5.1 Cost, Area, and the Definite Integral

Def’n The area of the region S under the graph of a positive function f is the limit of the sum of approximating rectangles, and is given by .

Def’n The definite integral of f from a to b is given by , where

, , and , and it represents a function’s net area

above the x–axis between a and b.

Rule Definite integrals of some curves can be calculated using geometric

formulas, and others can be approximated using left or right endpoints.

C Approximate Integration Def’n The Midpoint Rule uses the midpoint of each rectangle to estimate the definite integral as follows: , where .

Def’n The Trapezoidal Rule uses trapezoids to estimate the definite integral as follows: , where

Rule The Trapezoidal Rule is an average of the left endpoint approximation and the right endpoint approximation.

Rule The size of the error in the Trapezoidal Rule is usually about twice the size of the error in the Midpoint Rule.

Def’n Simpson’s Rule uses parabolas to estimate the definite integral as follows:

, where n is even.

Rule Simpson’s Rule is a weighted average of the Midpoint Rule and the Trapezoidal Rule .

Rule The size of the error in Simpson’s Rule is much smaller than the size of the error in either the Midpoint Rule or the Trapezoidal Rule. 5.2 The Fundamental Theorem of Calculus Def’n If , then the function F is an antiderivative of f.

Rule If F is an antiderivative of f, then so is .

Def’n The indefinite integral of f is .

Rule If , then .

Rule The integral of a sum or difference of functions is given by .

Rule The integral of a constant times a function is given by .

Rule The definite integral with the endpoints of the interval reversed is given

Rule The definite integral over adjacent intervals is given by .

5.3 The Net Change Theorem and Average Value

Rule The integral of a rate of change is the net change and is given by . Def’n The average value of a function on the interval is given by .

5.4 The Substitution Rule

Rule If , then the chain rule can be reversed, resulting in .

Rule For definite integrals, .

5.5 Integration by Parts

Rule The product rule can be reversed, resulting in or .

Rule For definite integrals, .

6.1 Areas Between Curves

Rule If on the interval , then the area between f and g is

given by .

6.2 Consumer Surplus and Producer Surplus Def’n If consumers are willing to buy a quantity of q units at a unit price of p dollars per unit, then the demand function D is given by .

Def’n The consumer surplus is the total difference in the price all consumers are willing to pay for a good and the actual selling price.

Rule The consumer surplus of a good when Q units are sold at price is given by .

Def’n If producers are willing to sell a quantity of q units at a unit price of p dollars per unit, then the supply function S is given by .

Def’n The producer surplus is the total difference in the actual selling price of a good and the price all producers are willing to sell it for.

Rule The producer surplus of a good when Q units are sold at price is given by .

Rule Total surplus is maximized when supply equals demand, or . 6.4 Differential Equations

Def’n A separable equation is a differential equation whose derivative can be expressed as a product of functions of the input and output variables.

Rule When solving a separable equation, separate the input and output variables, then integrate the equation, and solve for the output variable.

Rule Populations with growth that is directly proportional to the population are modeled by the differential equation , resulting in exponential

growth given by .

6.5 Improper Integrals

Def’n An improper integral is an integral with one or both endpoints at infinity.

Rule Improper integrals are given by or

Def’n The improper integral is convergent if its limit exists and divergent if its limit does not exist. Rule Integrals over the real line are given by

6.2, 6.5 Income Streams

Def’n The total amount of income earned from an income stream after a given period of time is the future value of the income stream.

Rule The future value of income earned at a rate of dollars per year and invested at interest rate r compounded continuously for T years is given by .

Def’n The amount of money invested now at interest rate r that would provide the same future value as an income stream is the present value. Rule The present value of income earned at a rate of dollars per year and invested at interest rate r compounded continuously for T years is given by .

Rule The present value of a perpetuity is given by .

7.1 Functions of Several Variables

Def’n A bivariate function is a rule that assigns an input pair to an output value , and its graph is the surface of all points in space such that .

Def’n A vertical trace is the intersection of a bivariate function with a vertical plane, and it is found by setting one of the input variables equal to a constant.

Def’n A horizontal trace is the intersection of a bivariate function with a horizontal plane, and it is found by setting the output variable equal to a constant. Def’n A level curve of a function f is a projection of a horizontal trace onto the xy–plane, and its equation is .

Def’n A contour map is a set of level curves with different values of the

output variable.

7.2 Partial Derivatives

Def’n The partial derivative of a bivariate function is the rate of change of the function with respect to one input, holding the other input constant.

Rule The partial derivatives of are given by and .

Note The following notations for partial derivatives are equivalent: .

Rule When finding , x is treated as a variable and y as a constant, and when finding , y is treated as a variable and x as a constant.

Rule The partial derivative is the slope of the tangent line to the surface of f parallel to the x–axis, and the partial derivative is the slope of the tangent line to the surface of f parallel to the y–axis.

Rule The second partial derivatives are given by

, , and .

7.2 Partial Derivative Applications

Rule The Cobb-Douglas production function is given by , where .

Def’n The marginal productivity of labor is the rate of change of

production with respect to labor.

Def’n The marginal productivity of capital is the rate of change of

production with respect to capital.

Def’n Substitute products are those for which an increase in demand for one

product results in a decrease in demand for the other product.

Def’n Complementary products are those for which an increase in demand for one product results in an increase in demand for the other product.

Rule For substitute products, and , and

for complementary products, and .

7.3 Maximum and Minimum Values

Def’n If for all near , then is a local maximum. If for all near , then is a local minimum.

Rule If is a local maximum or local minimum, then and .

Rule Given a critical point of a multivariate function and let .

(1) If and , then is a local minimum. (2) If and , then is a local maximum. (3) If then is a saddle point. (4) If then the test is inconclusive. 7.4 Lagrange Multipliers

Def’n The Lagrange function is used to

find extreme values subject to a constraint.

Rule The minimum and maximum values of subject to are found by solving , , and .

Rule The Lagrange multiplier is the ratio of the change in the optimal value of f to the change in the constant k.

Rule At the optimum level of a production function, and . D Double Integrals

Def’n The double integral of a multivariate function over a rectangular region is given by .

Rule The double integral represents a multivariate function’s volume above the xy–plane.

Def’n The double integral of a multivariate function over a non-rectangular

region is given by

Def’n The average value of a multivariate function over a region R is given by . 10.1 Geometric Series

Def’n A finite series with n terms is given by .

Def’n A series written in sigma notation is given by .

Def’n An infinite series is given by .

Def’n A finite geometric series is given by ,

where r is the common ratio.

Rule The sum of a finite geometric series is given by .

Def’n An infinite geometric series is given by .

Rule The sum of an infinite geometric series is given by , for .

Rule If , then the infinite geometric series is divergent. 10.2 Taylor Polynomials

Def’n Factorial notation is given by .

Def’n The first Taylor polynomial at of is given by

Def’n The n th Taylor polynomial at of is given by

Def’n The error of the nth Taylor polynomial at of is given by

, where .

Def’n The n th Taylor polynomial at of is given by

Def’n The error of the nth Taylor polynomial at of is given by

, where . 10.3 Taylor Series

Def’n A power series is an infinite geometric series with variable terms, and

it takes the form .

Def’n The radius of convergence R is a number such that a power series

converges if and diverges if .

Def’n The ratio of terms in a power series is given by .

Rule A power series converges if and diverges if .

Def’n The Taylor series at of is given by

Rule The Taylor series expansions of some common functions are given by:

at , for

10.4 Newton’s Method Rule The solution to can be approximated using the following

procedure with an initial guess :

(1) replace the function with its first Taylor polynomial at ,

(2) solve to get ,

(3) use an improved guess and repeat steps 1 and 2.

Def’n The internal rate of return r is the interest rate for which the present

values of all payments add up to the loan amount.

Rule Newton’s Method may not work if one of the following conditions is true:

(1) the derivative is zero at any approximation

(2) an inflection point exists between two successive approximations

(3) a critical point exists near an approximation