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### exploration3

Course: MATH 1400, Spring 2010
School: North Texas
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Word Count: 636

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3 Introduction Exploration to Definite Integrals As you drive on the highway you accelerate to 100 feet per second to pass a truck. After you have passed, you slow down to a more moderate 60 ft/sec. The diagram shows the graph of your velocity, v(t), as a function of the number of seconds, t, since you started slowing. v(t) 100 Name_______________ Period_______ 4. How many feet does each small square on the...

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3 Introduction Exploration to Definite Integrals As you drive on the highway you accelerate to 100 feet per second to pass a truck. After you have passed, you slow down to a more moderate 60 ft/sec. The diagram shows the graph of your velocity, v(t), as a function of the number of seconds, t, since you started slowing. v(t) 100 Name_______________ Period_______ 4. How many feet does each small square on the graph represent? How far, therefore, did you go in the time interval [0, 20]? Problem 3 and 4 involved finding the product of the x-value and the y-value for a function where y may vary with x. Such a product is called the definite integral of y with respect to x. Based on the units of t and v(t), explain why the definite integral of v(t) with respect to t in Problem 4 has feet for its units. The graph shows the cross-sectional area, y square inches, of a football as a function of the distance, x inches, from one of its ends. Estimate the definite integral of y with respect to x. y 5. 60 6. t 0 10 20 30 40 50 1. What does your velocity seem to be between t = 30 and t = 50 seconds? How far do you travel in the time interval [30, 50]? Explain why the answer to Problem 1 can be represented as the area of a rectangle region of the graph. Shade this region. The distance you travel between t = 0 and t = 20 can also be represented as the area of a region bounded by the (curved) graph. Count the number of squares in this region. Estimate the area of parts of squares to the nearest 0.1 square space. For instance, how would you count this partial square? 30 20 10 2. 0 6 12 x 3. 7. What are the units of the definite integral in Problem 6? What, therefore, do you suppose the definite integral represents? What did you learn as a result of doing this Exploration that you did not know 3 before? 8. Exploration Assignment 3. h(x ) = sin x Name________________ Period_______ For Problems 1 - 4, estimate the definite integral by counting squares on a graph. 1. 2. f(x ) = -0.1x 2 + 7 x = 0 to x = 5 f(x ) = -0.2x 2 + 8 x = -2 to x = 5 x = 0 to x = 4. g(x ) = 2x + 5 x = -1 to x = 1 For Problems 5 - 6, estimate the derivative of the function at the given value of x. 5. 6. f(x ) = tan x , x = 1 h(x ) = -7x + 100 , x = 5 7. Sports Car Problem: You have been hired by an automobile manufacturer to analyze the predicted motion of a new sports car they are building. When accelerated hard from a standing start, the velocity of the car, v(t) ft/sec, is expected to vary exponentially with time, t seconds, according to the equation v(t) = 100(1 - 0.9 t ) . a. b. c. d. e. f. Draw the graph of the function v in the domain [0, 10]. What is the range of the velocity function? range for the domain? Approximately how many seconds will it take the car to reach 60 ft/sec? Approximately how far will the car have traveled when it reaches 60 ft/sec? At approximately what rate is the velocity changing when t = 5? What special name is given to the rate of change of velocity? 8. Slide Problem: Phoebe sits atop the swimming pool slide. At time t = 0 sec she pushes off. Calvin ascertains that her velocity, v(t), is given by v(t) = 10 sin 0.3t . where v(t) is in feet per second. Phoebe splashes into the water at time t = 4 sec. a. b. c. d. e. f. Plot the graph of function v. (Dont forget to set your calculator to radian mode.) What are the domain and range of the velocity function? How fast was she going when she hit the water? Approximately how long is the slide? At approximately what rate was her velocity changing at t = 3? What special name is given to the rate of change of velocity?
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North Texas - MATH - 1400
Exploration 4Definite Integrals by Trapezoidal RuleRocket Problem: Ella Vader (Darths daughter) is driving in her rocket ship. At time t = 0 minutes she fires her rocket engine. The ship speeds up for a while, then slows down as Alderaans gravity takes
North Texas - MATH - 1400
Explorations 5Introduction to Limits1. Plot on your calculator the graph of this function. 5.Name_ Period_Between what two numbers does f(x) stay when x is kept in the open interval (2.5, 3.5)?Use a friendly window with x = 3 as a grid point. Sketch
North Texas - MATH - 1400
Chapter 3Worksheet 3AName_ Period_Directions: On a separate sheet of graph paper sketch each function. Make sure you number and label each graph. Plot at least 2 points for accuracy. 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. y = x2 + 5 y = x+1+ 5 y = x3 +
North Texas - MATH - 1400
Chapter 3Worksheet 3BName_ Period_Directions: On a separate sheet of graph paper sketch each function. Make sure you number and label each graph. Plot at least 2 points for accuracy. 1. 3. 5. 7. 9. 11. 13. 15. y = -x+2 y= 1 -3 x 2. 4. 6. 8. 10.3y = -
North Texas - MATH - 1400
Chapter 3Worksheet 3DName_ Period_Directions: Determine whether each function below is even, odd, or neither. Explain. 1. y = 5x 1 0 - 3x 4 + 2 2. w = 10 x 9 - 2x 3 + x3.f(x ) = x4.g(x ) =1 xDirections: Determine which graphs below are even, odd
North Texas - MATH - 1400
Chapter 3Worksheet 3EName_ Period_Directions: Determine whether each function below is even, odd, or neither. Explain. 1 1. y= 2 2. f(x ) = -7x 4 + 6x - 3 x -13.4.5,6.Directions: For questions 7 and 8 the graphs below are portions of completed gra
North Texas - MATH - 1400
Chapter 3Worksheet 3ZName_ Period_Directions: Graph the following equations on your calculator. Adjust the window so you see the important parts of the graph. Make a small sketch of your screen next to the equation. Label your window. x 1 1. y= 2. y= 2
North Texas - MATH - 1400
Chapter 4WorksheetName_ Period_Directions: Find all of the zeros of the functions. Show all work. 1. f(x ) = 2x 3 - 3 x 2 - 4x + 6 2. f(x ) = x 3 + 3x 2 - 3 x - 93.f(x ) = x 3 + x 2 - 8x - 64.f(x ) = x 3 - 6x 2 + 7 x + 45.f(x ) = x 4 - 3x 3 - 6 x
North Texas - MATH - 1400
WORKSHEET 1 1. Below is a list of some simple algebra problems. Some of the solutions are correct and some of them are wrong! For each problem: A. determine if the answer is correct; B. determine if there are any mistakes made in solving the problem and l
North Texas - MATH - 1400
WORKSHEET 2 - Fall 1995 1. For each graph below of a function f (x), sketch a graph of its derivative f (x): a) b) c)d)e)f)g)h)i)2. Without using the concept of a limit (and thus derivative), write the equations of all lines through the point (a, a
North Texas - MATH - 1400
WORKSHEET 4 - Fall 1995 1. The graph of f (x) is given below. Use it to graph the following: a) f (3x) e) f (x) + 1 b) f (x) f) 5f (x) c) f (2x) g) f (x + 2) d) f (x 1) h) 5f (3x + 2) + 1i) In your own words, describe the manner in which the graph of f (
North Texas - MATH - 1400
WORKSHEET 5 - Fall 19951. Use a calculator to ll in the table given and then make a guess at the given limit. x 3 a) lim = x3 x3x- 3 x-3 x 2 2.5 2.75 2.9 2.99 limitb) limx23x = 3x + 23x 3x + 2 x 1 1.5 1.75 1.9 1.99 limitc) lim1 = x0 x21 x2 x 1 0.
North Texas - MATH - 1400
WORKSHEET 6 - Fall 1995 1. A function is said to be continuous at a point x0 if i. f (x0 ) is dened; ii. lim f (x) exists;xx0iii. and lim f (x) = f (x0 ).xx0Determine whether the following functions are continuous at the points given. At discontinuous
North Texas - MATH - 1400
WORKSHEET 7 - Fall 1995 1. Let f (x) = x3 + 3x2 3x. a) At any point (x0 , y0 ) on the graph, what is the slope of the tangent line to the graph? b) The graph of f (x) has two tangent lines parallel to the line y = 6x + 100. Find the equations of these two
North Texas - MATH - 1400
WORKSHEET 8 - Fall 19951. Give examples of functions satisfying the following conditions. If no such function exists, explain why. a) Continuous at x = c but not dierentiable there; b) Dierentiable at x = c but not continuous there; c) Not continuous at
North Texas - MATH - 1400
WORKSHEET 9 - Fall 1995 1. Recall that sin2 (x) + cos2 (x) = 1. Write an expression for each trig function squared in terms of one of the other trig functions. a) sin2 (x) 2. Compute the limits. sin2 2x x0 x2 1 + cot2 3x d) lim x0 4x2 a) lim b) lim e) 1 c
North Texas - MATH - 1400
WORKSHEET 10 - Fall 1995 1. a) Explain geometrically what it means for a function to be continuous. Do the same for dierentiability. b) Draw graphs to describe all possible ways a function can fail to be continuous at a point. Also draw graphs showing how
North Texas - MATH - 1400
WORKSHEET 11 - Fall 1995 1. The curve y = ax2 + bx + c passes through the point (1, 2) and is tangent to the line y = x at the origin. Find a, b, and c. 2. If f is even (or odd), is f necessarily one or the other? Construct a proof. 3. A particle travels
North Texas - MATH - 1400
WORKSHEET 12 - Fall 1995 1. Write f (x) as a composition of two functions in two dierent ways. Write f (x) as a composition of three functions. Dierentiate f (x). a) f (x) = x2 + 1 b) f (x) = sin x x x2 x+2 c) f (x) = (2x2 x)5/22. a) The radius of a bal
North Texas - MATH - 1400
WORKSHEET 13 - Fall 1995 1. Find the domain and range of the following functions. a) f (x) = ln x2 2. b) f (x) = 2 ln x c) f (x) = ln | sin x| lim (1 + s)1/s . d) f (x) = ln(sin x).a) Use a calculator to estimates0This limit exists and its value is ver
North Texas - MATH - 1400
WORKSHEET 14 - Fall 1995 1. Reiview properties of logs (if necessary) and then solve these for x: a) b) c) d) e) f) g) h) i) x = log4 2 log4 x = 5/2 1 3 logx 8 = 2 log4 (x 2) log4 (2x + 3) = 0 log10 x + log10 (x 15) = 2 3 logx 4 = 2 log2 x = log4 5 + 3 lo
North Texas - MATH - 1400
WORKSHEET 15 - Fall 1995 1. The Intermediate-Value Theorem Let f be continuous throughout the closed interval [a, b]. Let m be any number between f (a) and f (b). a) Draw several pictures of functions satisfying this hypotheses. Make sure you include both
North Texas - MATH - 1400
WORKSHEET 16 - Fall 1995dy dx1. Computefor the following.a) y = (arccos x)(cos x) b) y = arccos(cos x) c) y 4 x7 = 8 d) y = sin3 x2 + e) y = arctan x tan x 1 x22. Suppose = arcsin x with 1 x 1. Find the following in terms of x: cos tan cot sec csc si
North Texas - MATH - 1400
WORKSHEET 17 - Fall 1995 1. Find y both implicitly and explicitly for each relation below. a) x1/2 + y 1/2 = 1 b) |x| + |y | = 1 c) x2 + y 2 = 1 d) x3 + y 3 = 1 2. Find y for each of the parts of problem 1. 3. Consider the curve x2 + y 2 xy + 3x 9 = 0. a)
North Texas - MATH - 1400
WORKSHEET 18 - Fall 1995 1. a) For each of the following equations, nd i) x2 + y 2 = 1dy dx :ii) y = cot2 xiii) y 2 sin x = x3 + 5x2 4b) For each equation above, nd dx . Explain the dierence between the meaning of this derivative dy and the one you fo
North Texas - MATH - 1400
WORKSHEET 19 - Fall 1995 1. In each of the following equations, suppose that each variable is actually a function of time t and dierentiate each expression with respect to t. a) x2 + y 2 = 100 s x+s = b) 5 1.5 c) 40y xy = 80 d) (x + 7)(7 gt2 ) = 9x, e) V
North Texas - MATH - 1400
WORKSHEET 21 - Fall 1995 1. Find f (x) in terms of g (x) and g (x), where g (x) &gt; 0 for all x. (Recall: If c is a constant, then g (c) is a constant.) a) f (x) = g (x)(x a) b) f (x) = g (a)(x a) c) f (x) = g (x + g (x) d) f (x) = g (x) xa 1 e) f (x) = g (
North Texas - MATH - 1400
WORKSHEET 22 - Fall 19951. The graph of the derivative of a function f (x) is given below.f2 1 3 4a) What are the critical points of f (x)? b) Which critical point(s) correspond to relative extrema? Are they maxima or minima? c) Can you determine any
North Texas - MATH - 1400
WORKSHEET 24 - Fall 19951.a) Find three distinct functions f (x) such that f (x) = 0 for all x. b) Find three distinct functions g (x) such that g (x) = 4x for all x. c) Find three distinct functions h(x) such that h (x) = 3x x2 for all x.2. Let f1 (x)
North Texas - MATH - 1400
WORKSHEET 25 - Fall 1995 In the theory of quantum mechanics, the Heisenberg uncertainty principle (also called the indeterminacy principle) states that experiment cannot simultaneously determine the exact value of a component of momentum, px say, of an ob
North Texas - MATH - 1400
WORKSHEET 26 - Fall 1995 1. What is area? a) List the geometric objects for which you know how to nd the area. Include the formula you would use (and, of course, the meaning of any variables used.) How do you know these formulas? b) Describe some geometri
North Texas - MATH - 1400
WORKSHEET 27 - Fall 1995 1. Below, the graph of f (x) = x2 is given. The line below the graph is the tangent line at x = 1. The lines above the graph connect the points (0, 0), (1, 1), and (2, 4).432112a) Find the area bounded by the tangent line,
North Texas - MATH - 1400
WORKSHEET 28 - Fall 1995 1. Consider the following denition: Denition. If f is a function dened on [a, b] and the sums i=1 f (ci )(xi xi1 ) approaches a certain number as the mesh of partitions of [a, b] shrinks toward 0 (no matter how the sampling number
North Texas - MATH - 1400
WORKSHEET 29 - Fall 1995 1. Label each of the following as TRUE or FALSE: a) c) e) (x2 + 1)9 (2x) dx = (x2 + 1)10 +C 10 sin2 (x) +C sin(x) cos(x) dx = 2 b) d) f) 2x dx = 2x+1 +C (x + 1) cos2 (x) +C sin(x) cos(x) dx = 22 tan(x) sec2 (x) dx = tan2 (x) + C
North Texas - MATH - 1400
WORKSHEET 31- Fall 19951.a) Give the denition of a denite integral. b) State the Fundamental Theorem of Calculus. c) What is wrong with the following argument?2 1dt 1 = 2 t t2 11 1 = 2 13 1 = 1= 2 22)a) Let f (x) = 4x x2 . Draw a graph of this fu
North Texas - MATH - 1400
WORKSHEET 32 - Fall 1995 1. Find all continuous functions f (x) satisfyingx 0f (t) dt = [f (x)]2 + C(Hint: Dierentiate both sides with respect to x.) 2. Let A be the average value of f (x) = x3 on the interval [1, 3]. Determine A, then graph y = f (x)
North Texas - MATH - 1400
WORKSHEET - Fall 1995 1. Write down the denition of a denite integral. Take turns explaining the dierent parts to the members of your group. 2. Use the denition of the denite integral to compute the following integrals:5 1 n3x2 14x + 11n2 07x.Hint:
North Texas - MATH - 1400
WORKSHEET 34 - Fall 1995 1. Sketch the nite area whose boundary is composed of pieces of the curves x = 1 y 4 and x = y 2 1. a) Find the area of this region by integrating with respect to y . b) Find the area of this region by integrating with respect to
North Texas - MATH - 1400
WORKSHEET 35 - Fall 1995 a) Let be the region bounded by the curves y = 3 x2 , y = 2x, and x = 0. Sketch the graphs labeling the point of intersection in the rst quadrant. b) Find the area of . c) Using discs nd the volume of hte solid obtained by revolvi
North Texas - MATH - 1400
WORKSHEET 36 - Fall 1995 1. Compute /2 /2sin(3x) cos(5x) dx.2. It is known that m parts (by weight) of chemical A combine with n parts of chemical B to produce a comound C. Suppose that the rate at which C is produced varies directly with the product o
North Texas - MATH - 1400
North Texas - MATH - 1400
North Texas - MATH - 1400
North Texas - MATH - 1400
North Texas - MATH - 1400
North Texas - MATH - 1400
North Texas - MATH - 1400
North Texas - MATH - 1400
CALCULUS 1. Dierentiate: f (x) = log2 sin x 2. Dierentiate: f (x) = cos sin x 3. Dierentiate: f (x) = log4 10x 4. Dierentiate: f (x) = cos 5x 5. Dierentiate: f (x) = ln cos x 6. Dierentiate: f (x) = sin sin x 7. Dierentiate: f (x) = 32xTrancedental Func
North Texas - MATH - 1400
Answers:1. f (x) = (sin x)cos xx 1 x ln sin x + cos x sin x 2x2. f (x) = (cos x) 3. f (x) = xx[ln cos x + 1] ( sin x) 1 ln x + 24. f (x) = (ln x)log x5. f (x) = (cos x) 6. f (x) = 7. f (x) = x xx1 1 log x ln ln x + (ln 10)x x ln x 1 x ln cos
North Texas - MATH - 1400
CALCULUSTrancedental Functions. Higher Derivatives1. Find f (x), f (x), f (x), and f (4) (x) for the following function: f (x) = 2 sin(3x) 2. Find f (x), f (x), f (x), and f (4) (x) for the following function: f (x) = 5 log(5x) 3. Find f (x), f (x), f (
North Texas - MATH - 1400
CALCULUSTrancedental Functions. Velocity and Acceleration1. Find the velocity, acceleration, and jerk functions for the following position function: s(t) = 3 log(t) 2. Find the velocity, acceleration, and jerk functions for the following position functi
North Texas - MATH - 1400
CALCULUSTrancedental Functions. Tangent Line1. Find the point of tangency and the equation of the tangent line passing through the point P (6, e6 ) to the graph of the function: f (x) = ex . 2. Find the point of tangency and the equation of the tangent
North Texas - MATH - 1400
Calculus and Vectors How to get an A+6.1 An Introduction to Vectors A Scalars and Vectors Scalars (in Mathematics and Physics) are quantities described completely by a number and eventually a measurement unit. Vectors are quantities described by a magnit
North Texas - MATH - 1400
Calculus and Vectors How to get an A+6.2 Addition and Subtraction of Geometric Vectors A Addition of two Vectors r r r The vector addition s of two vectors a and b is rr denoted by a + b and is called the sum or resultant of the two vectors. So: rrr s =
North Texas - MATH - 1400
Calculus and Vectors How to get an A+6.3 Multiplication of a Vector by a Scalar B Properties A Multiplication of a Vector by a Scalar r By multiplying a vector v by a scalar k we obtain The following properties apply for multiplication of a vector r a ne
North Texas - MATH - 1400
Calculus and Vectors How to get an A+6.4 Properties of Vectors A Properties of Vectors rrrr a +b =b +a rrrrr a +0 = 0+a = a r r r rr a + (a ) = (a ) + a = 0 rr rr rr (a + b ) + c = a + (b + c ) r r | ka |=| k | | a | r r r r k (a + b ) = ka + kb r r r (k
North Texas - MATH - 1400
Calculus and Vectors How to get an A+6.5 Vectors in R2 and R3 A Polar Coordinates Given a Cartesian system of coordinates, a 2D r r vector v may be defined by its magnitude | v | and the counter-clockwise angle between the positive direction of the x-axi
North Texas - MATH - 1400
Calculus and Vectors How to get an A+6.6 Operations with Algebraic Vectors in R2 A 2D Algebraic Vectors A 2D Algebraic Vector may be written in components form as: r v = (v x , v y ) or in terms of unit vectors as: r r r v = vxi + v y j and has a magnitu
North Texas - MATH - 1400
Calculus and Vectors How to get an A+6.7 Operations with Algebraic Vectors in R3 A 3D Algebraic Vectors A 3D Algebraic Vector may be written in components form as: r v = (v x , v y , v z ) or in terms of unit vectors as: r r r r v = vxi + v y j + vz k an
North Texas - MATH - 1100
Direct Variation DIRECT VARIATION Learning Targets 1. Identify direct variation from an equation, a table, and a graph. 2. Write linear equations from context (including direct variation).Algebra 2Example 1: The price of a movie is \$8 for a general admi
North Texas - MATH - 1100
5.1 Working with Simple Quadratic Functions 5.1 WORKING WITH SIMPLE QUADRATIC FUNCTIONS Learning Targets 1. Recognize and graph quadratic functions of the form y = ax2 2. Solve quadratic equations of the form ax2 = c. 3. Write the equation of a quadratic