction Rule If f(x) and g(x) are continuous on the closed interval [a, b], then: Z Z Z [(f(x) ± g(x))dx] = [f(x)dx] ± [g(x)dx] + C Zb Zb [(f(x) ± g(x))dx] = a Zb [f(x)dx] ± a [g(x)dx] a As a result, one can take an equation, break it up into terms, figure out the definite integrals individually, and build the answer back up. Constant Multiplier Rule Z Z [c × f(x)dx] = c [f(x)dx] Zb Zb [c × f(x)dx] = c a [f(x)dx] a Power Rule Z [xn dx] = Zb [xn dx] = a xn+1 +C n+1 bn+1 − an+1 n+1 where n is a constant exponent not equal to −1 and x 6= 0. Expressions containing roots (i.e. square roots) can be intregrated by using a fractional value for n √ a ( b xa = x /b ). Expressions containing algebraic monomials in the denominator of a fraction can be integrated by inverting the sign of n ( x1n = x−n ). Logarithms 1 x Rule Z  dx = ln |x| + C x Zb   dx = ln |b| − ln |a| x a where x 6= 0. ex Rule Zh Zb h a where k is a constant. i ekx ekx dx = +C k i ekb eka ekx dx = − k k ax Rule Z [ax dx] = ax +C ln a Trigonometry • integrating the derivatives of the six trigonometric functions • integrating the derivatives of the inverse trigonometric functions See the the trigonometric section of the derivatives chapter on page 20 for more information. Constant If the constant is outside the trigonometric function, use the constant multiplier rule (Section ). If the constant is inside the trigonometric function, use the following rule. R Z ( [trig]kx) [(trig kx)dx] = +C k where k is a constant. Definite Integrals Additivity Rule The area under the graph of f(x) between a and b is the area between a and c plus the area between c and b. Zb Zc Zb [f(x)dx] = [f(x)dx] + [f(x)dx] a a c Zero Rule Za [f(x)dx] = 0 a Order of Integration Rule Za Zb [f(x)dx] = − b [f(x)dx] a Mean Value of Definite Integrals Mean Value The average (arithmetic mean) y-value of a function over an interval is the integral over the interval divided by the length of the interval. Rb a [f(x)dx] favg = b−a Mean Value Theorem If f is continuous on the closed interval [a, b], then at some point c in [a, b] there exists the following: Rb a [f(x)dx] f(c) = b−a Initial Value Problems Introduction An equation that contains a derivative is called a differential equation. For example, dy dx = 2x is a differential equation. Every differential equation of a function corresponds to a specific equation at a particular point (referred to as a particular solution), assuming the point is in the function’s domain. An initial value problem provides a differential equation and a particular point through which the function passes through. The specific equation is determined by calculating the value of C. Example dy = 2x, dx Z dy dx Z  y 6 6 C y y(1) = 6 = [2xdx] = = = = = x2 + C (1)2 + C 1+C 5 x2 + 5 Slope Fields Slope fields (also known as direction fields) are a logical extension to initial value problems as they provide a sketch of the differential equation for any value of C. A table containing the value of slope field. dy dx (the function’s slope) at different x and y values is used to create a Approaches These approaches reduce the time required to make or analyze slope fields and the possibility of making errors. Patterns Horizontal Pattern When the differential equation only contains the letter y (e.g. a horizontal pattern. Vertical Pattern vertical pattern. When the differential equation only contains the letter x (e.g. dy dx dy dx = y), there is = x), there is a Direction of Slope Determining whether the slopes of points in a certain vicinity are positive or negative is useful for comparing slope fields. Zero/No Slope Determining where the slopes of points are infinity (vertical and undefined) and where they are zero is useful for comparing slope fields. Separation of Variables Separation of variables is one method to isolate variables in a differentiable equation. The separated variables can then be integrated. If dy dx = g(x)h(y), then Basically, dy dx dy h(y) = g(x)dx. is being treated as a fraction, which can be can be separated. Example dy dx dy y = yx = xdx ln |y| = 1 2 x 2 eln |y| = e 2 x 1 2 y = e 2 x +C 1 2 Integration By Substitution Integration by substitution is a method for integrating a composition of function, when the entire integral can be expressed in terms of constants, u, and du. Integration by substitution may be used in combination with rules for inverse trigonometric functions. Trigonometric Identities Trigonometric identities (page 42) can be used to simplify expressions before or after integrating. Appendix Trigonometric Identities Pythagorean Identities 1. 2. 3. sin2 θ + cos2 θ = 1 1 + tan2 θ = sec2 θ 1 + cot2 θ = csc2 θ Quotient Identities 1. tan θ = sin θ cos θ cot θ = cos θ sin θ 2. Sum of Two Angles 1. sin(A + B) = sin A cos B + cos A sin B 2. cos(A + B) = cos A cos B − sin A sin B