Unit 5 Analytic Applications Of Derivativeap Calculus

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Unit 5: Analytical Applications of Differentiation. Class Resources. Calculus is primarily the study of rates of change. However, there are numerous applications of derivatives beyond just finding rates and velocities. In this review article, we will highlight the most important applications of derivatives for the AP Calculus AB/BC exams. Applications of Derivatives Here is a list of some of the essential derivative applications. (Instantaneous) Rates of Change. Unit 5 Review – Analytical Applications of Differentiation This review summarizes everything from Unit 5 along with examples but contains no problems to work through. DEFINITIONS Extrema: The maximum and minimum points. Extrema can be absolute or relative. Critical Points: Where the first derivative is zero or DNE. These are possible maximum. Graphical analysis cannot tell us easily exactly at what level of output, profits will be maximum, for it takes time to draw a graph and conclude from it. However, it is easier to use differential calculus to find the profit-maximising output. For this we simply find the first derivative of the profit function and set it equal to zero.

Version #1
​The course below follows CollegeBoard's Course and Exam Description. Lessons will begin to appear starting summer 2020.
BC Topics are listed, but there will be no lessons available for SY 2020-2021

Unit 0 - Calc Prerequisites (Summer Work)
0.1 Summer Packet
Unit 1 - Limits and Continuity
1.1 Can Change Occur at an Instant?
1.2 Defining Limits and Using Limit Notation
1.3 Estimating Limit Values from Graphs
1.4 Estimating Limit Values from Tables
1.5 Determining Limits Using Algebraic Properties
(1.5 includes piecewise functions involving limits)
1.6 Determining Limits Using Algebraic Manipulation
1.7 Selecting Procedures for Determining Limits
(1.7 includes rationalization, complex fractions, and absolute value)
1.8 Determining Limits Using the Squeeze Theorem
1.9 Connecting Multiple Representations of Limits
Mid-Unit Review - Unit 1
1.10 Exploring Types of Discontinuities
1.11 Defining Continuity at a Point
1.12 Confirming Continuity Over an Interval
1.13 Removing Discontinuities
1.14 Infinite Limits and Vertical Asymptotes
1.15 Limits at Infinity and Horizontal Asymptotes
1.16 Intermediate Value Theorem (IVT)
Review - Unit 1
Unit 2 - Differentiation: Definition and Fundamental Properties
2.1 Defining Average and Instantaneous Rate of
Change at a Point
2.2 Defining the Derivative of a Function and Using
Derivative Notation
(2.2 includes equation of the tangent line)
2.3 Estimating Derivatives of a Function at a Point
2.4 Connecting Differentiability and Continuity
2.5 Applying the Power Rule
2.6 Derivative Rules: Constant, Sum, Difference, and
Constant Multiple
(2.6 includes horizontal tangent lines, equation of the
normal line, and differentiability of piecewise)
2.7 Derivatives of cos(x), sin(x), e^x, and ln(x)
2.8 The Product Rule
2.9 The Quotient Rule
2.10 Derivatives of tan(x), cot(x), sec(x), and csc(x)
Review - Unit 2
Unit 3 - Differentiation: Composite, Implicit, and Inverse Functions
3.1 The Chain Rule
3.2 Implicit Differentiation
3.3 Differentiating Inverse Functions
3.4 Differentiating Inverse Trigonometric Functions
3.5 Selecting Procedures for Calculating Derivatives
3.6 Calculating Higher-Order Derivatives
Review - Unit 3
Unit 4 - Contextual Applications of Differentiation
4.1 Interpreting the Meaning of the Derivative in Context
4.2 Straight-Line Motion: Connecting Position, Velocity,
and Acceleration
4.3 Rates of Change in Applied Contexts Other Than
Motion
4.4 Introduction to Related Rates
4.5 Solving Related Rates Problems
4.6 Approximating Values of a Function Using Local
Linearity and Linearization
4.7 Using L'Hopital's Rule for Determining Limits of
Indeterminate Forms
Review - Unit 4
Unit 5 - Analytical Applications of Differentiation
5.1 Using the Mean Value Theorem
5.2 Extreme Value Theorem, Global Versus Local
Extrema, and Critical Points
5.3 Determining Intervals on Which a Function is
Increasing or Decreasing
5.4 Using the First Derivative Test to Determine Relative
Local Extrema
5.5 Using the Candidates Test to Determine Absolute
(Global) Extrema
5.6 Determining Concavity of Functions over Their
Domains
5.7 Using the Second Derivative Test to Determine
Extrema
Mid-Unit Review - Unit 5
5.8 Sketching Graphs of Functions and Their Derivatives
5.9 Connecting a Function, Its First Derivative, and Its
Second Derivative
(5.9 includes a revisit of particle motion and
determining if a particle is speeding up/down.)
5.10 Introduction to Optimization Problems
5.11 Solving Optimization Problems
5.12 Exploring Behaviors of Implicit Relations
Review - Unit 5
Unit 6 - Integration and Accumulation of Change
6.1 Exploring Accumulation of Change
6.2 Approximating Areas with Riemann Sums
6.3 Riemann Sums, Summation Notation, and Definite
Integral Notation
6.4 The Fundamental Theorem of Calculus and
Accumulation Functions
6.5 Interpreting the Behavior of Accumulation Functions
​ Involving Area
Mid-Unit Review - Unit 6
6.6 Applying Properties of Definite Integrals
6.7 The Fundamental Theorem of Calculus and Definite
Integrals
6.8 Finding Antiderivatives and Indefinite Integrals:
Basic Rules and Notation
6.9 Integrating Using Substitution
6.10 Integrating Functions Using Long Division
​ and Completing the Square
6.11 Integrating Using Integration by Parts (BC topic)
6.12 Integrating Using Linear Partial Fractions (BC topic)
6.13 Evaluating Improper Integrals (BC topic)
6.14 Selecting Techniques for Antidifferentiation
Review - Unit 6
Unit 7 - Differential Equations
7.1 Modeling Situations with Differential Equations
7.2 Verifying Solutions for Differential Equations
7.3 Sketching Slope Fields
7.4 Reasoning Using Slope Fields
7.5 Euler's Method (BC topic)
7.6 General Solutions Using Separation of Variables
7.7 Particular Solutions using Initial Conditions and
Separation of Variables
7.8 Exponential Models with Differential Equations
7.9 Logistic Models with Differential Equations (BC topic)
Review - Unit 7
Unit 8 - Applications of Integration
8.1 Average Value of a Function on an Interval
8.2 Position, Velocity, and Acceleration Using Integrals
8.3 Using Accumulation Functions and Definite Integrals
in Applied Contexts
8.4 Area Between Curves (with respect to x)
8.5 Area Between Curves (with respect to y)
8.6 Area Between Curves - More than Two Intersections
Mid-Unit Review - Unit 8
8.7 Cross Sections: Squares and Rectangles
8.8 Cross Sections: Triangles and Semicircles
8.9 Disc Method: Revolving Around the x- or y- Axis
8.10 Disc Method: Revolving Around Other Axes
8.11 Washer Method: Revolving Around the x- or y- Axis
8.12 Washer Method: Revolving Around Other Axes
8.13 The Arc Length of a Smooth, Planar Curve and
Distance Traveled (BC topic)
Review - Unit 8
Unit 9 - Parametric Equations, Polar Coordinates, and Vector-Valued Functions (BC topics)
9.1 Defining and Differentiating Parametric Equations
9.2 Second Derivatives of Parametric Equations
9.3 Arc Lengths of Curves (Parametric Equations)
9.4 Defining and Differentiating Vector-Valued Functions
9.5 Integrating Vector-Valued Functions
9.6 Solving Motion Problems Using Parametric and
Vector-Valued Functions
9.7 Defining Polar Coordinates and Differentiating in
Polar Form
9.8 Find the Area of a Polar Region or the Area Bounded
by a Single Polar Curve
9.9 Finding the Area of the Region Bounded by Two
Polar Curves
Review - Unit 9
Unit 10 - Infinite Sequences and Series (BC topics)
10.1 Defining Convergent and Divergent Infinite Series
10.2 Working with Geometric Series
10.3 The nth Term Test for Divergence
10.4 Integral Test for Convergence
10.5 Harmonic Series and p-Series
10.6 Comparison Tests for Convergence
10.7 Alternating Series Test for Convergence
10.8 Ratio Test for Convergence
10.9 Determining Absolute or Conditional Convergence
10.10 Alternating Series Error Bound
10.11 Finding Taylor Polynomial Approximations of
Functions
10.12 Lagrange Error Bound
10.13 Radius and Interval of Convergence of Power
Series
10.14 Finding Taylor Maclaurin Series for a Function
10.15 Representing Functions as a Power Series
Review - Unit 8

Version #2
​The course below covers all topics for the AP Calculus AB exam, but was built for a 90-minute class that meets every other day.
Lessons and packets are longer because they cover more material.

Unit 0 - Calc Prerequisites (Summer Work)
0.1 Things to Know for Calc
0.2 Summer Packet
0.3 Calculator Skillz
Unit 1 - Limits
1.1 Limits Graphically
1.2 Limits Analytically
1.3 Asymptotes
1.4 Continuity
Review - Unit 1
Unit 2 - The Derivative
2.1 Average Rate of Change
2.2 Definition of the Derivative
2.3 Differentiability [Calculator Required]
​ Review - Unit 2
Unit 3 - Basic Differentiation
3.1 Power Rule
3.2 Product and Quotient Rules
3.3 Velocity and other Rates of Change
3.4 Chain Rule
3.5 Trig Derivatives
Review - Unit 3
Unit 4 - More Deriviatvies
4.1 Derivatives of Exp. and Logs
4.2 Inverse Trig Derivatives
4.3 L'Hopital's Rule
Review - Unit 4
Unit 5 - Curve Sketching
5.1 Extrema on an Interval
5.2 First Derivative Test
5.3 Second Derivative Test
Review - Unit 5
Unit 6 - Implicit Differentiation
6.1 Implicit Differentiation
6.2 Related Rates
6.3 Optimization
​ Review - Unit 6
Unit 7 - Approximation Methods
7.1 Rectangular Approximation Method
7.2 Trapezoidal Approximation Method
Review - Unit 7
Unit 8 - Integration
8.1 Definite Integral
8.2 Fundamental Theorem of Calculus (part 1)
8.3 Antiderivatives (and specific solutions)
Review - Unit 8
Unit 9 - The 2nd Fundamental Theorem of Calculus
9.1 The 2nd FTC
9.2 Trig Integrals
9.3 Average Value (of a function)
9.4 Net Change
Review - Unit 9
Unit 10 - More Integrals
10.1 Slope Fields
10.2 u-Substitution (indefinite integrals)
10.3 u-Substitution (definite integrals)
10.4 Separation of Variables
Review - Unit 10
Unit 11 - Area and Volume
11.1 Area Between Two Curves
11.2 Volume - Disc Method
11.3 Volume - Washer Method
​ 11.4 Perpendicular Cross Sections
Review - Unit 11

The process of optimisation often requires us to determine the maximum or minimum value of a function.

For a function to be a maximum (or minimum) its first derivative is zero. Derivative of a function measures its slope.

Therefore, maximization of a function occurs where its derivative is equal to zero. Thus, an important optimisation problem facing a business manager is to produce a level of output which maximises firm’s profits. Similarly, optimum use of resources requires that cost be minimised for producing a given level of output. These problems of maximisation and minimisation can be solved with the use of the concept of derivative.

1. Use in Profit Maximisation:

Unit 5 analytic applications of derivativeap calculus pdf

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For example, consider the following profit function:

π = -100+ 160Q-10Q2

Where π = profits and Q is units of output

For the profit (π) function to be maximum, its first derivative must be equal to zero.

Therefore, to find the profit-maximising level of output we find the derivative of the given profit functions and set it equal to zero. Thus

At 8 units of output profits will be maximum. Maximisation of profits through the use of derivative is graphically shown in Figure 5.9. It will be seen that profit maximisation curve reaches its maximum point at point H. Therefore, at point H, the slope of the tangent (which measures the value of derivative dπ / dQ) drawn to the profit curve at this point is equal to zero.

It will be seen that corresponding to maximum profit point H on the profit function level of output is 8 units.

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Total profits made at 8 units of output can be obtained by substituting 8 for Q in the given profit function. Thus

Unit 5 Analytic Applications Of Derivativeap Calculus 1

π = – 100 + 160 x 8 – 10 (8)2

= 1280 – 740 = 540

Thus at output level of 8 units profits are equal to 540.

Graphical analysis cannot tell us easily exactly at what level of output, profits will be maximum, for it takes time to draw a graph and conclude from it. However, it is easier to use differential calculus to find the profit-maximising output. For this we simply find the first derivative of the profit function and set it equal to zero.

2. Second Derivative and Second Order Condition for Optimisation:

A problem arises when we use the first derivative of a function to determine its maximum or minimum value. For setting the first derivative of a function equal to zero and solving the resulting equation for the optimum value of the independent variable does not guarantee that optimum value (maximum or minimum as the case may be) will in fact he obtained. For the optimum value, the first derivative being equal to zero is a necessary condition for maximum or minimum, but it is not a sufficient condition. For example, in a profit function, first derivative is equal to zero, both it at maximum and minimum profit levels.

To ensure that the derivative is zero at the profit maximising level of the decision variable (i.e. output in the present case), we require to apply the second order condition. According to the second order condition, for profit maximisation, the second derivative of the profit function must be negative, that is, d2π / dQ2 < 0. Thus, if optimisation requires maximisation of function say, y = f (x), then the second derivative, which is written as d2y / dx2 must be negative.

It should be noted that the second derivative of a function is obtained by differenting the first derivative with respect to the independent variable. In case optimisation requires minimisation of a function as in case of minimisation of cost for producing a given level of output, the second derivative must be positive that is, d2y / dx2 > 0.

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Consider again the case of profit maximisation explained above. A profit function curve such as the one drawn in Figure 5.10 may have both minimum point and maximum points. It will be seen from Figure 5.10 that point L represents the minimum point and H represents the maximum point of the profit curve. Important thing to note is that at both minimum point L and maximum point H, first order condition, that is, first derivative dπ/dQ be zero is satisfied at both points, L and H, corresponding to OQ1 and OQ2 levels of output.

However, at point L profits are minimum and at point H profits are maximum. It is with the help of the second derivative of a function that we can distinguish between maximum and minimum along a function. Whereas the first derivative measures the slope of a function, the second derivative measures the slope of the first derivative. Thus, in case of profit function, whereas first derivative, dπ / dQ measures the slope of the profit function curve. That is, marginal profit, its second derivative, d2π / dQ2 measures slope of the marginal profit function curve.

Since the second derivative of a function when measured at the maximisation level is always negative and when measured at the minimisation level is always positive, it can be used to distinguish between points of maximum and minimum. For example, if the second derivative in our profit function curve is negative, it implies that profits are maximum at the level where first derivative is equal to zero.

On the other hand, if the second derivative at a point on a profit function where first derivative is zero is positive, it shows profits are in fact minimum rather than maximum. It can be easily known from having a look at Figure 5.10. It will be seen from this figure that up to point L, marginal profit (dπ / dQ) that is, slope of the total profit curve, is negative and has been causing the total profits to fall (in fact, up to L, due to negative marginal profits loses have been increasing. At point L, marginal profit I and thereafter it becomes positive and therefore it will causes the total profit to increase.

Hence, point L beyond which the second derivative (i.e. the slope of the first derivative) is positive, and since profits will be increasing beyond this point, it cannot be point of maximum profits.

Now consider point H on the total profit corresponding to output level OQ2. At point H, first derivative (dπ / dQ) is again equal to zero but after that marginal profit dπ / dQ becomes negative as the slope of total profit curve is negative as output is expanded beyond Q2. This causes the total profits to fall. This shows that point H at which first derivative, dπ / dQ is zero and also beyond which second derivative (d2π / dQ2), that is, slope of the first derivative becomes negative is indeed the point of maximum profit.

To conclude, we get a following general test for maximum and minimum:

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(1) If the second derivative d2y / dx2 of a function is negative (< 0) at the point where the first derivative (dy / dx) is zero, it will represent a point of maximum.

(2) If the second derivative (d2y / dx2) of a function is positive (> 0) at the point where first derivative is zero, it will represent a point of minimum.

Coming back to our profit function (π = – 100 + 160 Q – 10 Q2) in which case the first derivative is zero at 8 units of output, we test for the sign of second derivative. Thus,

dπ / dQ = 160 – 20Q

d2π / d2Q = -20.

Thus, we find that at 8 units of output profits will in fact be maximum.

3. Minimisation Problem:

In some decision making problems the objective of a manager is to minimise the objective function. For example, efficiency in the use of resources requires that a firm should produce at the minimum possible cost per unit of output.

For example, the following average cost function of a firm is given:

AC = 25,000 – 180Q + 0.50Q2

A manager is interested to find what level of output the firm will minimise its average cost. This can be obtained by differentiating the AC function with respect to output (Q) and setting it equal to zero, Thus

d(AC)/ dQ = – 180 + 1.0Q

Setting it equal to zero and solving for Q we have:

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-180 +1.0 Q = 0

Q = 180

Applying the second order condition to ensure whether it is really minimum we take the second derivative of AC function

Unit 5 Analytic Applications Of Derivativeap Calculus Pdf

d2(AC)/ dQ2 = + 1.0

Since second derivative of AC function is positive, d2 (AC)/ dQ2 > 0, output of 180 units of output is one that minimises average cost of production.

4. Multivariate Optimisation:

Unit 5 Analytic Applications Of Derivativeap Calculus 2

When a dependent variable is a function of many independent variables we use the concept of a partial derivative. Partial derivatives are therefore used to find optimal solution to maximisation or minimisation problem in case of two or more independent variables. Rules for finding maximisation and minimisation problems are the same as described above in case of one independent variable. To maximise or minimise a multivariate function we set partial derivative with respect to each independent variable equal to zero and solve the resulting set of simultaneous equations.

Consider that a firm is producing two products X and Y. Its profit function may be written as

π = f (X, Y).

Consider a firm producing the two products whose function is given below

π = 50X -2X2 –XY-4Y2 +75Y

Where X and Y are two independent variables representing levels of outputs of two products. It is to decide what levels of output of the two products will maximise profits.

Differentiating the profit function with respect to X while holding Y constant we have

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dπ/ dX = 50 – 4X – Y

Differentiating the profit function with respect to y while holding X constant we have

dπ/ dY = – X – 8Y+ 75

For maximisation of profits we must set each partial derivative equal to zero and then solve the resulting set of simultaneous equations for optimal values of independent variables x and Y. Thus,

∂π/ dX = 50 – 4X- y= 0 ….(i)

∂π/ dy =75 – X – 8y = 0 ….(ii)

Unit 5 Analytic Applications Of Derivativeap Calculus Answers

To solve the above two equations simultaneously we multiply equation (i) by – 8 and adding (i) and (ii) we have

Thus the firm will maximise profits of it produces and sells 10.45 units of product X and 8.2 units of product Y1.