This document covers key topics related to differentiation including:
- The difference quotient and how it relates to differentiability.
- Conditions for a function to be differentiable and continuous at a point.
- Critical points of a function including where the derivative does not exist or equals zero.
- Points of inflection where the graph of a function changes concavity.
- Concepts related to velocity, speed, and acceleration and how they are defined and related.
This document discusses differentiation and how it can be used to meet the needs of diverse learners. Differentiation is meant to make instruction multi-purpose and goal-oriented to meet the needs of all students. It involves using flexible grouping, formative assessments, matched resources, student choice, and exit points. Differentiation also considers a student's readiness, interests, and learning profile. The document outlines various models for differentiation including content, process, product, and affect. It also discusses learning profiles and intelligences using Gardner's multiple intelligences.
This document discusses differentiation rules and differentials. It explains that the differential form of differentiation rules can be used to find differentials of functions. It provides examples of finding differentials of products, composite functions, and using differentials to approximate function values. The key is choosing a value for x that makes calculations easier. Exercises are provided for using differentials to find derivatives of composite functions and to approximate function values.
This document covers key topics related to differentiation including:
- The difference quotient and how it relates to differentiability.
- Conditions for a function to be differentiable and continuous at a point.
- Critical points of a function including where the derivative does not exist or equals zero.
- Points of inflection where the graph of a function changes concavity.
- Concepts related to velocity, speed, and acceleration and how they are defined and related.
This document discusses differentiation and how it can be used to meet the needs of diverse learners. Differentiation is meant to make instruction multi-purpose and goal-oriented to meet the needs of all students. It involves using flexible grouping, formative assessments, matched resources, student choice, and exit points. Differentiation also considers a student's readiness, interests, and learning profile. The document outlines various models for differentiation including content, process, product, and affect. It also discusses learning profiles and intelligences using Gardner's multiple intelligences.
This document discusses differentiation rules and differentials. It explains that the differential form of differentiation rules can be used to find differentials of functions. It provides examples of finding differentials of products, composite functions, and using differentials to approximate function values. The key is choosing a value for x that makes calculations easier. Exercises are provided for using differentials to find derivatives of composite functions and to approximate function values.
The document outlines several basic rules and theorems for differentiation including definitions for differentiation, linearization, Leibniz' rule for repeated differentiation, standard inverse trigonometric derivatives, the intermediate value theorem, Rolle's theorem, the mean value theorem, and the constant difference theorem. Key concepts covered are definitions of continuity and differentiability, functions taking on values within a range, existence of critical points and points where the derivative is equal to the difference quotient, and conditions where the difference of derivatives is constant.
The document discusses differentiation in education. It defines differentiation as an approach to teaching that ensures all learners can learn despite their differences. Effective differentiation includes teaching methods that make the content accessible to all students, allowing all students to succeed. The document emphasizes that differentiation is important for students to benefit from instruction and can impact whether students pass or fail. It also notes that differentiation can influence students' future jobs and lives.
The document provides an overview of basic differentiation rules including:
- The constant rule
- The power rule
- The constant multiple rule
- Sum and difference rules
- Rules for deriving sine and cosine
It includes examples of applying these rules to find derivatives of functions such as f(x)=x^4 and the derivative of sine. The document also discusses using derivatives to find the equation of a tangent line.
This document provides formulas and examples for evaluating indefinite integrals involving various types of functions. It covers:
1) The integral of a variable to a power;
2) Integrals involving constants;
3) Integrals of sums and algebraic expressions;
4) Integrals of functions raised to a power;
5) Integrals of quotients using three different methods.
It also provides shortcuts for integrals involving expressions of ax + b, integrals of exponential functions, and the integral of x-1. Examples are included to demonstrate each formula and concept.
This document outlines rules for differentiating functions including the constant rule, power rule, constant multiple rule, sum and difference rule, product rule, and quotient rule. It provides examples of applying each rule to differentiate simple functions containing constants, powers, sums, differences, products, and quotients.
A PowerPoint presentation on the Derivative as a Function. Includes example problems on finding the derivative using the definition, Power Rule, examining graphs of f(x) and f'(x), and local linearity.
This document discusses basic differentiation rules and rates of change. It introduces the constant rule, power rule, constant multiple rule, sum and difference rules, derivatives of sine and cosine functions, and using derivatives to find rates of change such as velocity and acceleration. Examples are provided to demonstrate finding derivatives of various functions and using derivatives to calculate average velocity and velocity from a position function.
This problem set covers differential calculus and requires students to find the derivatives of given functions and solve the equation 2=x+e-x. Students are instructed to write their solutions neatly and orderly.
This document is a section from a Calculus I course at New York University that discusses basic differentiation rules. It provides objectives for understanding rules like the derivative of a constant function, the constant multiple rule, sum rule, and derivatives of sine and cosine functions. It then gives examples of finding the derivative of a squaring function using the definition, and introduces the concept of the second derivative.
Managerial economics applies economic theory and analysis to business decision-making. It uses theoretical models to predict firm behavior and outcomes. The key aspects covered in the document are:
1. Managerial economics models firm behavior and analyzes how changes impact equilibrium using comparative statics.
2. Models make simplifying assumptions and aim to generate testable predictions, not describe reality perfectly. Useful models have predictions supported by evidence.
3. Managerial economics can help decision-making by providing a logical framework, but unrealistic assumptions limit its prescriptive power for managers.
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 outlines several basic rules and theorems for differentiation including definitions for differentiation, linearization, Leibniz' rule for repeated differentiation, standard inverse trigonometric derivatives, the intermediate value theorem, Rolle's theorem, the mean value theorem, and the constant difference theorem. Key concepts covered are definitions of continuity and differentiability, functions taking on values within a range, existence of critical points and points where the derivative is equal to the difference quotient, and conditions where the difference of derivatives is constant.
The document discusses differentiation in education. It defines differentiation as an approach to teaching that ensures all learners can learn despite their differences. Effective differentiation includes teaching methods that make the content accessible to all students, allowing all students to succeed. The document emphasizes that differentiation is important for students to benefit from instruction and can impact whether students pass or fail. It also notes that differentiation can influence students' future jobs and lives.
The document provides an overview of basic differentiation rules including:
- The constant rule
- The power rule
- The constant multiple rule
- Sum and difference rules
- Rules for deriving sine and cosine
It includes examples of applying these rules to find derivatives of functions such as f(x)=x^4 and the derivative of sine. The document also discusses using derivatives to find the equation of a tangent line.
This document provides formulas and examples for evaluating indefinite integrals involving various types of functions. It covers:
1) The integral of a variable to a power;
2) Integrals involving constants;
3) Integrals of sums and algebraic expressions;
4) Integrals of functions raised to a power;
5) Integrals of quotients using three different methods.
It also provides shortcuts for integrals involving expressions of ax + b, integrals of exponential functions, and the integral of x-1. Examples are included to demonstrate each formula and concept.
This document outlines rules for differentiating functions including the constant rule, power rule, constant multiple rule, sum and difference rule, product rule, and quotient rule. It provides examples of applying each rule to differentiate simple functions containing constants, powers, sums, differences, products, and quotients.
A PowerPoint presentation on the Derivative as a Function. Includes example problems on finding the derivative using the definition, Power Rule, examining graphs of f(x) and f'(x), and local linearity.
This document discusses basic differentiation rules and rates of change. It introduces the constant rule, power rule, constant multiple rule, sum and difference rules, derivatives of sine and cosine functions, and using derivatives to find rates of change such as velocity and acceleration. Examples are provided to demonstrate finding derivatives of various functions and using derivatives to calculate average velocity and velocity from a position function.
This problem set covers differential calculus and requires students to find the derivatives of given functions and solve the equation 2=x+e-x. Students are instructed to write their solutions neatly and orderly.
This document is a section from a Calculus I course at New York University that discusses basic differentiation rules. It provides objectives for understanding rules like the derivative of a constant function, the constant multiple rule, sum rule, and derivatives of sine and cosine functions. It then gives examples of finding the derivative of a squaring function using the definition, and introduces the concept of the second derivative.
Managerial economics applies economic theory and analysis to business decision-making. It uses theoretical models to predict firm behavior and outcomes. The key aspects covered in the document are:
1. Managerial economics models firm behavior and analyzes how changes impact equilibrium using comparative statics.
2. Models make simplifying assumptions and aim to generate testable predictions, not describe reality perfectly. Useful models have predictions supported by evidence.
3. Managerial economics can help decision-making by providing a logical framework, but unrealistic assumptions limit its prescriptive power for managers.
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.
P
4
3
2
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
7. Rules For Differentiation
(1) y c
cc
f x c f x lim
h0 h
f x h c 0
lim
h 0 h
lim 0
h0
0
8. Rules For Differentiation
(1) y c
cc
f x c f x lim
h0 h
f x h c 0
lim
h 0 h
lim 0
h0
0
(2) y kx
9. Rules For Differentiation
(1) y c
cc
f x c f x lim
h0 h
f x h c 0
lim
h 0 h
lim 0
h0
0
(2) y kx
f x kx
10. Rules For Differentiation
(1) y c
cc
f x c f x lim
h0 h
f x h c 0
lim
h 0 h
lim 0
h0
0
(2) y kx
f x kx
f x h k x h
kx kh
11. Rules For Differentiation
(1) y c
cc
f x c f x lim
h0 h
f x h c 0
lim
h 0 h
lim 0
h0
0
(2) y kx
kx kh kx
f x kx f x lim
h0 h
f x h k x h
kx kh
12. Rules For Differentiation
(1) y c
cc
f x c f x lim
h0 h
f x h c 0
lim
h 0 h
lim 0
h0
0
(2) y kx
kx kh kx
f x kx f x lim
h0 h
f x h k x h kh
lim
kx kh h0 h
13. Rules For Differentiation
(1) y c
cc
f x c f x lim
h0 h
f x h c 0
lim
h 0 h
lim 0
h0
0
(2) y kx
kx kh kx
f x kx f x lim
h0 h
f x h k x h kh
lim
kx kh h0 h
lim k
h0
k
16. (3) y x n
a 2 b 2 a b a b
a 3 b3 a b a 2 ab b 2
17. (3) y x n
a 2 b 2 a b a b
a 3 b3 a b a 2 ab b 2
a 4 b 4 a b a 3 a 2b ab 2 b3
18. (3) y x n
a 2 b 2 a b a b
a 3 b3 a b a 2 ab b 2
a 4 b 4 a b a 3 a 2b ab 2 b3
a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1
19. (3) y x n
a 2 b 2 a b a b
a 3 b3 a b a 2 ab b 2
a 4 b 4 a b a 3 a 2b ab 2 b3
a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1
f x xn
20. (3) y x n
a 2 b 2 a b a b
a 3 b3 a b a 2 ab b 2
a 4 b 4 a b a 3 a 2b ab 2 b3
a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1
f x xn f x h x h
n
21. (3) y x n
a 2 b 2 a b a b
a 3 b3 a b a 2 ab b 2
a 4 b 4 a b a 3 a 2b ab 2 b3
a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1
f x xn f x h x h
n
x h xn
n
f x lim
h0 h
22. (3) y x n
a 2 b 2 a b a b
a 3 b3 a b a 2 ab b 2
a 4 b 4 a b a 3 a 2b ab 2 b3
a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1
f x xn f x h x h
n
x h xn
n
f x lim
h0 h
lim
x h x x h x h x x h x n2 x n1
n 1 n2
h0 h
23. (3) y x n
a 2 b 2 a b a b
a 3 b3 a b a 2 ab b 2
a 4 b 4 a b a 3 a 2b ab 2 b3
a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1
f x xn f x h x h
n
x h xn
n
f x lim
h0 h
lim
x h x x h x h x x h x n2 x n1
n 1 n2
h0 h
lim
h x h x h x x h x n2 x n1
n 1 n2
h0 h
24. (3) y x n
a 2 b 2 a b a b
a 3 b3 a b a 2 ab b 2
a 4 b 4 a b a 3 a 2b ab 2 b3
a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1
f x xn f x h x h
n
x h xn
n
f x lim
h0 h
lim
x h x x h x h x x h x n2 x n1
n 1 n2
h0 h
lim
h x h x h x x h x n2 x n1
n 1 n2
h0 h
lim x h x h x x h x n2 x n1
n 1 n2
h0
25. f x lim x h x h x x h x n2 x n1
n 1 n2
h0
26. f x lim x h x h x x h x n2 x n1
n 1 n2
h0
x n1 x n1 x n1 x n1
27. f x lim x h x h x x h x n2 x n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
28. f x lim x h x h x x h x n2 x n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
1
(4) y
x
29. f x lim x h x h x x h x n2 x n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
1
(4) y
x
1
f x
x
30. f x lim x h x h x x h x n2 x n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
1
(4) y
x
1
f x
x
1
f x h
xh
31. f x lim x h x h x x h x n2 x n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
1
(4) y
x 1 1
1
f x f x lim x h x
x h0 h
1
f x h
xh
32. f x lim x h x h x x h x n2 x n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
1
(4) y
x 1 1
1
f x f x lim x h x
x h0 h
1 xxh
f x h lim
xh h0 hx x h
33. f x lim x h x h x x h x n2 x n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
1
(4) y
x 1 1
1
f x f x lim x h x
x h0 h
1 xxh
f x h lim
xh h0 hx x h
h
lim
h0 hx x h
34. f x lim x h x h x x h x n2 x n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
1
(4) y
x 1 1
1
f x f x lim x h x
x h0 h
1 xxh
f x h lim
xh h0 hx x h
h
lim
h0 hx x h
1
lim
h0 x x h
35. f x lim x h x h x x h x n2 x n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
1
(4) y
x 1 1
1
f x f x lim x h x
x h0 h
1 xxh
f x h lim
xh h0 hx x h
h
lim
h0 hx x h
1
lim
h0 x x h
1
2
x
36. f x lim x h x h x x h x n2 x n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
1
(4) y
x 1 1
1 Note:
f x f x lim x h x
x h0 h
1 xxh
f x h lim
xh h0 hx x h
h
lim
h0 hx x h
1
lim
h0 x x h
1
2
x
37. f x lim x h x h x x h x n2 x n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
1
(4) y
x 1 1
1 Note:
f x f x lim x h x
x h0 h f x x 1
1 xxh
f x h lim
xh h0 hx x h
h
lim
h0 hx x h
1
lim
h0 x x h
1
2
x
38. f x lim x h x h x x h x n2 x n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
1
(4) y
x 1 1
1 Note:
f x f x lim x h x
x h0 h f x x 1
1 xxh
f x h lim
xh h0 hx x h f x x 2
h
lim
h0 hx x h
1
lim
h0 x x h
1
2
x
39. f x lim x h x h x x h x n2 x n1
n 1 n2
h0
x n1 x n1 x n1 x n1
nx n1
1
(4) y
x 1 1
1 Note:
f x f x lim x h x
x h0 h f x x 1
1 xxh
f x h lim
xh h0 hx x h f x x 2
h 1
lim 2
h0 hx x h x
1
lim
h0 x x h
1
2
x
43. (5) y x
f x x
xh x
f x h x h f x lim
h 0 h
44. (5) y x
f x x
xh x xh x
f x h x h f x lim
h 0 h xh x
45. (5) y x
f x x
xh x xh x
f x h x h f x lim
h 0 h xh x
xhx
lim
h0
h xh x
46. (5) y x
f x x
xh x xh x
f x h x h f x lim
h 0 h xh x
xhx
lim
h0
h xh x
h
lim
h0
h xh x
47. (5) y x
f x x
xh x xh x
f x h x h f x lim
h 0 h xh x
xhx
lim
h0
h xh x
h
lim
h0
h xh x
1
lim
h0
xh x
48. (5) y x
f x x
xh x xh x
f x h x h f x lim
h 0 h xh x
xhx
lim
h0
h xh x
h
lim
h0
h xh x
1
lim
h0
xh x
1
x x
49. (5) y x
f x x
xh x xh x
f x h x h f x lim
h 0 h xh x
xhx
lim
h0
h xh x
h
lim
h0
h xh x
1
lim
h0
xh x
1
x x
1
2 x
50. (5) y x
f x x
xh x xh x
f x h x h f x lim
h 0 h xh x
xhx
lim
h0
h xh x
h
lim
h0
h xh x Note:
1
lim
h0
xh x
1
x x
1
2 x
51. (5) y x
f x x
xh x xh x
f x h x h f x lim
h 0 h xh x
xhx
lim
h0
h xh x
h
lim
h0
h xh x Note:
1
1 f x x2
lim
h0
xh x
1
x x
1
2 x
52. (5) y x
f x x
xh x xh x
f x h x h f x lim
h 0 h xh x
xhx
lim
h0
h xh x
h
lim
h0
h xh x Note:
1
1 f x x2
lim
h0
xh x 1 1
f x x 2
1 2
x x
1
2 x
53. (5) y x
f x x
xh x xh x
f x h x h f x lim
h 0 h xh x
xhx
lim
h0
h xh x
h
lim
h0
h xh x Note:
1
1 f x x2
lim
h0
xh x 1 1
f x x 2
1 2
1
x x
2 x
1
2 x
57. e.g. i y 7
dy
0
dx
ii y 37 x
dy
37
dx
58. e.g. i y 7
dy
0
dx
ii y 37 x
dy
37
dx
iii y x10
59. e.g. i y 7
dy
0
dx
ii y 37 x
dy
37
dx
iii y x10
dy
10 x 9
dx
60. e.g. i y 7
dy
0
dx
ii y 37 x
dy
37
dx
iii y x10
dy
10 x 9
dx
iv y 3x 2 6 x 2
61. e.g. i y 7
dy
0
dx
ii y 37 x
dy
37
dx
iii y x10
dy
10 x 9
dx
iv y 3x 2 6 x 2
dy
6x 6
dx
62. e.g. i y 7 v y 2 x 1
2
dy
0
dx
ii y 37 x
dy
37
dx
iii y x10
dy
10 x 9
dx
iv y 3x 2 6 x 2
dy
6x 6
dx
63. e.g. i y 7 v y 2 x 1
2
dy
0 4x2 4 x 1
dx
ii y 37 x
dy
37
dx
iii y x10
dy
10 x 9
dx
iv y 3x 2 6 x 2
dy
6x 6
dx
64. e.g. i y 7 v y 2 x 1
2
dy
0 4x2 4 x 1
dx dy
8x 4
dx
ii y 37 x
dy
37
dx
iii y x10
dy
10 x 9
dx
iv y 3x 2 6 x 2
dy
6x 6
dx
65. e.g. i y 7 v y 2 x 1
2
dy
0 4x2 4 x 1
dx dy
8x 4
dx
ii y 37 x 1
dy vi y 3x 2
37 x
dx
iii y x10
dy
10 x 9
dx
iv y 3x 2 6 x 2
dy
6x 6
dx
66. e.g. i y 7 v y 2 x 1
2
dy
0 4x2 4 x 1
dx dy
8x 4
dx
ii y 37 x 1
dy vi y 3x 2
37 x
dx 3 x x 2
iii y x10
dy
10 x 9
dx
iv y 3x 2 6 x 2
dy
6x 6
dx
67. e.g. i y 7 v y 2 x 1
2
dy
0 4x2 4 x 1
dx dy
8x 4
dx
ii y 37 x 1
dy vi y 3x 2
37 x
dx 3 x x 2
dy
3 2 x 3
iii y x10 dx
dy
10 x 9
dx
iv y 3x 2 6 x 2
dy
6x 6
dx
68. e.g. i y 7 v y 2 x 1
2
dy
0 4x2 4 x 1
dx dy
8x 4
dx
ii y 37 x 1
dy vi y 3x 2
37 x
dx 3 x x 2
dy
3 2 x 3
iii y x10 dx
dy 2
10 x 9 3 3
dx x
iv y 3x 2 6 x 2
dy
6x 6
dx
69. e.g. i y 7 v y 2 x 1
2
vii y x 2 x
dy
0 4x2 4 x 1
dx dy
8x 4
dx
ii y 37 x 1
dy vi y 3x 2
37 x
dx 3 x x 2
dy
3 2 x 3
iii y x10 dx
dy 2
10 x 9 3 3
dx x
iv y 3x 2 6 x 2
dy
6x 6
dx
70. e.g. i y 7 v y 2 x 1
2
vii y x 2 x
dy
0 4x2 4 x 1
5
dx x 2
dy
8x 4
dx
ii y 37 x 1
dy vi y 3x 2
37 x
dx 3 x x 2
dy
3 2 x 3
iii y x10 dx
dy 2
10 x 9 3 3
dx x
iv y 3x 2 6 x 2
dy
6x 6
dx
71. e.g. i y 7 v y 2 x 1
2
vii y x 2 x
dy
0 4x2 4 x 1
5
dx x 2
dy
8x 4 dy 5 3
dx x2
ii y 37 x 1 dx 2
dy vi y 3x 2
37 x
dx 3 x x 2
dy
3 2 x 3
iii y x10 dx
dy 2
10 x 9 3 3
dx x
iv y 3x 2 6 x 2
dy
6x 6
dx
72. e.g. i y 7 v y 2 x 1
2
vii y x 2 x
dy
0 4x2 4 x 1
5
dx x 2
dy
8x 4 dy 5 3
dx x2
ii y 37 x 1 dx 2
dy vi y 3x 2 5
37 x x x
dx 3 x x 2 2
dy
3 2 x 3
iii y x10 dx
dy 2
10 x 9 3 3
dx x
iv y 3x 2 6 x 2
dy
6x 6
dx
73. e.g. i y 7 v y 2 x 1
2
vii y x 2 x
dy
0 4x2 4 x 1
5
dx x 2
dy
8x 4 dy 5 3
dx x2
ii y 37 x 1 dx 2
dy vi y 3x 2 5
37 x x x
dx 3 x x 2 2
dy
3 2 x 3
iii y x10 dx viii If f x x3 3,
dy
10 x 9 3 3
2 find f 2
dx x
iv y 3x 2 6 x 2
dy
6x 6
dx
74. e.g. i y 7 v y 2 x 1
2
vii y x 2 x
dy
0 4x2 4 x 1
5
dx x 2
dy
8x 4 dy 5 3
dx x2
ii y 37 x 1 dx 2
dy vi y 3x 2 5
37 x x x
dx 3 x x 2 2
dy
3 2 x 3
iii y x10 dx viii If f x x3 3,
dy
10 x 9 3 3
2 find f 2
dx x
f x x3 3
iv y 3x 2 6 x 2
dy
6x 6
dx
75. e.g. i y 7 v y 2 x 1
2
vii y x 2 x
dy
0 4x2 4 x 1
5
dx x 2
dy
8x 4 dy 5 3
dx x2
ii y 37 x 1 dx 2
dy vi y 3x 2 5
37 x x x
dx 3 x x 2 2
dy
3 2 x 3
iii y x10 dx viii If f x x3 3,
dy
10 x 9 3 3
2 find f 2
dx x
f x x3 3
iv y 3x 2 6 x 2 f x 3x 2
dy
6x 6
dx
76. e.g. i y 7 v y 2 x 1
2
vii y x 2 x
dy
0 4x2 4 x 1
5
dx x 2
dy
8x 4 dy 5 3
dx x2
ii y 37 x 1 dx 2
dy vi y 3x 2 5
37 x x x
dx 3 x x 2 2
dy
3 2 x 3
iii y x10 dx viii If f x x3 3,
dy
10 x 9 3 3
2 find f 2
dx x
f x x3 3
iv y 3x 2 6 x 2 f x 3x 2
dy
6x 6
f 2 3 2
2
dx
77. e.g. i y 7 v y 2 x 1
2
vii y x 2 x
dy
0 4x2 4 x 1
5
dx x 2
dy
8x 4 dy 5 3
dx x2
ii y 37 x 1 dx 2
dy vi y 3x 2 5
37 x x x
dx 3 x x 2 2
dy
3 2 x 3
iii y x10 dx viii If f x x3 3,
dy
10 x 9 3 3
2 find f 2
dx x
f x x3 3
iv y 3x 2 6 x 2 f x 3x 2
dy
6x 6
f 2 3 2
2
dx
12
78. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2
at the point 1,1
79. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2
at the point 1,1
y 5 x3 6 x 2 2
80. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2
at the point 1,1
y 5 x3 6 x 2 2
dy
15 x 2 12 x
dx
81. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2
at the point 1,1
y 5 x3 6 x 2 2
dy
15 x 2 12 x
dx
dy
when x 1, 15 1 12 1
2
dx
82. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2
at the point 1,1
y 5 x3 6 x 2 2
dy
15 x 2 12 x
dx
dy
when x 1, 15 1 12 1
2
dx
3
83. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2
at the point 1,1
y 5 x3 6 x 2 2
dy
15 x 2 12 x
dx
dy
when x 1, 15 1 12 1
2
dx
3
required slope 3
84. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2
at the point 1,1
y 5 x3 6 x 2 2 y 1 3 x 1
dy
15 x 2 12 x
dx
dy
when x 1, 15 1 12 1
2
dx
3
required slope 3
85. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2
at the point 1,1
y 5 x3 6 x 2 2 y 1 3 x 1
dy y 1 3x 3
15 x 2 12 x
dx
dy
when x 1, 15 1 12 1
2
dx
3
required slope 3
86. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2
at the point 1,1
y 5 x3 6 x 2 2 y 1 3 x 1
dy y 1 3x 3
15 x 2 12 x
dx
dy
when x 1, 15 1 12 1
2 3x y 2 0
dx
3
required slope 3
87. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2
at the point 1,1
y 5 x3 6 x 2 2 y 1 3 x 1
dy y 1 3x 3
15 x 2 12 x
dx
dy
when x 1, 15 1 12 1
2 3x y 2 0
dx
3
required slope 3
x Find the points on the curve y x3 12 x where the tangents
are horizontal
88. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2
at the point 1,1
y 5 x3 6 x 2 2 y 1 3 x 1
dy y 1 3x 3
15 x 2 12 x
dx
dy
when x 1, 15 1 12 1
2 3x y 2 0
dx
3
required slope 3
x Find the points on the curve y x3 12 x where the tangents
are horizontal
y x 3 12 x
89. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2
at the point 1,1
y 5 x3 6 x 2 2 y 1 3 x 1
dy y 1 3x 3
15 x 2 12 x
dx
dy
when x 1, 15 1 12 1
2 3x y 2 0
dx
3
required slope 3
x Find the points on the curve y x3 12 x where the tangents
are horizontal
y x 3 12 x
dy
3 x 2 12
dx
90. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2
at the point 1,1
y 5 x3 6 x 2 2 y 1 3 x 1
dy y 1 3x 3
15 x 2 12 x
dx
dy
when x 1, 15 1 12 1
2 3x y 2 0
dx
3
required slope 3
x Find the points on the curve y x3 12 x where the tangents
are horizontal dy
tangents are horizontal when 0
y x 3 12 x dx
dy
3 x 2 12
dx
91. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2
at the point 1,1
y 5 x3 6 x 2 2 y 1 3 x 1
dy y 1 3x 3
15 x 2 12 x
dx
dy
when x 1, 15 1 12 1
2 3x y 2 0
dx
3
required slope 3
x Find the points on the curve y x3 12 x where the tangents
are horizontal dy
tangents are horizontal when 0
y x 3 12 x dx
i.e. 3 x 12 0
2
dy
3 x 2 12
dx
92. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2
at the point 1,1
y 5 x3 6 x 2 2 y 1 3 x 1
dy y 1 3x 3
15 x 2 12 x
dx
dy
when x 1, 15 1 12 1
2 3x y 2 0
dx
3
required slope 3
x Find the points on the curve y x3 12 x where the tangents
are horizontal dy
tangents are horizontal when 0
y x 3 12 x dx
i.e. 3 x 12 0
2
dy
3 x 2 12 x2 4
dx
x 2
93. xix Find the equation of the tangent to the curve y 5 x3 6 x 2 2
at the point 1,1
y 5 x3 6 x 2 2 y 1 3 x 1
dy y 1 3x 3
15 x 2 12 x
dx
dy
when x 1, 15 1 12 1
2 3x y 2 0
dx
3
required slope 3
x Find the points on the curve y x3 12 x where the tangents
are horizontal dy
tangents are horizontal when 0
y x 3 12 x dx
i.e. 3 x 12 0
2
dy
3 x 2 12 x2 4
dx
x 2
tangents are horizontal at 2,16 and 2, 16
94. A normal is a line perpendicular to the tangent at the point of contact
95. A normal is a line perpendicular to the tangent at the point of contact
y
y f x
x
96. A normal is a line perpendicular to the tangent at the point of contact
y
y f x
x
tangent
97. A normal is a line perpendicular to the tangent at the point of contact
y
y f x
x
normal tangent
98. A normal is a line perpendicular to the tangent at the point of contact
y
y f x
x
normal tangent
xi Find the equation of the normal to the curve y 4 x 2 3 x 2 at
the point 3, 29
99. A normal is a line perpendicular to the tangent at the point of contact
y
y f x
x
normal tangent
xi Find the equation of the normal to the curve y 4 x 2 3 x 2 at
the point 3, 29
y 4 x 2 3x 2
100. A normal is a line perpendicular to the tangent at the point of contact
y
y f x
x
normal tangent
xi Find the equation of the normal to the curve y 4 x 2 3 x 2 at
the point 3, 29
y 4 x 2 3x 2
dy
8x 3
dx
101. A normal is a line perpendicular to the tangent at the point of contact
y
y f x
x
normal tangent
xi Find the equation of the normal to the curve y 4 x 2 3 x 2 at
the point 3, 29
y 4 x 2 3x 2
dy
8x 3
dx
dy
when x 3, 8 3 3
dx
102. A normal is a line perpendicular to the tangent at the point of contact
y
y f x
x
normal tangent
xi Find the equation of the normal to the curve y 4 x 2 3 x 2 at
the point 3, 29
y 4 x 2 3x 2
dy
8x 3
dx
dy
when x 3, 8 3 3
dx
21
103. A normal is a line perpendicular to the tangent at the point of contact
y
y f x
x
normal tangent
xi Find the equation of the normal to the curve y 4 x 2 3 x 2 at
the point 3, 29
y 4 x 2 3x 2
dy
8x 3
dx
dy
when x 3, 8 3 3
dx
21
1
required slope
21
104. A normal is a line perpendicular to the tangent at the point of contact
y
y f x
x
normal tangent
xi Find the equation of the normal to the curve y 4 x 2 3 x 2 at
the point 3, 29
1
y 4 x 3x 2
2
y 29 x 3
dy 21
8x 3
dx
dy
when x 3, 8 3 3
dx
21
1
required slope
21
105. A normal is a line perpendicular to the tangent at the point of contact
y
y f x
x
normal tangent
xi Find the equation of the normal to the curve y 4 x 2 3 x 2 at
the point 3, 29
1
y 4 x 3x 2
2
y 29 x 3
dy 21
8x 3 21 y 609 x 3
dx
dy
when x 3, 8 3 3
dx
21
1
required slope
21
106. A normal is a line perpendicular to the tangent at the point of contact
y
y f x
x
normal tangent
xi Find the equation of the normal to the curve y 4 x 2 3 x 2 at
the point 3, 29
1
y 4 x 3x 2
2
y 29 x 3
dy 21
8x 3 21 y 609 x 3
dx
dy
when x 3, 8 3 3 x 21 y 612 0
dx
21
1
required slope
21