This document provides information about polynomials including definitions, types, terms, and relationships between coefficients and zeros. It begins with acknowledging those who helped create the presentation. It then defines a polynomial as an expression with variable terms raised to whole number powers. The main types discussed are linear, quadratic, and cubic polynomials. Linear polynomials have one zero while quadratics have two zeros and cubics have three. Relationships are defined between the zeros and coefficients. Graphs of linear and quadratic polynomials are presented. The division algorithm for polynomials is also explained.
The Cobb-Douglas production function models the relationship between an output and inputs like labor and capital. It assumes outputs increase with inputs but at a decreasing rate. The formula relates the natural log of output to the natural log of inputs with elasticity coefficients representing the percentage change in output from a 1% change in an input. If the coefficients sum to 1 there are constant returns to scale, less than 1 is decreasing returns, and more than 1 is increasing returns. An example using Taiwan agricultural data from 1958-1972 estimated elasticities of 1.5 for labor and 0.4 for capital, indicating increasing returns to scale.
The document discusses the chain rule and how to use it to find derivatives of more complex equations. It provides examples of using the chain rule to take derivatives of functions involving exponents, trigonometric functions, radicals, and combinations of these. Key steps include identifying the inner and outer functions, taking the derivative of the inner function, and plugging into the chain rule formula. The document also contrasts using the chain rule method versus the inside-outside method for some problems.
This document provides information about polynomials including definitions, types, terms, and relationships between coefficients and zeros. It begins with acknowledging those who helped create the presentation. It then defines a polynomial as an expression with variable terms raised to whole number powers. The main types discussed are linear, quadratic, and cubic polynomials. Linear polynomials have one zero while quadratics have two zeros and cubics have three. Relationships are defined between the zeros and coefficients. Graphs of linear and quadratic polynomials are presented. The division algorithm for polynomials is also explained.
The Cobb-Douglas production function models the relationship between an output and inputs like labor and capital. It assumes outputs increase with inputs but at a decreasing rate. The formula relates the natural log of output to the natural log of inputs with elasticity coefficients representing the percentage change in output from a 1% change in an input. If the coefficients sum to 1 there are constant returns to scale, less than 1 is decreasing returns, and more than 1 is increasing returns. An example using Taiwan agricultural data from 1958-1972 estimated elasticities of 1.5 for labor and 0.4 for capital, indicating increasing returns to scale.
The document discusses the chain rule and how to use it to find derivatives of more complex equations. It provides examples of using the chain rule to take derivatives of functions involving exponents, trigonometric functions, radicals, and combinations of these. Key steps include identifying the inner and outer functions, taking the derivative of the inner function, and plugging into the chain rule formula. The document also contrasts using the chain rule method versus the inside-outside method for some problems.
The document describes the Cobb-Douglas production function and the results of estimating its parameters. It finds that labor (L) and capital (K) explain 95% of the variation in output (Q) according to the estimated equation Q=-0.31+0.20K+0.95L. Diagnostic tests show the data and estimated model meet the assumptions of regression analysis. Specifically, the variables are stationary and there is no multicollinearity while the model and estimated coefficients are statistically significant with a goodness of fit of 95%. Therefore, the Cobb-Douglas production function appropriately captures the relationship between L, K, and Q in the data.
11 x1 t09 03 rules for differentiation (2013)Nigel Simmons
The document outlines differentiation rules:
1) The derivative of a constant function is 0.
2) The derivative of a function with respect to x multiplied by a constant k is the derivative of the function multiplied by k.
3) The derivative of a polynomial function is found by taking the derivative of each term.
4) The derivative of a function divided by x is the derivative of the function minus the function divided by x squared.
The derivative of a composition of functions is the product of the derivatives of those functions. This rule is important because compositions are so powerful.
This document discusses rules for finding derivatives of products and quotients using the product rule and quotient rule. It provides examples of applying these rules to functions like f(x)=(x3)(x+x2) and f(x)=(4x2+5)/(x5). It notes that in some cases, it may be easier to first distribute and simplify the expression before finding the derivative.
The document summarizes key aspects of the Cobb-Douglas production function model, including:
1) The functional form of the Cobb-Douglas production function, which exhibits constant returns to scale properties based on labor and capital inputs.
2) How total factor productivity can be measured from the production function as a residual, providing a better measure than partial productivity.
3) A growth accounting formula derived from the logarithmic form of the production function that partitions GDP growth into contributions from labor, capital, and productivity.
4) How the exponent 'a' represents labor's share of output based on marginal productivity theory.
This document discusses using the Chain Rule to differentiate more complicated functions. It introduces the Chain Rule and provides Gottfried Leibniz as the inventor of the notation for derivatives. Two examples are given to demonstrate differentiating functions using the Chain Rule without showing the steps of the work.
This document provides examples and instructions for differentiating products using the product rule. It introduces Gottfried Leibniz and his discovery of the product rule in 1675. Examples are given to differentiate products and find the coordinates of a stationary point on a curve using differentiation. The exercises are from a textbook and ask the reader to start on question 3.
The document discusses finding the equation of the normal line to a curve at a given point using differentiation. It introduces the Quotient Rule for differentiating quotients of functions and notes that the Product Rule is often easier to use. It also attributes the discovery of the Quotient Rule to Gottfried Leibniz in 1677 and provides an example of using it to find stationary points on a curve.
The Building Block of Calculus - Chapter 4 The Product Rule and Quotient RuleTenri Ashari Wanahari
The chapter discusses the product rule and quotient rule for derivatives. The product rule states that the derivative of the product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function. The quotient rule gives the formula for finding the derivative of the quotient of two functions.
This document discusses rules for taking derivatives of various functions including:
1. The derivative of a constant function is 0.
2. The power rule states that the derivative of x^n is nx^{n-1}.
3. Higher derivatives can be found by taking additional derivatives, and the nth derivative is written as f^(n).
It also covers the product rule, quotient rule, and applying rules to polynomials and exponential functions.
The document discusses various chain rules for derivatives, including:
- The power chain rule: [up]' = pup−1(u)'
- Trigonometric chain rules: [sin(u)]' = cos(u)(u)', [cos(u)]' = −sin(u)(u)'
- Examples are provided to demonstrate applying the chain rules to find derivatives of more complex functions like y = sin(x3) and y = sin3(x). Repeated application of the appropriate chain rule at each step is often required.
The document discusses estimating the parameters of a Cobb-Douglas production function using econometric methods. It provides the equation Q=A*K^α + L^β and estimates the parameters as α = 0.206925 and β = 0.952008 using least squares regression. It finds that changes in K and L explain 95% of the variation in Q. Finally, it determines that the industry exhibits increasing returns to scale since α + β is greater than 1.
This document discusses the chain rule for functions of multiple variables. It begins by reviewing the chain rule for single-variable functions, then extends it to functions of more variables. The chain rule is presented for cases where the dependent variable z is a function of intermediate variables x and y, which are themselves functions of independent variables s and t. General formulas are given using partial derivatives. Examples are worked out, such as finding the derivative of a function defined implicitly by an equation. Diagrams are used to illustrate the relationships between variables.
This document provides an overview of lessons on the chain rule in calculus. It introduces the chain rule for functions of one variable and then extends it to functions of multiple variables. Examples are provided to demonstrate how to use the chain rule to calculate derivatives of composite functions. Formulas for the chain rule are stated for reference. The document also discusses using tree diagrams to visualize applications of the chain rule and introduces matrix expressions of the chain rule.
Presentation on CVP Analysis, Break Even Point & Applications of Marginal Cos...Leena Kakkar
CVP analysis helps managers understand the relationship between cost, volume, and profit by examining how price, volume, variable costs, fixed costs, and product mix interact. It is used to determine what products to make/sell, pricing policies, marketing strategies, and facility investments. The break-even point is where total costs and revenues are equal, and no profit or loss has occurred. Marginal costing is used to set optimal prices, evaluate price reductions, choose product mixes, calculate safety margins, and set different prices for different customers.
The Cobb-Douglas production function is widely used to model the relationship between output and two inputs, labor and capital. It takes the form of P(L,K) = B*L^α*K^β, where P is total production, L is labor input, K is capital input, B is total factor productivity, and α and β are output elasticities. The function was formulated by Cobb and Douglas based on statistical evidence showing how U.S. output and the two inputs changed together from 1889-1920. It has since been widely applied despite some criticisms around its lack of microeconomic foundations.
The document discusses production functions and their relationships. It shows that a production function relates the maximum quantity of output (Q) that can be produced from given amounts of inputs (capital K and labor L). The production function is represented as Q=f(K,L). It then derives and graphs the equations for total product (Q), average product (APL), and marginal product (MPL) based on the Cobb-Douglas production function of Q=K^0.3 L^0.8. It finds that average product is maximized when average product equals marginal product.
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
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The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
3. Calculus Rules
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2. Product Rule uv uv vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
4. Calculus Rulesd
2. Product Rule uv uv vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
e.g. i y x 7 x 9 6
5. Calculus Rules
d
2. Product Rule uv uv vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
e.g. i y x 7 x 9 6
x 7 9 x8 x 9 6 7 x 6
dy
dx
6. Calculus Rules
d
2. Product Rule uv uv vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
e.g. i y x 7 x 9 6
x 7 9 x8 x 9 6 7 x 6
dy
dx
9 x15 7 x15 42 x 6
7. Calculus Rules
d
2. Product Rule uv uv vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
e.g. i y x 7 x 9 6
x 7 9 x8 x 9 6 7 x 6
dy
dx
9 x15 7 x15 42 x 6
16 x15 42 x 6
8. Calculus Rules
d
2. Product Rule uv uv vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
e.g. i y x 7 x 9 6 ii y x 2 2 x 3
x 7 9 x8 x 9 6 7 x 6
dy
dx
9 x15 7 x15 42 x 6
16 x15 42 x 6
9. Calculus Rules
d
2. Product Rule uv uv vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
e.g. i y x 7 x 9 6 ii y x 2 2 x 3
dy
x 9 x x 6 7 x x 2 2 2 x 31
dy 7 8 9 6
dx dx
9 x15 7 x15 42 x 6
16 x15 42 x 6
10. Calculus Rules
d
2. Product Rule uv uv vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
e.g. i y x 7 x 9 6 ii y x 2 2 x 3
dy
x 9 x x 6 7 x x 2 2 2 x 31
dy 7 8 9 6
dx dx
9 x15 7 x15 42 x 6 2x 4 2x 3
16 x15 42 x 6
11. Calculus Rules
d
2. Product Rule uv uv vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
e.g. i y x 7 x 9 6 ii y x 2 2 x 3
dy
x 9 x x 6 7 x x 2 2 2 x 31
dy 7 8 9 6
dx dx
9 x15 7 x15 42 x 6 2x 4 2x 3
16 x15 42 x 6 4x 7
12. Calculus Rules
d
2. Product Rule uv uv vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
e.g. i y x 7 x 9 6 ii y x 2 2 x 3
dy
x 9 x x 6 7 x x 2 2 2 x 31
dy 7 8 9 6
dx dx
9 x15 7 x15 42 x 6 2x 4 2x 3
16 x15 42 x 6 4x 7
iii
d
dx
x 7 x 3 3 x 2 7
13. Calculus Rules
d
2. Product Rule uv uv vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
e.g. i y x 7 x 9 6 ii y x 2 2 x 3
dy
x 9 x x 6 7 x x 2 2 2 x 31
dy 7 8 9 6
dx dx
9 x15 7 x15 42 x 6 2x 4 2x 3
16 x15 42 x 6 4x 7
iii
d
dx
x 7 x 3 3 x 2 7
x 7 x 3 6 x 3 x 2 7 7 x 6 3 x 2
14. Calculus Rules
d
2. Product Rule uv uv vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
e.g. i y x 7 x 9 6 ii y x 2 2 x 3
dy
x 9 x x 6 7 x x 2 2 2 x 31
dy 7 8 9 6
dx dx
9 x15 7 x15 42 x 6 2x 4 2x 3
16 x15 42 x 6 4x 7
iii
d
dx
x 7 x 3 3 x 2 7
x 7 x 3 6 x 3 x 2 7 7 x 6 3 x 2
6 x8 6 x 4 21x8 9 x 4 49 x 6 21x 2
15. Calculus Rules
d
2. Product Rule uv uv vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
e.g. i y x 7 x 9 6 ii y x 2 2 x 3
dy
x 9 x x 6 7 x x 2 2 2 x 31
dy 7 8 9 6
dx dx
9 x15 7 x15 42 x 6 2x 4 2x 3
16 x15 42 x 6 4x 7
iii
d
dx
x 7 x 3 3 x 2 7
x 7 x 3 6 x 3 x 2 7 7 x 6 3 x 2
6 x8 6 x 4 21x8 9 x 4 49 x 6 21x 2
27 x8 49 x 6 15 x 4 21x 2
