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School: Kentucky
Course: Business Calculus
MA 123 - Business Calculus Summer 2007 P. Busse Quiz 1 Solutions 1. (2 pts) An open box is to be constructed from a 30 30 square piece of posterboard by removing a small square from each corner and folding up the aps to form the sides. Express the volume
School: Kentucky
Course: Calculus IV
MA 214 Calculus IV Quiz 2 Name: 1. Consider the initial value problem y + t2 t y = et t, 9 with y(1) = 4. Determine the largest interval on which the existence and uniqueness theorem for rst order linear dierential equations guarantees the existence of a
School: Kentucky
Course: Calculus IV
MA 214 Calculus IV Quiz 3 Name: 1. Initially 10 grams of salt are dissolved into 40 liters of water. Brine with concentration of salt 3 grams per liter is added at a rate of 4 liters per minute. The tank is well mixed and drained at 4 liters per minute. L
School: Kentucky
Course: Calculus IV
MA 214 Calculus IV Exam 1 Name: 1. (16 points) Find all solutions to xy dy = 2 . dx (x + 1)(y + 1) y Solution: First y+1 = 0 when y = 0. Then y = 0 is a solution. y Dividing y+1 we have y + 1 dy x = 2 . y dx (x + 1) Integrate x dx. (x2 + 1) y+1 dy = y to
School: Kentucky
Course: Calculus IV
MA 214 Calculus IV Quiz 5 Name: 1. Use Eulers method with step size 0.25 to compute the approximate y-valuesy1 , y2 , y3 , and y4 of the solution of the initial-value problem y = 1 4x + 4y, y(0) = 1. Solution: With the step size h = 0.25, Eulers method ge
School: Kentucky
Course: Calculus IV
MA 214 Calculus IV Quiz 2 Name: 1. Find all solutions y = y(x) to the dierential equation y dy = + x. dx x+1 dy 1 dx + x+1 y = x. To nd an integrating eC |x + 1| = eC (x + 1) and pick (x) = Solution: First rewrite the equation as 1 dx x+1 (x), we calculat
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Algebra and Geometry Review We will make use of the Pythagorean Theorem repeatedly throughout the semester. We will also occasionally use facts about similar triangles. The following problems preview some of the ways we will use these geometric facts. 1.
School: Kentucky
Exponential and logarithmic functions Supplement: Practice/review problems The collection of problems listed below contains questions taken from previous MA123 exams. [1]. Suppose that f (x) = ln(g(x). Assume that g(5) = 3 and g (5) = 4. Find f (5). (a) 5
School: Kentucky
Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus Chapter 10: Practice/review problems The collection of problems listed below comprises questions taken from previous MA123 exams. x [1]. Let t2 + 3 dt. h(x) = Find
School: Kentucky
Computing some integrals Chapter 9: Practice/review problems The collection of problems listed below comprises questions taken from previous MA123 exams. [1]. Evaluate the sum 3 + 6 + 9 + 12 + + 30 (a) 55 (b) 550 (c) 110 (d) 275 (e) 165 (e) 2544 (e) 67239
School: Kentucky
The Twenty-first Lecture on March 5 4.2 The Simplex Method: Standard Minimization Problems 1. Minimization with <= Constraints The objective function is to be minimization All the variables involved are nonnegative. Each linear constraint may be w
School: Kentucky
Course: Business Calculus
MA 123 - Business Calculus Summer 2007 P. Busse Quiz 1 Solutions 1. (2 pts) An open box is to be constructed from a 30 30 square piece of posterboard by removing a small square from each corner and folding up the aps to form the sides. Express the volume
School: Kentucky
Course: Calculus IV
MA 214 Calculus IV Quiz 2 Name: 1. Consider the initial value problem y + t2 t y = et t, 9 with y(1) = 4. Determine the largest interval on which the existence and uniqueness theorem for rst order linear dierential equations guarantees the existence of a
School: Kentucky
Course: Calculus IV
MA 214 Calculus IV Quiz 3 Name: 1. Initially 10 grams of salt are dissolved into 40 liters of water. Brine with concentration of salt 3 grams per liter is added at a rate of 4 liters per minute. The tank is well mixed and drained at 4 liters per minute. L
School: Kentucky
Course: Calculus IV
MA 214 Calculus IV Exam 1 Name: 1. (16 points) Find all solutions to xy dy = 2 . dx (x + 1)(y + 1) y Solution: First y+1 = 0 when y = 0. Then y = 0 is a solution. y Dividing y+1 we have y + 1 dy x = 2 . y dx (x + 1) Integrate x dx. (x2 + 1) y+1 dy = y to
School: Kentucky
Course: Calculus IV
MA 214 Calculus IV Quiz 5 Name: 1. Use Eulers method with step size 0.25 to compute the approximate y-valuesy1 , y2 , y3 , and y4 of the solution of the initial-value problem y = 1 4x + 4y, y(0) = 1. Solution: With the step size h = 0.25, Eulers method ge
School: Kentucky
Course: Calculus IV
MA 214 Calculus IV Quiz 2 Name: 1. Find all solutions y = y(x) to the dierential equation y dy = + x. dx x+1 dy 1 dx + x+1 y = x. To nd an integrating eC |x + 1| = eC (x + 1) and pick (x) = Solution: First rewrite the equation as 1 dx x+1 (x), we calculat
School: Kentucky
Course: Business Calculus
MA 123 - Business Calculus Summer 2007 P. Busse Quiz 1 Solutions 1. (2 pts) An open box is to be constructed from a 30" 30" square piece of posterboard by removing a small square from each corner and folding up the flaps to form the sides. Express the vol
School: Kentucky
Summation Notation and Summation Formulas You may nd the summation formulas useful: n n n(n + 1) k= 2 k=1 k2 = k=1 n(n + 1)(2n + 1) 6 1. Write the sum 1 + 4 + 9 + 16 + 25 + + 196 + 225 in summation notation. Then use a summation formula to nd the value of
School: Kentucky
Optimization Word Problems 1. The product of two positive real numbers, x and y, is 24. (a) Find the minimal sum of these two numbers. (b) Find the minimal value of the expression 3x + 2y. 2. Find the point on the curve y = x which is closest to the point
School: Kentucky
Maximum and Minimum Values 1. On the same graph, plot both f (x) = x3 3x 5 and its derivative. What do you notice? (In particular, what appears to be true about f (x) when the derivative is zero? What appears to be true about f (x) when the derivative is
School: Kentucky
Increasing and Decreasing Functions, Concavity 1. Suppose f (x) = (x 1)(x 4)(x 9) = x3 14x2 + 49x 36. (a) Find the intervals on which f (x) is increasing and the intervals on which f (x) is decreasing. (b) Find the intervals on which f (x) is concave up a
School: Kentucky
Using the Fundamental Theorem of Calculus 1. Compute the integrals (a) 9 dt t (b) 5 3 9 dt t 2. Compute the integrals (a) t3 + t2 + t + 1 dt (b) 3 t3 + t2 + t + 1 dt 3 3. Find the derivative F (x) where x F (x) = 0 e5s ds (s2 + 2s + 19) 4. Find the deri
School: Kentucky
Chapter 7 Two types: (1) maximum-minimum problems and (2) related rate problems 7.2 Max-Min Word Problems o Ask you to find the largest or the smallest value a function has on an interval o Steps Read the problem once over quickly Read the problem
School: Kentucky
Course: Business Calculus
MA 123 - Business Calculus Summer 2007 P. Busse Quiz 1 Solutions 1. (2 pts) An open box is to be constructed from a 30 30 square piece of posterboard by removing a small square from each corner and folding up the aps to form the sides. Express the volume
School: Kentucky
Course: Calculus IV
MA 214 Calculus IV Quiz 2 Name: 1. Consider the initial value problem y + t2 t y = et t, 9 with y(1) = 4. Determine the largest interval on which the existence and uniqueness theorem for rst order linear dierential equations guarantees the existence of a
School: Kentucky
Course: Calculus IV
MA 214 Calculus IV Quiz 3 Name: 1. Initially 10 grams of salt are dissolved into 40 liters of water. Brine with concentration of salt 3 grams per liter is added at a rate of 4 liters per minute. The tank is well mixed and drained at 4 liters per minute. L
School: Kentucky
Course: Calculus IV
MA 214 Calculus IV Exam 1 Name: 1. (16 points) Find all solutions to xy dy = 2 . dx (x + 1)(y + 1) y Solution: First y+1 = 0 when y = 0. Then y = 0 is a solution. y Dividing y+1 we have y + 1 dy x = 2 . y dx (x + 1) Integrate x dx. (x2 + 1) y+1 dy = y to
School: Kentucky
Course: Calculus IV
MA 214 Calculus IV Quiz 5 Name: 1. Use Eulers method with step size 0.25 to compute the approximate y-valuesy1 , y2 , y3 , and y4 of the solution of the initial-value problem y = 1 4x + 4y, y(0) = 1. Solution: With the step size h = 0.25, Eulers method ge
School: Kentucky
Course: Calculus IV
MA 214 Calculus IV Quiz 2 Name: 1. Find all solutions y = y(x) to the dierential equation y dy = + x. dx x+1 dy 1 dx + x+1 y = x. To nd an integrating eC |x + 1| = eC (x + 1) and pick (x) = Solution: First rewrite the equation as 1 dx x+1 (x), we calculat
School: Kentucky
Course: Calculus IV
MA 214 Calculus IV Quiz 1 NAME: 1. Consider the differential equation y = y(t m y). (a) What is the order of the equation? (b) Is the equation linear or nonlinear? (c) Draw the direction field for this equation in the (t, y)plane at the following points (
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Business Calculus
11.4 123 - Busi~ess Csaladus Summer 2007 P.Busse Name QUIZ7 June 12, 2007 All work mvsl be clmrhy s h o ~ t nin o d ~ to r ~ r ~ i j1111 cwdit. r vr 1. (2 pts each) Find the ~ m r soll~tions o t h e following rqiintiotifi. ? f (a) 5 - 2e3" = - I 9 - c= -a
School: Kentucky
Course: Business Calculus
!VIA 123 - Businws Calculus Summer 2007 P. Busse rill ~cnrl: m f hF rl~nrI?~ m ~Ilournin order lo wc<:i?:r l l / rwdat. fi 1. ( 2 pts each) Find tht. tlrrivativr ,2~-' d.r (a) J,! = PC <'% O!/ \; .l ^E /, ? d->c 2. (2 pts) The slope a t each point ( r , ~
School: Kentucky
Course: Business Calculus
MA 123 - Business Calculus Summer 2007 P. Busse Quiz 1 Solutions 1. (2 pts) An open box is to be constructed from a 30" 30" square piece of posterboard by removing a small square from each corner and folding up the flaps to form the sides. Express the vol
School: Kentucky
Course: Business Calculus
MA 123 - Business Calculus Summer 2007 P. Busse Name Quiz 5 June 1, 2007 All work must be clearly shown in order to receive full credit. 1. Let f (x) = 2 3 x - 2x2 - 6x - 21. 3 (a) (2 pts) Determine the intervals where f (x) is increasing and decreasing.
School: Kentucky
Course: Business Calculus
MA 123 - Business Calculus Summer 2007 P. Busse Quiz 2 1. (3 pts) Evaluate lim x2 + 3x x3 x2 + x - 12 x3 x2 lim x2 + 3x x(x + 3) 18 = lim + x - 12 x3 (x + 4)(x - 3) 0 The expression cannot be simplified, so the limit does not exist. 2. (2 pts) Evaluate li
School: Kentucky
Course: Business Calculus
MA 123 - Business Calculus Summer 2007 P. Busse Quiz 3 Solutions 1. (3 pts) Use the definition to find the derivative of f (x) = 3x2 - x. Show all steps and include limit signs where needed. f (x) = = = = = = f (x + h) - f (x) (definition of the derivativ
School: Kentucky
Course: Business Calculus
MA 123 - Business Calculus Summer 2007 P. Busse Quiz 4 Solutions 1. (2 pts each) Dierentiate the following functions. Do not simplify. (a) f (x) = 2x4 6 1 = 6(2x4 + 7x 9) 2 + 7x 9 3 3 1 f (x) = 6 (2x4 + 7x 9) 2 (8x3 + 7) = 3(8x3 + 7)(2x4 + 7x 9) 2 . 2 (b)
School: Kentucky
Course: Business Calculus
MA 123 - Business Calculus Summer 2007 P. Busse Quiz 1 Solutions 1. (4 pts) Completely simplify the difference quotient for the function f (x) = 2x2 - 3x + 4. f (x + h) - f (x) h 2(x + h)2 - 3(x + h) + 4 - (2x2 - 3x + 4) h 2(x2 + 2hx + h2 ) - 3x - 3h + 4
School: Kentucky
MA 123 - Elementary Calculus SECOND MIDTERM FALL 2008 10/22/2008 Name: Sec.: Do not remove this answer page - you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calcul
School: Kentucky
MA 123 Elementary Calculus SECOND MIDTERM FALL 2007 10/17/2007 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator
School: Kentucky
MA 123 - Elementary Calculus FIRST MIDTERM SPRING 2008 02/06/2008 Name: Sec.: Do not remove this answer page - you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calcu
School: Kentucky
Algebra and Geometry Review We will make use of the Pythagorean Theorem repeatedly throughout the semester. We will also occasionally use facts about similar triangles. The following problems preview some of the ways we will use these geometric facts. 1.
School: Kentucky
MA 123 Elementary Calculus FIRST MIDTERM FALL 2006 09/20/2006 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator
School: Kentucky
MA 123 Elementary Calculus FIRST MIDTERM SPRING 2009 02/11/2009 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculato
School: Kentucky
Summation Notation and Summation Formulas You may nd the summation formulas useful: n n n(n + 1) k= 2 k=1 k2 = k=1 n(n + 1)(2n + 1) 6 1. Write the sum 1 + 4 + 9 + 16 + 25 + + 196 + 225 in summation notation. Then use a summation formula to nd the value of
School: Kentucky
Optimization Word Problems 1. The product of two positive real numbers, x and y, is 24. (a) Find the minimal sum of these two numbers. (b) Find the minimal value of the expression 3x + 2y. 2. Find the point on the curve y = x which is closest to the point
School: Kentucky
Maximum and Minimum Values 1. On the same graph, plot both f (x) = x3 3x 5 and its derivative. What do you notice? (In particular, what appears to be true about f (x) when the derivative is zero? What appears to be true about f (x) when the derivative is
School: Kentucky
Increasing and Decreasing Functions, Concavity 1. Suppose f (x) = (x 1)(x 4)(x 9) = x3 14x2 + 49x 36. (a) Find the intervals on which f (x) is increasing and the intervals on which f (x) is decreasing. (b) Find the intervals on which f (x) is concave up a
School: Kentucky
Using the Fundamental Theorem of Calculus 1. Compute the integrals (a) 9 dt t (b) 5 3 9 dt t 2. Compute the integrals (a) t3 + t2 + t + 1 dt (b) 3 t3 + t2 + t + 1 dt 3 3. Find the derivative F (x) where x F (x) = 0 e5s ds (s2 + 2s + 19) 4. Find the deri
School: Kentucky
MA 123 Elementary Calculus FINAL EXAM Spring 2009 05/06/2009 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator d
School: Kentucky
MA 123 Elementary Calculus FINAL EXAM FALL 2009 12/14/2009 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator dur
School: Kentucky
Exponential and logarithmic functions Supplement: Practice/review problems The collection of problems listed below contains questions taken from previous MA123 exams. [1]. Suppose that f (x) = ln(g(x). Assume that g(5) = 3 and g (5) = 4. Find f (5). (a) 5
School: Kentucky
MA 123 Elementary Calculus FINAL EXAM Spring 2010 05/06/2010 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator d
School: Kentucky
School: Kentucky
MA 123 Elementary Calculus THIRD MIDTERM EXAM Spring 2010 4/14/2010 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calcu
School: Kentucky
MA 123 Elementary Calculus SECOND MIDTERM EXAM Spring 2010 03/10/2010 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing cal
School: Kentucky
MA 123 Elementary Calculus FIRST MIDTERM EXAM Spring 2010 02/10/2010 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calc
School: Kentucky
MA 123 Elementary Calculus THIRD MIDTERM SPRING 2009 4/15/2009 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator
School: Kentucky
MA 123 Elementary Calculus THIRD MIDTERM FALL 2009 11/18/2009 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator
School: Kentucky
MA 123 Elementary Calculus SECOND MIDTERM SPRING 2009 03/11/2009 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculat
School: Kentucky
MA 123 Elementary Calculus SECOND MIDTERM FALL 2009 10/21/2009 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator
School: Kentucky
MA 123 Elementary Calculus FIRST MIDTERM FALL 2009 09/23/2009 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator
School: Kentucky
Derivatives From Graphs 1. The graph of the function y = f (x) is shown below, along with the graph of the tangent line to this curve at x = 2. Determine f (2). 2. Determine the x coordinates of all points of nondierentiability for the function graphed be
School: Kentucky
Computing Derivatives Using the Denition of the Derivative 1. You would like to know f (2). Suppose you dont have a formula for f (x) (Thus, none of the shortcut formulas can be applied ) but you happen to know 3 x2 h + 15 x h2 + 19 h f (x + h) f (x) = x2
School: Kentucky
Computing Some Integrals 1. Compute the integral 5 3 x2 dx 0 via the following steps: (a) Write down a sum which approximates the integral. The sum should use n rectangles of equal width and the height of each rectangle should be determined by the right e
School: Kentucky
Exponential and logarithmic functions Supplement: Practice/review problems The collection of problems listed below contains questions taken from previous MA123 exams. [1]. Suppose that f (x) = ln(g(x). Assume that g(5) = 3 and g (5) = 4. Find f (5). (a) 5
School: Kentucky
MA 123 - Elementary Calculus FINAL EXAM SPRING 2008 04/29/2008 Name: Sec.: Do not remove this answer page - you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculat
School: Kentucky
MA 123 - Elementary Calculus FINAL EXAM SPRING 2007 04/30/2007 Name: Sec.: Do not remove this answer page - you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculat
School: Kentucky
MA 123 - Elementary Calculus FINAL EXAM FALL 2006 12/12/2006 Name: Sec.: Do not remove this answer page - you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator
School: Kentucky
MA 123 Elementary Calculus THIRD MIDTERM FALL 2008 11/19/2008 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator
School: Kentucky
MA 123 Elementary Calculus THIRD MIDTERM FALL 2006 11/15/2006 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Algebra and Geometry Review We will make use of the Pythagorean Theorem repeatedly throughout the semester. We will also occasionally use facts about similar triangles. The following problems preview some of the ways we will use these geometric facts. 1.
School: Kentucky
Exponential and logarithmic functions Supplement: Practice/review problems The collection of problems listed below contains questions taken from previous MA123 exams. [1]. Suppose that f (x) = ln(g(x). Assume that g(5) = 3 and g (5) = 4. Find f (5). (a) 5
School: Kentucky
Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus Chapter 10: Practice/review problems The collection of problems listed below comprises questions taken from previous MA123 exams. x [1]. Let t2 + 3 dt. h(x) = Find
School: Kentucky
Computing some integrals Chapter 9: Practice/review problems The collection of problems listed below comprises questions taken from previous MA123 exams. [1]. Evaluate the sum 3 + 6 + 9 + 12 + + 30 (a) 55 (b) 550 (c) 110 (d) 275 (e) 165 (e) 2544 (e) 67239
School: Kentucky
The idea of the integral Chapter 8: Practice/review problems The collection of problems listed below comprises questions taken from previous MA123 exams. [1]. Estimate the area under the graph of f (x) = x2 + 2 on the interval [0, 2] by dividing the inter
School: Kentucky
Word Problems Chapter 7: Practice/review problems The collection of problems listed below contains questions taken from previous MA123 exams. Max-min problems [1]. A eld has the shape of a rectangle with two semicircles attached at opposite sides. Find th
School: Kentucky
Formulas for derivatives Chapter 5: Practice/review problems The collection of problems listed below contains questions taken from previous MA123 exams. Derivatives [1]. If f (x) = 6x2 + 3x 1, nd f (x). (a) 6x + 1 (b) 12x + 3 (c) 12x 1 (d) 2x + 3 (e) 2x +
School: Kentucky
Extreme values, the Mean Value Theorem, curve sketching and concavity Chapter 6: Practice/review problems The collection of problems listed below contains questions taken from previous MA123 exams. Extreme values problems on a closed interval [1]. Suppose
School: Kentucky
Limits and continuity Chapter 3: Practice/review problems The collection of problems listed below contains questions taken from previous MA123 exams. Limits and one-sided limits [1]. Suppose H(t) = t2 + 5t + 1. Find the limit (a) 15 (b) 1 (c) 9 lim H(t).
School: Kentucky
School: Kentucky
12 Introduction to Trigonometry Concepts: Angles Initial Side and Terminal Side Standard Position Coterminal Angles Measuring Angles Radian Measure vs. Degree Measure Radian Measure as a Distance on the Unit Circle Converting between Radian Measur
School: Kentucky
16 Inverse Trigonometric Functions Concepts: Restricting the Domain of the Trigonometric Functions The Inverse Sine Function The Inverse Cosine Function The Inverse Tangent Function Using the Inverse Trigonometric Functions and the Periodicity Identi
School: Kentucky
15 More Trigonometric Identities Concepts: The Addition and Subtraction Identities for Sine and Cosine The Cofunction Identities for Sine and Cosine Identities That You Should Learn 15.1 Addition and Subtraction Identities for Sine and Cosine Theorem 1
School: Kentucky
13 Trigonometric Graphs Concepts: Period The Graph of the sin, cos, tan, csc, sec, and cot Functions Applying Graph Transformations to the Graphs of the sin, cos, tan, csc, sec, and cot Functions Using Graphical Evidence to Make Conjectures about Iden
School: Kentucky
14 Simplifying Trigonometric Expressions and Proving Trigonometric Identities Concepts: Expressions vs. Identities Simplifying Trigonometric Expressions Proving Trigonmetric Identities Disproving Trigonmetric Identities (Section 7.1) 14.1 Simplifying
School: Kentucky
17 Triangle Trigonometry Concepts: Trigonometry for Acute Angles - The Right Triangle Perspective Trigonometry for Acute and Obtuse Angles The Law of Cosines The Law of Sines The Problem with SSA Solving Application Problems that Involve Acute and O
School: Kentucky
19 Parametric Equations And Polar Coordinates Concepts: Sketching Graphs of Parametric Equations Converting Between Polar Coordinates and Cartesian Coordinates Sketching Graphs of Polar Equations (Sections 10.5-10.6) There are a lot of graphs which do
School: Kentucky
11 Exponential and Logarithmic Functions Concepts: Exponential Functions Power Functions vs. Exponential Functions The Denition of an Exponential Function Graphing Exponential Functions Exponential Growth and Exponential Decay The Irrational Number
School: Kentucky
Course: Elementary Calculus
MA123, Chapter 1: Equations, functions, and graphs (pp. 1-15, Gootman) Chapter Goals: Assignments: Solve an equation for one variable in terms of another. What is a function? Find inverse functions. What is a graph? Understand linear and quadratic fu
School: Kentucky
Course: Elementary Calculus
MA123, Chapter 4: Computing Some Derivatives (pp. 69-82, Gootman) Chapter Goals: Assignments: Understand the derivative as the slope of the tangent line at a point. Investigate further the notions of continuity and dierentiability. Use the denition to
School: Kentucky
Course: Elementary Calculus
MA123, Chapter 3: The idea of limits (pp. 47-67, Gootman) Chapter Goals: Assignments: Evaluate limits. Evaluate one-sided limits. Understand the concepts of continuity and dierentiability and their relationship. Assignment 04 Assignment 05 Earlier, the
School: Kentucky
Course: Elementary Calculus
MA123, Chapter 2: Change, and the idea of the derivative (pp. 17-45, Gootman) Chapter Goals: Assignments: Understand average rates of change. Understand the ideas leading to instantaneous rates of change. Understand the connection between instantaneous ra
School: Kentucky
Course: Elementary Calculus
MA123, Chapter 5: Formulas for Derivatives (pp. 83-102, Gootman) Chapter Goals: Know and be able to apply the formulas for derivatives. Understand the chain rule and be able to apply it. Know how to compute higher derivatives. Assignment 08 Assignment
School: Kentucky
Course: Elementary Calculus
MA123, Chapter 6: Extreme values, Mean Value Theorem, Curve sketching, and Concavity Chapter Goals: Apply the Extreme Value Theorem to nd the global extrema for continuous function on closed and bounded interval. Understand the connection between critic
School: Kentucky
Course: Elementary Calculus
MA123, Supplement: Exponential and logarithmic functions (pp. 315-319, Gootman) Chapter Goals: Review properties of exponential and logarithmic functions. Learn how to dierentiate exponential and logarithmic functions. Learn about exponential growth an
School: Kentucky
Course: Elementary Calculus
MA123, Chapter 8: Idea of the Integral (pp. 155-187, Gootman) Chapter Understand the relationship between the area under a curve and the denite integral. Understand the relationship between velocity (speed), distance and the denite integral. Estimate t
School: Kentucky
Course: Elementary Calculus
MA123, Chapter 7: Word Problems (pp. 125-153, Gootman) Chapter Goals: In this Chapter we learn a general strategy on how to approach the two main types of word problems that one usually encounters in a rst Calculus course: Max-Min problems Related Rates
School: Kentucky
Course: Elementary Calculus
MA123, Chapter 2: Change, and the idea of the derivative (pp. 17-45, Gootman) Chapter Goals: Assignments: Understand average rates of change. Understand the ideas leading to instantaneous rates of change. Understand the connection between instantaneous ra
School: Kentucky
Course: Elementary Calculus
Algebra review Chapter 1: Practice/review problems The collection of problems listed below contains questions taken from previous MA123 exams. Warm ups [1]. A rectangular solid has edges of lengths 4 ft, 5 ft, and 8 ft. Suppose we double the length of two
School: Kentucky
Course: Elementary Calculus
Rates of change and derivatives Chapter 2: Practice/review problems The collection of problems listed below contains questions taken from previous MA123 exams. Average rates of change (Word Problems) [1]. A train travels from A to B to C. The distance fro
School: Kentucky
Course: Elementary Calculus
MA123, Chapter 1: Equations, functions, and graphs (pp. 1-15, Gootman) Chapter Goals: Assignments: Solve an equation for one variable in terms of another. What is a function? Find inverse functions. What is a graph? Understand linear and quadratic fu
School: Kentucky
Course: Precalculus
Precalculus Notes University of Kentucky Fall 2011 1 1.1 Some Things To Know About This Course The Syllabus If you are in sections 001-008 the syllabus is located at www.ms.uky.edu/mshaw. It is a contract between you and your instructor. Read it. Referenc
School: Kentucky
Course: Finite Mathematics
3 3.1 More on Accumulation and Discount Functions Introduction In previous section, we used (1.03)# of years as the accumulation factor. This section looks at other accumulation factors, including various forms of compound interest. 3.2 Interest earned ve
School: Kentucky
The Twenty-first Lecture on March 5 4.2 The Simplex Method: Standard Minimization Problems 1. Minimization with <= Constraints The objective function is to be minimization All the variables involved are nonnegative. Each linear constraint may be w
School: Kentucky
MA109, Activity 1: Modeling the Real World (Section P.1, pp. 4-9) Today's Goal: Date: Assignments: Distribution of the syllabus & discussion of course policy. We then study simple examples of how the methods of Algebra allow us to describe various
School: Kentucky
Course: Business Calculus
MA 123 - Business Calculus Summer 2007 P. Busse Quiz 1 Solutions 1. (2 pts) An open box is to be constructed from a 30 30 square piece of posterboard by removing a small square from each corner and folding up the aps to form the sides. Express the volume
School: Kentucky
Course: Calculus IV
MA 214 Calculus IV Quiz 2 Name: 1. Consider the initial value problem y + t2 t y = et t, 9 with y(1) = 4. Determine the largest interval on which the existence and uniqueness theorem for rst order linear dierential equations guarantees the existence of a
School: Kentucky
Course: Calculus IV
MA 214 Calculus IV Quiz 3 Name: 1. Initially 10 grams of salt are dissolved into 40 liters of water. Brine with concentration of salt 3 grams per liter is added at a rate of 4 liters per minute. The tank is well mixed and drained at 4 liters per minute. L
School: Kentucky
Course: Calculus IV
MA 214 Calculus IV Exam 1 Name: 1. (16 points) Find all solutions to xy dy = 2 . dx (x + 1)(y + 1) y Solution: First y+1 = 0 when y = 0. Then y = 0 is a solution. y Dividing y+1 we have y + 1 dy x = 2 . y dx (x + 1) Integrate x dx. (x2 + 1) y+1 dy = y to
School: Kentucky
Course: Calculus IV
MA 214 Calculus IV Quiz 5 Name: 1. Use Eulers method with step size 0.25 to compute the approximate y-valuesy1 , y2 , y3 , and y4 of the solution of the initial-value problem y = 1 4x + 4y, y(0) = 1. Solution: With the step size h = 0.25, Eulers method ge
School: Kentucky
Course: Calculus IV
MA 214 Calculus IV Quiz 2 Name: 1. Find all solutions y = y(x) to the dierential equation y dy = + x. dx x+1 dy 1 dx + x+1 y = x. To nd an integrating eC |x + 1| = eC (x + 1) and pick (x) = Solution: First rewrite the equation as 1 dx x+1 (x), we calculat
School: Kentucky
Course: Calculus IV
MA 214 Calculus IV Quiz 1 NAME: 1. Consider the differential equation y = y(t m y). (a) What is the order of the equation? (b) Is the equation linear or nonlinear? (c) Draw the direction field for this equation in the (t, y)plane at the following points (
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Elementary Calculus And Its Applications
School: Kentucky
Course: Business Calculus
11.4 123 - Busi~ess Csaladus Summer 2007 P.Busse Name QUIZ7 June 12, 2007 All work mvsl be clmrhy s h o ~ t nin o d ~ to r ~ r ~ i j1111 cwdit. r vr 1. (2 pts each) Find the ~ m r soll~tions o t h e following rqiintiotifi. ? f (a) 5 - 2e3" = - I 9 - c= -a
School: Kentucky
Course: Business Calculus
!VIA 123 - Businws Calculus Summer 2007 P. Busse rill ~cnrl: m f hF rl~nrI?~ m ~Ilournin order lo wc<:i?:r l l / rwdat. fi 1. ( 2 pts each) Find tht. tlrrivativr ,2~-' d.r (a) J,! = PC <'% O!/ \; .l ^E /, ? d->c 2. (2 pts) The slope a t each point ( r , ~
School: Kentucky
Course: Business Calculus
MA 123 - Business Calculus Summer 2007 P. Busse Name Quiz 5 June 1, 2007 All work must be clearly shown in order to receive full credit. 1. Let f (x) = 2 3 x - 2x2 - 6x - 21. 3 (a) (2 pts) Determine the intervals where f (x) is increasing and decreasing.
School: Kentucky
Course: Business Calculus
MA 123 - Business Calculus Summer 2007 P. Busse Quiz 2 1. (3 pts) Evaluate lim x2 + 3x x3 x2 + x - 12 x3 x2 lim x2 + 3x x(x + 3) 18 = lim + x - 12 x3 (x + 4)(x - 3) 0 The expression cannot be simplified, so the limit does not exist. 2. (2 pts) Evaluate li
School: Kentucky
Course: Business Calculus
MA 123 - Business Calculus Summer 2007 P. Busse Quiz 3 Solutions 1. (3 pts) Use the definition to find the derivative of f (x) = 3x2 - x. Show all steps and include limit signs where needed. f (x) = = = = = = f (x + h) - f (x) (definition of the derivativ
School: Kentucky
Course: Business Calculus
MA 123 - Business Calculus Summer 2007 P. Busse Quiz 4 Solutions 1. (2 pts each) Dierentiate the following functions. Do not simplify. (a) f (x) = 2x4 6 1 = 6(2x4 + 7x 9) 2 + 7x 9 3 3 1 f (x) = 6 (2x4 + 7x 9) 2 (8x3 + 7) = 3(8x3 + 7)(2x4 + 7x 9) 2 . 2 (b)
School: Kentucky
Course: Business Calculus
MA 123 - Business Calculus Summer 2007 P. Busse Quiz 1 Solutions 1. (4 pts) Completely simplify the difference quotient for the function f (x) = 2x2 - 3x + 4. f (x + h) - f (x) h 2(x + h)2 - 3(x + h) + 4 - (2x2 - 3x + 4) h 2(x2 + 2hx + h2 ) - 3x - 3h + 4
School: Kentucky
MA 123 - Elementary Calculus SECOND MIDTERM FALL 2008 10/22/2008 Name: Sec.: Do not remove this answer page - you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calcul
School: Kentucky
MA 123 Elementary Calculus SECOND MIDTERM FALL 2007 10/17/2007 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator
School: Kentucky
MA 123 - Elementary Calculus FIRST MIDTERM SPRING 2008 02/06/2008 Name: Sec.: Do not remove this answer page - you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calcu
School: Kentucky
MA 123 Elementary Calculus FIRST MIDTERM FALL 2006 09/20/2006 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator
School: Kentucky
MA 123 Elementary Calculus FIRST MIDTERM SPRING 2009 02/11/2009 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculato
School: Kentucky
MA 123 Elementary Calculus FINAL EXAM Spring 2009 05/06/2009 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator d
School: Kentucky
MA 123 Elementary Calculus FINAL EXAM FALL 2009 12/14/2009 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator dur
School: Kentucky
MA 123 Elementary Calculus FINAL EXAM Spring 2010 05/06/2010 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator d
School: Kentucky
School: Kentucky
MA 123 Elementary Calculus THIRD MIDTERM EXAM Spring 2010 4/14/2010 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calcu
School: Kentucky
MA 123 Elementary Calculus SECOND MIDTERM EXAM Spring 2010 03/10/2010 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing cal
School: Kentucky
MA 123 Elementary Calculus FIRST MIDTERM EXAM Spring 2010 02/10/2010 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calc
School: Kentucky
MA 123 Elementary Calculus THIRD MIDTERM SPRING 2009 4/15/2009 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator
School: Kentucky
MA 123 Elementary Calculus THIRD MIDTERM FALL 2009 11/18/2009 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator
School: Kentucky
MA 123 Elementary Calculus SECOND MIDTERM SPRING 2009 03/11/2009 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculat
School: Kentucky
MA 123 Elementary Calculus SECOND MIDTERM FALL 2009 10/21/2009 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator
School: Kentucky
MA 123 Elementary Calculus FIRST MIDTERM FALL 2009 09/23/2009 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator
School: Kentucky
MA 123 - Elementary Calculus FINAL EXAM SPRING 2008 04/29/2008 Name: Sec.: Do not remove this answer page - you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculat
School: Kentucky
MA 123 - Elementary Calculus FINAL EXAM SPRING 2007 04/30/2007 Name: Sec.: Do not remove this answer page - you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculat
School: Kentucky
MA 123 - Elementary Calculus FINAL EXAM FALL 2006 12/12/2006 Name: Sec.: Do not remove this answer page - you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator
School: Kentucky
MA 123 Elementary Calculus THIRD MIDTERM FALL 2008 11/19/2008 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator
School: Kentucky
MA 123 Elementary Calculus THIRD MIDTERM FALL 2006 11/15/2006 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator
School: Kentucky
MA 123 Elementary Calculus SECOND MIDTERM SPRING 2007 03/07/2007 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculat
School: Kentucky
MA123 Exam 1 February 06 2008 NAME _ Section _ Problem 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Answer b c d b c d b c d b c d b c d b c d b c d b c d b c d b c d b c d b c d b c d b c d b c d a a a a a a a a a a a a a a a e e e e e e e e e e e e e e e Instruc
School: Kentucky
MA 123 Elem. Calculus FINAL EXAM SPRING 2007 30 April 2007 Name: Sec.: Do not remove this answer page you will turn in the entire exam. You have two hours to do this exam. No books or notes may be used. You may use a graphing calculator during the exam, b
School: Kentucky
MA 123 Elem. Calculus 3rd MIDTERM SPRING 2007 11 April 2007 Name: Sec.: Do not remove this answer page you will turn in the entire exam. You have two hours to do this exam. No books or notes may be used. You may use a graphing calculator during the exam,
School: Kentucky
MA 123 Elem. Calculus FIRST MIDTERM SPRING 2007 02/07/2007 Name: Sec.: Do not remove this answer page you will turn in the entire exam. You have two hours to do this exam. No books or notes may be used. You may use a graphing calculator during the exam, b
School: Kentucky
School: Kentucky
MA 123 Elementary Calculus FINAL EXAM Spring 2009 05/06/2009 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator d
School: Kentucky
MA 123 Elementary Calculus FINAL EXAM Fall 2009 12/14/2009 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator dur
School: Kentucky
MA 123 - Elementary Calculus FINAL EXAM FALL 2008 12/17/2008 Name: Sec.: Do not remove this answer page - you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator
School: Kentucky
MA123 Final Exam December 12 2007 NAME _ Section _ Problem 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 a a a a a a a a a a a a a a a Answer b b b b b b b b b b b b b b b c c c c c c c c c c c c c c c d d d d d d d d d d d d d d d e e e e e e e e e e e e e e e Ins
School: Kentucky
MA123 Final Exam 12 December 2006 NAME _ Section _ Problem 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Answer b c d b c d b c d b c d b c d b c d b c d b c d b c d b c d b c d b c d b c d b c d b c d a a a a a a a a a a a a a a a e e e e e e e e e e e e e e e Ins
School: Kentucky
MA123 Exam 1 September 19 2007 Instructions. Circle your answer in ink on the page containing the problem and on the cover sheet. After the exam begins, you may not ask a question about the exam. Be sure you have all pages (containing 15 problems) before
School: Kentucky
MA123 Exam 2 October 17 2007 NAME _ Section _ Problem 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 a a a a a a a a a a a a a a a Answer b b b b b b b b b b b b b b b c c c c c c c c c c c c c c c d d d d d d d d d d d d d d d e e e e e e e e e e e e e e e Instruct
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 a a a a a a a a a a a a a a a b b b b b b b b b b b b b b b c c c c c c c c c c c c c c c d d d d d d d d d d d d d d d e e e e e e e e e e e e e e e (0, 0) y = 2 x2 2 3 8 3 8 3 4 3 2 3 8 3 2 3 4 3 2 3 4 3
School: Kentucky
Course: Business Calculus
MA 123 - Business Calculus Summer 2007 P. Busse Quiz 1 Solutions 1. (2 pts) An open box is to be constructed from a 30" 30" square piece of posterboard by removing a small square from each corner and folding up the flaps to form the sides. Express the vol
School: Kentucky
Summation Notation and Summation Formulas You may nd the summation formulas useful: n n n(n + 1) k= 2 k=1 k2 = k=1 n(n + 1)(2n + 1) 6 1. Write the sum 1 + 4 + 9 + 16 + 25 + + 196 + 225 in summation notation. Then use a summation formula to nd the value of
School: Kentucky
Optimization Word Problems 1. The product of two positive real numbers, x and y, is 24. (a) Find the minimal sum of these two numbers. (b) Find the minimal value of the expression 3x + 2y. 2. Find the point on the curve y = x which is closest to the point
School: Kentucky
Maximum and Minimum Values 1. On the same graph, plot both f (x) = x3 3x 5 and its derivative. What do you notice? (In particular, what appears to be true about f (x) when the derivative is zero? What appears to be true about f (x) when the derivative is
School: Kentucky
Increasing and Decreasing Functions, Concavity 1. Suppose f (x) = (x 1)(x 4)(x 9) = x3 14x2 + 49x 36. (a) Find the intervals on which f (x) is increasing and the intervals on which f (x) is decreasing. (b) Find the intervals on which f (x) is concave up a
School: Kentucky
Using the Fundamental Theorem of Calculus 1. Compute the integrals (a) 9 dt t (b) 5 3 9 dt t 2. Compute the integrals (a) t3 + t2 + t + 1 dt (b) 3 t3 + t2 + t + 1 dt 3 3. Find the derivative F (x) where x F (x) = 0 e5s ds (s2 + 2s + 19) 4. Find the deri
School: Kentucky
Exponential and logarithmic functions Supplement: Practice/review problems The collection of problems listed below contains questions taken from previous MA123 exams. [1]. Suppose that f (x) = ln(g(x). Assume that g(5) = 3 and g (5) = 4. Find f (5). (a) 5
School: Kentucky
Derivatives From Graphs 1. The graph of the function y = f (x) is shown below, along with the graph of the tangent line to this curve at x = 2. Determine f (2). 2. Determine the x coordinates of all points of nondierentiability for the function graphed be
School: Kentucky
Computing Derivatives Using the Denition of the Derivative 1. You would like to know f (2). Suppose you dont have a formula for f (x) (Thus, none of the shortcut formulas can be applied ) but you happen to know 3 x2 h + 15 x h2 + 19 h f (x + h) f (x) = x2
School: Kentucky
Computing Some Integrals 1. Compute the integral 5 3 x2 dx 0 via the following steps: (a) Write down a sum which approximates the integral. The sum should use n rectangles of equal width and the height of each rectangle should be determined by the right e
School: Kentucky
Summation Notation and Summation Formulas (page 24), Solutions You may nd the summation formulas useful: n n n(n + 1) k= 2 k=1 k2 = k=1 n(n + 1)(2n + 1) 6 1. Write the sum 1 + 4 + 9 + 16 + 25 + + 196 + 225 in summation notation. Then use a summation formu
School: Kentucky
Related Rates (page 21), Solutions 1. Two trains leave a train station at 1:00 PM. One train travels north at 70 miles per hour. The other train travels east at 50 miles per hour. How fast is the distance between the two trains changing at 4:00 PM? Soluti
School: Kentucky
Optimization Word Problems (page 20), Solutions 1. The product of two positive real numbers, x and y, is 24. (a) Find the minimal sum of these two numbers. (b) Find the minimal value of the expression 3x + 2y. Solution: (a) Let S be the sum, S = x + y. Bu
School: Kentucky
Related Rates 1. Two trains leave a train station at 1:00 PM. One train travels north at 70 miles per hour. The other train travels east at 50 miles per hour. How fast is the distance between the two trains changing at 4:00 PM? 2. Suppose the height of a
School: Kentucky
Using the Fundamental Theorem of Calculus Some More (pages 28-29), Solutions 1. Compute the denite integral 2 1 6x5 dx x6 + 1 (Hint: Youll need to use a substitution) Solution: Let u = x6 + 1. Then du = 6x5 dx. Now, when x = 1, then u = 16 + 1 = 2, and wh
School: Kentucky
Computing Derivatives With Formulas Some More (pages 14-15), Solutions This worksheet focuses on computing derivatives using the shortcut formulas, including the power rule, product rule, quotient rule, chain rule, and exponential functions. We will make
School: Kentucky
Using the Fundamental Theorems of Calculus Some More 1. Compute the denite integral 2 1 6x5 dx x6 + 1 (Hint: Youll need to use a substitution) 2. Compute the denite integral 2 3 3x2 ex dx 1 (Hint: Youll need to use a substitution) 3. Compute the integral
School: Kentucky
More Derivative Formulas This worksheet focuses on computing derivatives using the shortcut formulas, including the power rule, product rule, quotient rule, chain rule, and exponential functions. We will make constant use of these techniques throughout th
School: Kentucky
Maximum and Minimum Values (page 18), Solutions 1. On the same graph, plot both f (x) = x3 3x 5 and its derivative. What do you notice? (In particular, what appears to be true about f (x) when the derivative is zero? What appears to be true about f (x) wh
School: Kentucky
Limits (pages 8-9), Solutions This worksheet focuses on limits and the related idea of continuity. Many limits can be computed numerically (through a table of values), graphically, and algebraically. When possible, try to compute each of the limits below
School: Kentucky
Limits This worksheet focuses on limits and the related idea of continuity. Many limits can be computed numerically (through a table of values), graphically, and algebraically. When possible, try to compute each of the limits below using all three methods
School: Kentucky
Introduction to Integration (page 22), Solutions 1. Estimate the area under the curve y = x2 on the interval [0, 4] in four dierent ways: (a) Divide [0, 4] into four equal subintervals, and use the left endpoint on each subinterval as the sample point. (b
School: Kentucky
MA123, Supplement: Exponential and logarithmic functions (pp. 315-319, Gootman) Chapter Goals: Review properties of exponential and logarithmic functions. Learn how to dierentiate exponential and logarithmic functions. Learn about exponential growth an
School: Kentucky
Increasing and Decreasing Functions, Concavity 1. Suppose f (x) = (x 1)(x 4)(x 9) = x3 14x2 + 49x 36. (a) Find the intervals on which f (x) is increasing and the intervals on which f (x) is decreasing. (b) Find the intervals on which f (x) is concave up a
School: Kentucky
Using the Fundamental Theorem of Calculus (pages 26-27), Solutions 1. Compute the integrals (a) 9 dt t (b) 5 3 9 dt t Solutions: 9 Notice t = 9 t1/2 so we may apply the power rule for integrals: 9 dt = 9 t t1/2 dt = 9 1 1 1 2 t1/2 + C = 18 t + C For th
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Introduction to Integration 1. Estimate the area under the curve y = x2 on the interval [0, 4] in four dierent ways: (a) Divide [0, 4] into four equal subintervals, and use the left endpoint on each subinterval as the sample point. (b) Divide [0, 4] into
School: Kentucky
School: Kentucky
Exponential Function Word Problems Exponential growth is modelled by y = y0 ekt There are four variables, the initial amount, y0 , the time t, the growth factor k, and the current amount y. You should be comfortable with nding any one of these four, given
School: Kentucky
MA123, Supplement: Exponential and logarithmic functions (pp. 315-319, Gootman) Chapter Goals: Review properties of exponential and logarithmic functions. Learn how to dierentiate exponential and logarithmic functions. Learn about exponential growth an
School: Kentucky
MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp. 207-233, Gootman) Chapter Goals: Understand the statement of the Fundamental Theorem of Calculus. Learn how to compute the antiderivative
School: Kentucky
MA123, Chapter 7: Word Problems (pp. 125-153, Gootman) Chapter Goals: In this Chapter we learn a general strategy on how to approach the two main types of word problems that one usually encounters in a rst Calculus course: Max-Min problems Related Rates
School: Kentucky
Computing Derivatives With Formulas (pages 12-13), Solutions This worksheet focuses on computing derivatives using the shortcut formulas, including the power rule, product rule, and quotient rule. We will make constant use of these techniques throughout t
School: Kentucky
Formulas for Derivative This worksheet focuses on computing derivatives using the shortcut formulas, including the power rule, product rule, and quotient rule. We will make constant use of these techniques throughout the rest of the semester. Invest the t
School: Kentucky
Computing Derivatives With Graphs (page 10), Solutions 1. The graph of the function y = f (x) is shown below, along with the graph of the tangent line to this curve at x = 2. Determine f (2). Solution: f (2) is the slope of the tangent line to y = f (x) a
School: Kentucky
MA123, Chapter 9: Computing some integrals (pp. 189-205, Gootman) Understand how to use basic summation formulas to evaluate more complex sums. Chapter Goals: Assignments: Understand how to compute limits of rational functions at innity. Understand how
School: Kentucky
Computing Derivatives Using the Denition of the Derivative (page 11), Solutions 1. You would like to know f (2). Suppose you dont have a formula for f (x) (Thus, none of the shortcut formulas can be applied ) but you happen to know 3 x2 h + 15 x h2 + 19 h
School: Kentucky
MA123, Chapter 8: Idea of the Integral (pp. 155-187, Gootman) Chapter Understand the relationship between the area under a curve and the denite integral. Understand the relationship between velocity (speed), distance and the denite integral. Estimate t
School: Kentucky
Computing Some Integrals (page 25), Solutions 1. Compute the integral 5 3 x2 dx 0 via the following steps: (a) Write down a sum which approximates the integral. The sum should use n rectangles of equal width and the height of each rectangle should be dete
School: Kentucky
MA123, Chapter 6: Extreme values, Mean Value Theorem, Curve sketching, and Concavity Chapter Goals: Apply the Extreme Value Theorem to nd the global extrema for continuous function on closed and bounded interval. Understand the connection between critic
School: Kentucky
School: Kentucky
Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus Chapter 10: Practice/review problems The collection of problems listed below comprises questions taken from previous MA123 exams. x [1]. Let t2 + 3 dt. h(x) = Find
School: Kentucky
MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp. 207-233, Gootman) Chapter Goals: Understand the statement of the Fundamental Theorem of Calculus. Learn how to compute the antiderivative
School: Kentucky
School: Kentucky
MA123, Chapter 5: Formulas for Derivatives (pp. 83-102, Gootman) Chapter Goals: Know and be able to apply the formulas for derivatives. Understand the chain rule and be able to apply it. Know how to compute higher derivatives. Assignment 08 Assignment
School: Kentucky
MA123, Chapter 9: Computing some integrals (pp. 189-205, Gootman) Understand how to use basic summation formulas to evaluate more complex sums. Chapter Goals: Understand how to compute limits of rational functions at innity. Understand how to use the b
School: Kentucky
School: Kentucky
The idea of the integral Chapter 8: Practice/review problems The collection of problems listed below comprises questions taken from previous MA123 exams. [1]. Estimate the area under the graph of f (x) = x2 + 2 on the interval [0, 2] by dividing the inter
School: Kentucky
School: Kentucky
MA123, Chapter 8: Idea of the Integral (pp. 155-187, Gootman) Chapter Understand the relationship between the area under a curve and the denite integral. Understand the relationship between velocity (speed), distance and the denite integral. Estimate t
School: Kentucky
MA123, Chapter 4: Computing Some Derivatives (pp. 69-82, Gootman) Chapter Goals: Assignments: Understand the derivative as the slope of the tangent line at a point. Investigate further the notions of continuity and dierentiability. Use the denition to
School: Kentucky
School: Kentucky
MA123, Chapter 7: Word Problems (pp. 125-153, Gootman) Chapter Goals: In this Chapter we learn a general strategy on how to approach the two main types of word problems that one usually encounters in a rst Calculus course: Max-Min problems Related Rates
School: Kentucky
MA123, Chapter 3: The idea of limits (pp. 47-67, Gootman) Chapter Goals: Assignments: Evaluate limits. Evaluate one-sided limits. Understand the concepts of continuity and dierentiability and their relationship. Assignment 04 Assignment 05 Earlier, the
School: Kentucky
MA123, Chapter 6: Extreme values, Mean Value Theorem, Curve sketching, and Concavity Chapter Goals: Apply the Extreme Value Theorem to nd the global extrema for continuous function on closed and bounded interval. Understand the connection between critic
School: Kentucky
School: Kentucky
Formulas for derivatives Chapter 5: Practice/review problems The collection of problems listed below contains questions taken from previous MA123 exams. Derivatives [1]. If f (x) = 6x2 + 3x 1, nd f (x). (a) 6x + 1 (b) 12x + 3 (c) 12x 1 (d) 2x + 3 (e) 2x +
School: Kentucky
MA123, Chapter 5: Formulas for Derivatives (pp. 83-102, Gootman) Chapter Goals: Know and be able to apply the formulas for derivatives. Understand the chain rule and be able to apply it. Know how to compute higher derivatives. Assignment 08 Assignment
School: Kentucky
School: Kentucky
Computing some derivatives Chapter 4: Practice/review problems The collection of problems listed below contains questions taken from previous MA123 exams. Computing some derivatives [1]. If f (x) = (x + 3)2 then f (x + h) f (x) = h (a) 2x + h (b) 2x + 3 +
School: Kentucky
School: Kentucky
Chapter 7 Two types: (1) maximum-minimum problems and (2) related rate problems 7.2 Max-Min Word Problems o Ask you to find the largest or the smallest value a function has on an interval o Steps Read the problem once over quickly Read the problem