3.7, 3.9, 4.1-4.3 Packet PDF

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AP Calculus AB

Chapter 3 Applications of Differentiation

3.7 Optimization Problems 3.9 Tangent Line Approximations &

Chapter 4 Integration

4.1 Antiderivatives and Indefinite Integration 4.2 Area 4.3 Riemann Sums & Definite Integral

3.7 Optimization Problems Optimization Problems – Problems that ask you to find the maxima or minima of something. I. GUIDELINES FOR SOLVING MAX/MIN PROBLEMS: (1) Read each problem slowly and carefully. Read the problem at least three times before trying to solve it. Sometimes words can be ambiguous. It is imperative to know exactly what the problem is asking. If you misread the problem, or hurry through it, you have NO chance of solving it correctly. (2) If appropriate, draw a sketch or diagram of the problem to be solved. Pictures are a great help in organizing and sorting out your thoughts. (3) Define variables to be used and carefully label your picture or diagram with these variables. This step is very important because it leads directly or indirectly to the creation of mathematical equations. (4) Write down all equations which are related to your problem or diagram. Clearly denote the equation which you are asked to maximize or minimize. Experience will show you that MOST optimization problems will begin with two equations. One equation is a “constraint” equation and the other is the “optimization” equation. The “constraint” equation is used to solve for one of the variables. This is then substituted into the “optimization” equation before differentiation occurs. Some problems may have no constraint equation, while some problems may have two or more constraint equations. (5) Before differentiating, make sure that the optimization equation is a function of only one variable. Then differentiate using the well-known rules of differentiation. (6) Verify that your result is a maximum or minimum value using the first or second derivative test for extrema.

II. TWO POSSIBLE TEST TO VERYIFY MAX OR MIN: 1. 1ST Derivative Test: f '(c)  0 or undefined (a) If f '  0 before c and f '  0 after c , then f ( x ) has a relative minimum at x  c (b) If f '  0 before c and f '  0 after c , then f ( x ) has a relative maximum at x  c 2. 2nd Derivative Test:

f '( c)  0 and (a) f ''( c)  0 at x  c , there is a relative minimum (b) f ''( c)  0 at x  c , there is a relative maximum (c) f ''( c)  0 at x  c , the test fails, must use the 1st Derivative Test

* So far we have two main types of word problems: (1) Related Rates: Solved with implicit differentiation on as many variables as given. (2) Optimization Problems: Solved with explicit differentiation after the problem is written in terms of one variable.

1) An open box is to be made out of a 10’’ x 12’’ piece of cardboard by cutting squares out of the corners. What dimensions will give the greatest volume?

2) A rectangular field is going to be fenced off along the bank of a river. The material for the fence cost $2 per foot for the two ends and $3 per foot for the side parallel to the river. Find the dimensions of the field of the largest area that can be enclosed with $900 of fencing.

3) A window is being built and at the bottom is a rectangle and at the top is a semicircle. If there is 12 meters of framing materials, what must the dimensions of the window be to let in the most light?

When instructed to “justify an answer”, students are expected to provide an explanation of the mathematical basis for their results or conclusions. For example, to justify the location of a relative extremum of a function, a student could invoke the First or Second Derivative Test accompanied by evidence that the hypotheses are satisfied – AP College Board

3.7 Homework Day 1

Warm Up Find two positive numbers that satisfy the given requirement: 1) The sum of the first number squared and the second number is 54 and the product is a maximum.

Section 3.7 Worksheet – Optimization Problems Day 2 Write a function for each problem, and justify your answers using Calculus. Give all decimal answers correct to three decimal places. 1.

Find two positive numbers such that the sum of the first and twice the second is 100 and their product is a maximum.

2.

A gardener wants to make a rectangular enclosure using a wall as one side and 120 m of fencing for the other three sides. Express the area in terms of x, and find the value of x that gives the greatest area.

3.

wall x

A rectangle has a perimeter of 80 cm. If its width is x, express its length and area in terms of x, and find the maximum area.

4.

Suppose you had 102 m of fencing to make two side-by-side enclosures as shown. What is the maximum area that you could enclose?

5.

Suppose you had to use exactly 200 m of fencing to make either one square enclosure or two separate square enclosures of any size you wished. What plan would give you the least area? What plan would give you the greatest area?

6.

A piece of wire 40 cm long is to be cut into two pieces. One piece will be bent to form a circle; the other will be bent to form a square. Find the lengths of the two pieces that cause the sum of the area of the circle and the area of the square to be a minimum.

7.

Four feet of wire is to be used to form a square and a circle. How much of the wire should be used for the square and how much should be used for the circle to enclose the maximum total area?

8.

The combined perimeter of an equilateral triangle and a square is 10. Find the dimensions of the triangle and square that produce a minimum total area.

9.

The combined perimeter of a circle and a square is 16. Find the dimensions of the circle and square that produce a minimum total area.

10. A tank with a rectangular base and rectangular sides is open at the top. It is to be constructed so that its height is 4 meters and its volume is 36 cubic meters. If building the tank costs $10/sq. m. for the base and $5/sq. m. for the sides, what is the cost of the least expensive tank, and what are its dimensions?

3.7 Day 3 Optimization Practice Worksheet 1.

The sum of two numbers is 6. The sum of their squares is a minimum. What are the two numbers?

2.

The product of two positive numbers is 16. Their sum is a minimum. What are the two numbers?

3.

The sum of a positive number and 4 times the square of its reciprocal is a minimum. What is the number?

4.

A rectangular piece of land borders a wall. The land is to be enclosed and divided by 200 feet of fencing, as shown below. (No fence is along the wall). What is the largest area that can be enclosed?

5.

An open rectangular box has a square base and a volume of 500 cubic inches. What dimensions minimize the amount of cardboard needed to make the box?

6.

A rectangle with horizontal and vertical sides has one vertex at the origin, one on the positive x-axis, one on the positive y-axis, and one on the line 2 x  y  100 . What is the maximum possible area of this rectangle?

7.

A right triangle with hypotenuse of 5 3 inches is revolved about one of its legs to generate a cone. What is volume of the cone of the greatest possible volume?

8.

The sum of two numbers is 10. The sum of their cubes is a minimum. What are the two numbers?

9.

A rectangle with perimeter of 36 inches is revolved about one of its sides to form a cylinder. What is the maximum possible volume that could be generated?

10. A cylinder is inscribed in a sphere with radius 3 inches. What is the volume of the cylinder with the greatest possible volume?

11. An open rectangular box is formed by cutting congruent squares from the corners of a piece of cardboard and folding the sides up. If the original piece of cardboard was 24 inches by 45 inches, what are the dimensions of the box with maximum volume?

12. A poster is to contain 50 square inches of print. The top and bottom margins are 4 inches each, and the left and right margins are 2 inches each. What dimensions minimize the area of the poster?

13. A rectangular piece of land has an area of 200 square feet, and is enclosed by fencing on 3 sides. (The fourth side is along a river). What dimensions minimize the amount of fence needed?

Warm Up Find the points of inflection and discuss the concavity of the graph of the function:

f ( x)  x 3  6 x 2  12 x

3.9 Tangent Line Approximations Notes To begin, consider a function f that is differentiable at c. The equation for the tangent line at point (c, f(c)) is given by

y  f  c   f ' c  x  c  y  f  c   f ' c  x  c  and is called the tangent line approximation or linear approximation of f at c. Because c is a constant, y is a linear function of x. Moreover, by restricting the values of x to be sufficiently close to c, the values of y can be used as approximations (to any desired accuracy) of the values of the function f. In other words, as x  c , the limit of y is f(c). Example 1 Find the tangent line approximation of f(x) = 1 + sin x at point (0, 1). Then use a table to compare the y-values of the linear function with those of f(x) on an open interval containing x = 0. a.

Find the derivative of f(x) = 1 + sin x.

b.

Find the equation of the tangent line at (0, 1).

c.

Graph f(x) and the tangent line in your calculator. Use the table below to compare the values of y given by this linear approximation with the values of f(x) near x = 0.

x

-0.5

-0.1

-0.01

0

0.01

0.1

0.5

f(x) = 1 + sin x

This linear approximation of f(x) = 1 + sin x depends on the point of tangency. At a different point on the graph of f, there would be different tangent line approximation. For x < 0, is the linear approximation values smaller or larger than the values of f(x)? Why?

For x > 0, is the linear approximation values smaller or larger than the values of f(x)? Why?

Example 2 2

a)

For f  x   3x

b)

Use the tangent line equation to approximate f(8.2).

c)

Find by f(8.2) by using the function of f(x).

d)

What is the error in your linear approximation?

e)

What does the error tell you about the sign of fꞌꞌ(x)?

3

, find an equation of the linear function that best fits f(x) at x = 8.

3.9 Tangent Line Approximation Homework 1(a)

For f(x) = 0.2x4, find an equation of the linear function that best fits f at 3.

(b)

Use the tangent line equation you found in (a) to approximate f(3.001).

(c)

Find f(3.001) by using the function f(x). What is the error in your linear approximation?

2(a)

For g(x) = sec x, find an equation of the linear function that best fits at x   . 3

(b)

Use the tangent line equation you found in (a) to approximate g   0.01 . 3

(c)

Find g   0.01 by using the function g(x). What is the error in your linear 3 approximation?









3

3.

Let f  x   1 tanx  2 .

(a)

Write an equation of the line tangent to the graph of f at the point where x = 0.

(b)

Use the equation you found in (a) to approximate f(0.02)

(c)

Find f(0.02) by using the function f(x). What is the error in your linear approximation? What does the error tell you about the sing of fꞌꞌ(x)?

4.

Consider the curve defined by -8x2 + 5xy + y3 = -149. Find dy . dx

(a)

(b)

Write an equation for the line tangent to the curve at the point (4, -1).

(c)

There is a number k such that the point (4.2, k) is on the curve. Using the tangent line found in part (b), approximate the value of k.

(d)

Write an equation that can be solved to find the actual value of k so that the point (4.2, k) is on the curve, and then use it to solve for k.

4.1 Anti-derivatives and Indefinite Integration Integration/Anti-differentiation – the opposite of taking a derivative. * Given a function, f ( x) , an anti-derivative of f ( x) is any function F ( x) such that F '( x )  f ( x) . * If F ( x ) is any anti-derivative of f ( x) then the most general anti-derivative of f ( x) is called an indefinite integral and denoted as:

 f ( x)dx  F ( x)  c, 

In this definition, the

where c is any constant

is called the integral symbol, f ( x) is called the integrand, x is called

the integration variable and the “c ” is called the constant of integration. Basic Integration Formulas: 1.

 0dx  c

2.

 Kdx  Kx  c

3.

EX:

5

dx ______________

 Kf ( x)dx  K f ( x)dx

EX:

3

2

4.

  f ( x)  g( x) dx  

EX:

 3

5.

 x dx  n  1 x

EX:



1)

3)

n

 4 x



3

1

( K is a constant )

n 1

 6 x 2  1 dx

x2  2x  3 dx x4

f ( x) dx   g ( x)dx

 c, n  1

x dx ______________ x 5  dx ______________

2

x dx ______________

5

 1  dx 4 

2)

  x

4)

 2t

2

 1  dt 2

Sometimes it may be easier to rewrite the equation, integrate, then simplify: Rewrite:

1

5)

x

6)

  3x

x

Integrate:

Simplify:

dx

1

2

dx

More Basic Integration Formulas: 6.

 cos xdx  sin x  c

7.

 sin xdx   cos x  c

8.

 sec

9.

 csc

10.

 sec x tan xdx  sec x  c

11.

 csc x cot xdx   csc x  c

7)

2

 t

2

xdx  tan x  c

 sin t  dt

2

xdx   cot x  c

8)

 

2

 sec2  d 

You do not know how to integrate a product or quotient, so you must simplify them to integrate with basic rules! 9)

 sec y tan y  sec y  dy

10)

cos x

 1  cos

2

x

dx

4.1 Day 1 Homework

Warm Up Given the acceleration function to be a(t)  6 feet per second per second, a) Find the velocity function if you know that v(0)  0 .

b) Find the position function if you know that s(3)  0 .

c) What is the velocity after 11 seconds?

4.1 Day 2 Position Function, Velocity & Acceleration - As previously mentioned, the derivative of a function representing the position of a particle along a line at time t is the instantaneous velocity. The derivative of the velocity, which is the second derivative of the position function, represents the instantaneous acceleration of the particle. Motion in Calculus (MEMORIZE) x( t)  position function  s( t) v( t)  velocity  x '(t ) De

ve

a( t)  acceleration  v'( t)  x ''(t)

Integrate

It may help to memorize the two basic starting points: (1) Given Understood Position Function Model: s(t )  16t2  v0t  s0 (2) Given Understood Position Acceleration: a(t )  32 ft / sec2 or a(t )   9.8m / sec2 * Positive velocity is moving right or up and Negative velocity is moving left or down. * Changes direction when velocity changes sign. Speed: Rate of change of distance with time, always positive, therefore: speed  v '(t )

If If If If

v( t ) Positive Negative Negative Positive

a( t ) Positive Negative Positive Negative

Speeding Up Speeding Up Slowing Down Slowing Down

If an object is speeding up, its acceleration vector, a(t ) , is in the same direction as its motion vector v( t ) . 1) A ball is thrown vertically upward from a height of 6 ft with an initial velocity of 60 ft/sec. How high will the ball go?

2) A balloon, rising vertically with a velocity of 16 ft/sec, releases a sandbag at the instant it is 64 feet above the ground. (a) How many seconds after its released will the bag strike the ground?

(b) At what velocity will it hit the ground?

3) A particle moving along the x -axis where s( t)  ( t 1)(t  3)2 represents its position at time t where 0  t  5 . (a) Find the velocity and acceleration of the particle at time t .

(b) Find the open t intervals on which the particle is moving to the right.

(c) Find the velocity of the particle when the acceleration is 0.

4.1 Day 2 Homework

Answer the following questions: 1. If x(t) represents the position of a particle along the x – axis at any time t then fill in the blanks to make the following statements true: (a) “Initially means when ____________ = 0.

(b) “At the origin” means ____________ = 0.

(c) “At rest” means ____________ = 0. (d) If the velocity of the particle is positive, then the particle is moving to the _______________. (e) If the velocity of the particle is _________________, then the particle is moving to the left. (f) _________________________ velocity is the velocity at a single moment in time. (g) If the acceleration of the particle is positive, then the ____________________is increasing. (h) If the acceleration of the particle is _____________________, then the velocity is decreasing. (i) In order for a particle to change directions, the ___________________ must change signs.

2. Let the position function be x t   1  t3 for t  0 2 t

(a) Determine the velocity at t  4

(a)_________________________

(b) Determine the acceleration at t  4

(b)_________________________

3. Let the position function be x t  3t2  5t  4 (a) When is position is increasing?

(b) When is velocity is increasing?

(a)_________________________

(b)_________________________

(c) Determine the smallest coordinate on the axis that the particle visits.

(c)______________________ (d) When does the particle change direction?

(d)______________________

4. Let the acceleration function be a t   6 t  3 if v 1  8 and x0  8 (a) Determine the position of the particle at t  2 .

(a)____________________________ (b) Determine when the velocity is increasing.

(c) Determine when the particle is speeding up.

(b)___________________________

(c)___________________________

5. A particle moves along the x- axis so that at any time, t  0, its acceleration is given by a t   6t  6 . At time t  0 , the velocity of the particle is  9 , and its position is  27 (a) Determine the velocity of the particle at any time ...