Math 1131Q Section 10
|
|
|
- Cornelia Page
- 5 years ago
- Views:
Transcription
1 Math 1131Q Section 10 Section 3.9 and 3.10 Oct 19, 2010
2 Find the derivative of ln 3 5 e 2 ln 3 5 e 2 = ln 3 + ln 5/2 + ln e 2 = 3 ln + ( 5 ) ln (ln 3 5 e 2 ) = Find the derivative of t 3 2t + 5 t t 3 2t + 5 t = t3 2t t1/2 t + 5 1/2 t = 1/2 t5/2 2t 1/2 + 5t 1/2 ( t3 2t + 5 ) = ( 5 t 2 )t3/2 t 1/2 5 2 t 3/2 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 2 / 25
3 Clicker Question Find dy d given 3 3y 2 + y 3 = 5 (a) y = y 2 (3y 2 6y) (b) y = y 2 (3y 2 6y) (c) y = 32 3y 2 (3y 2 + 6y) (d) y = y 2 (3y 2 + 6y) Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 3 / 25
4 3 3y 2 + y 3 = (y 2 + 2yy ) + 3y 2 y = y 2 6yy ) + 3y 2 y = 0 6yy + 3y 2 y = y 2 (3y 2 6y)y = y 2 y = y 2 (3y 2 6y) Answer = a Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 4 / 25
5 d d (sin 1 ) = Two views y = sin 1 = sin y d d () = d (sin y) d 1 = cos y dy d dy d = 1 cos y Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 5 / 25
6 d d (sin 1 ) = Two views y = sin 1 = sin y d d () = d (sin y) d 1 = cos y dy d dy d = 1 cos y sin (arcsin ) = Take the derivative of both sides: cos (arcsin ) d (arcsin ) = 1 d Solve for d (arcsin ) d d d (arcsin ) = 1 cos (arcsin ) Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 5 / 25
7 dy d = 1 cos y d d (arcsin ) = 1 cos (arcsin ) θ y = sin 1 is the angle with cos y = cos (sin 1 ) = 1 2 dy d = 1 cos y d d (arcsin ) = 1 cos (arcsin ) dy d = d d (arcsin ) Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 6 / 25
8 Derivatives of f 1 () The general form y = f 1 () = f(y) d d () = d d (f(y)) 1 = f (y) dy d dy d = 1 f (y) where y = f 1 () d d (f 1 ()) = 1 f (f 1 ()) Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 7 / 25
9 Derivatives of f 1 () The general form y = f 1 () = f(y) d d () = d d (f(y)) 1 = f (y) dy d If f(a) = b then f 1 (b) = a dy d = 1 f (y) where y = f 1 () d d (f 1 ()) = 1 f (f 1 ()) and d d (f 1 (b)) = 1 f (f 1 (b)) = 1 f (a) Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 7 / 25
10 d d (f 1 ()) = 1 f (f 1 ()) If f() = find the slope of the tangent line to the graph of f 1 at the point (2, 4). Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 8 / 25
11 d d (f 1 ()) = 1 f (f 1 ()) If f() = find the slope of the tangent line to the graph of f 1 at the point (2, 4). d d (f 1 (2)) = f () = f (f 1 (2)) = 1 f (4) Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 8 / 25
12 d d (f 1 ()) = 1 f (f 1 ()) If f() = find the slope of the tangent line to the graph of f 1 at the point (2, 4). d d (f 1 (2)) = f () = 1 2 d d (f 1 (2)) = f (f 1 (2)) = 1 f (4) 2 4 = 2 4 = 4 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 8 / 25
13 Clicker Question A boat is drawn close to a dock by pulling in the rope at a constant rate. The closer the boat gets to the dock, the faster it is moving. (a) TRUE (b) FALSE (c) Not enough information Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 9 / 25
14 Clicker Question A boat is drawn close to a dock by pulling in the rope at a constant rate. The closer the boat gets to the dock, the faster it is moving. (a) TRUE (b) FALSE (c) Not enough information a) True - The closer the boat gets to the dock, the faster it is moving. Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 9 / 25
15 Clicker Question A boat is drawn close to a dock by pulling in a rope as shown. How is the rate at which the rope is pulled in related to the rate at which the boat approaches the dock? (a) One is a constant multiple of the other. (b) They are equal. (c) It depends on how close the boat is to the dock. h z dz d h is constant Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
16 Clicker Question A boat is drawn close to a dock by pulling in a rope as shown. How is the rate at which the rope is pulled in related to the rate at which the boat approaches the dock? (a) One is a constant multiple of the other. (b) They are equal. (c) It depends on how close the boat is to the dock. h (c) The relationship between d boat is to the dock. z and dz dz d h is constant depends on how close the Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
17 Let h be the height of the dock above the point on the boat to which the rope is tied. h z Then, using the Pythagorean Theorem, h 2 + ((t)) 2 = (z(t)) 2 and differentiating, one gets 2(t) d = 2z(t)dz d = z(t) dz (t) dz = constant As (t) 0 we have z(t) h. Hence z(t) (t) d and Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
18 A streetlight is mounted at the top of a 12 ft pole. A man 5 ft tall walks away from the pole with a speed of 4 ft/sec. How fast is the tip of his shadow moving when he is 24 feet from the pole? y Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
19 A streetlight is mounted at the top of a 12 ft pole. A man 5 ft tall walks away from the pole with a speed of 4 ft/sec. How fast is the tip of his shadow moving when he is 24 feet from the pole? y We are given that the rate of change of y is 4 ft/sec and we want to find the rate of change of z = + y. Notice that d is how fast the length of the shadow is increasing, which is slower by the speed of the man walking. Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
20 We first draw a figure and determine the relationship between, y and z. We are given dy dz = 4 ft/sec and want to find. y 7 12 y z 5 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
21 We first draw a figure and determine the relationship between, y and z. We are given dy dz = 4 ft/sec and want to find. y 7 12 y z 5 z 12 = y 7 z = 12 7 y Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
22 We first draw a figure and determine the relationship between, y and z. We are given dy dz = 4 ft/sec and want to find. y 7 12 y z 5 z 12 = y 7 z = 12 7 y Take the derivative of both sides dz = 12 dy 17 dz = (4) = 7 7 ft/sec Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
23 Clicker Question A streetlight is mounted at the top of a 12 ft pole. A man 5 ft tall walks away from the pole with a speed of 4 ft/sec. How fast is the length of his shadow increasing when he is 24 feet from the pole? (a) 48/7 (b) 20/7 (c) 68/7 (d) 4 (e) None of these y 7 12 y z 5 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
24 Clicker Question A streetlight is mounted at the top of a 12 ft pole. A man 5 ft tall walks away from the pole with a speed of 4 ft/sec. How fast is the length of his shadow increasing when he is 24 feet from the pole? (a) 48/7 (b) 20/7 (c) 68/7 (d) 4 (e) None of these y 7 12 y z 5 b) d = dz dy d since = z y so = 48/7 4 = 20/7 Or 5 = y = = y = d = 5 dy 7 = 20 7 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
25 Related rate problems Draw a picture and label variables h z Epress the relationship between the variables. h 2 + ((t)) 2 = (z(t)) 2 Take the derivative with respect to t to find the relationship between the derivatives. 2(t) d = 2z(t) dz Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
26 Clicker Question The volume of a cube is increasing at the rate of 10 cubic cm per min. How fast is the surface area increasing when the length of an edge is 30 cm? (a) 1/3 (b) 2/3 (c) 4/3 (d) 5/3 (e) None of these Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
27 Clicker Question The volume of a cube is increasing at the rate of 10 cubic cm per min. How fast is the surface area increasing when the length of an edge is 30 cm? (a) 1/3 (b) 2/3 (c) 4/3 (d) 5/3 (e) None of these dv ds = 10 cu cm/min =? when = 30 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
28 S = 6 2 V = 3 ds = 12d dv When = 30 = 32 d 10 = 3(30) 2 d d = 10 3(30) 2 When = 30 ds = 12(30) 10 3(30) 2 ds = 4 3 cm3 /min Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
29 Clicker Question Two cars start moving from the same point. One travels south at 28 mph and the other one west at 70 mph. At what rate is the distance between the two cars increasing two hours later? Round the result to the nearest hundreth. (a) mph (b) mph (c) mph (d) 76.4 mph (e) mph Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
30 Clicker Question Two cars start moving from the same point. One travels south at 28 mph and the other one west at 70 mph. At what rate is the distance between the two cars increasing two hours later? Round the result to the nearest hundreth. Steps (a) mph (b) mph (c) mph (d) 76.4 mph (e) mph 1 Draw a figure and label 2 Find epressions relating variables 3 Differentiate with respect to t to find the relationship among the rates of change 4 Solve for the desired rate of change Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
31 Step 1. z y Step y 2 = z 2 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
32 Step 1. z y Step 2. Step y 2 = z 2 d (2 + y 2 ) = d (z2 ) 2 d + 2y dy = 2z dz Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
33 Step 1. z Step 2. Step 3. y 2 + y 2 = z 2 d (2 + y 2 ) = d (z2 ) 2 d + 2y dy = 2z dz Step 4. d dz d dz = + y dy z = 70 and dy = 28 = 140(70) + 28(56) 22, 736 dz mph Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
34 A plane is flying horizontally at an altitude of 4 miles and at a speed of 465 miles per hour as it passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 10 miles away from the station Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
35 A plane is flying horizontally at an altitude of 4 miles and at a speed of 465 miles per hour as it passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 10 miles away from the station Step 1. y Step = y 2 d = 465 mph Find dy when y = 10 miles. Step 3. Differentiate Step 4. Solve for dy when y = 10 miles Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
36 Clicker Question A plane is flying horizontally at an altitude of 4 miles and at a speed of 465 miles per hour as it passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 10 miles away from the station (a) About 426 mph (b) About 436 mph (c) About 406 mph (d) More than 427 mph (e) Less than 405 mph y Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
37 Clicker Question A plane is flying horizontally at an altitude of 4 miles and at a speed of 465 miles per hour as it passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 10 miles away from the station (a) About 426 mph (b) About 436 mph (c) About 406 mph (d) More than 427 mph (e) Less than 405 mph Answer = a, about mph y Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
38 Step y d = 465 mph Find dy when y = 10 miles. Step = y 2 Step 3. Differentiate 2 d 4 = 2y dy 2 d = 2y dy Step 4. Solve for dy when y = 10 miles = (10) 2 = = 84 d = 465 dy = Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
39 A water trough is 100 cm long and a cross-section has the shape of an isosceles triangle that is 75 cm wide at the top, and has a height of 55 cm. Find the volume of water in the trough when the depth of the water is h? Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
40 A water trough is 100 cm long and a cross-section has the shape of an isosceles triangle that is 75 cm wide at the top, and has a height of 55 cm. Find the volume of water in the trough when the depth of the water is h? 75 b h 55 Area of end = hb/2 b h = 75 similar triangles 55 A = 1 2 hb = 1 2 (15 11 )h2 Volume = area of end 100 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
41 A water trough is 100 cm long and a cross-section has the shape of an isosceles triangle that is 75 cm wide at the top, and has a height of 55 cm. Find the volume of water in the trough when the depth of the water is h? 75 b h 55 Area of end = hb/2 b h = 75 similar triangles 55 A = 1 2 hb = 1 2 (15 11 )h2 Volume = area of end 100 V = ( )h2 (100) V = ( )h2 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
42 Clicker Question A water trough is 100 cm long and a cross-section has the shape of an isosceles triangle that is 75 cm wide at the top, and has a height of 55 cm. If the trough is being filled with water at the rate of 750 cm 3 per min, how fast is the water level rising when the water is 45 cm deep? Round the result to the nearest (a) 1.03 cm 3 per min (b) 10 cm 3 per min (c) 0.27 cm 3 per min (d) cm 3 per min (e) None of these Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
43 V = h2 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
44 V = h2 dv = hdh 750 = (45)dh dh = 750 2( 1500)(45) 22 dh = 1.22 cm3 /min Answer = d Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section / 25
Chapter 3.4 Practice Problems
EXPECTED SKILLS: Chapter.4 Practice Problems Be able to solve related rates problems. It may be helpful to remember the following strategy:. Read the problem carefully. 2. Draw a diagram, if possible,
Chapter 3.5: Related Rates
Expected Skills: Chapter.5: Related Rates Be able to solve related rates problems. It may be helpful to remember the following strategy:. Read the problem carefully. 2. Draw a diagram, if possible, representing
Math 131. Related Rates Larson Section 2.6
Math 131. Related Rates Larson Section 2.6 There are many natural situations when there are related variables that are changing with respect to time. For example, a spherical balloon is being inflated
Math 103 Selected Homework Solutions, Section 3.9
Math 103 Selected Homework Solutions, Section 3.9 9. Let s be the distance from the base of the light pole to the top of the man s shadow, and the distance from the light pole to the man. 15 s 6 s We know:
(a) At what rate is the circumference of the circle changing when the radius is 10 inches? =2inches per minute and we want to find. c =2 r.
3.11 Related Rates Problem 1 The radius of a circle is increasing at a rate of 2 inches per minute. (a) At what rate is the circumference of the circle changing when the radius is 10 inches? We know: dr
AP Calculus AB Chapter 4 Packet Implicit Differentiation. 4.5: Implicit Functions
4.5: Implicit Functions We can employ implicit differentiation when an equation that defines a function is so complicated that we cannot use an explicit rule to find the derivative. EXAMPLE 1: Find dy
Math 2413 t2rsu14. Name: 06/06/ Find the derivative of the following function using the limiting process.
Name: 06/06/014 Math 413 trsu14 1. Find the derivative of the following function using the limiting process. f( x) = 4x + 5x. Find the derivative of the following function using the limiting process. f(
AP Calculus AB Semester 1 Practice Final
Class: Date: AP Calculus AB Semester 1 Practice Final Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the limit (if it exists). lim x x + 4 x a. 6
Solution: It could be discontinuous, or have a vertical tangent like y = x 1/3, or have a corner like y = x.
1. Name three different reasons that a function can fail to be differentiable at a point. Give an example for each reason, and explain why your examples are valid. It could be discontinuous, or have a
6.2 Related Rates Name: Notes
Calculus Write your questions and thoughts here! 6.2 Related Rates Name: Notes Guidelines to solving related rate problems 1. Draw a picture. 2. Make a list of all known and unknown rates and quantities.
x f(x)
1. Name three different reasons that a function can fail to be differentiable at a point. Give an example for each reason, and explain why your examples are valid. 2. Given the following table of values,
AP Calculus AB Chapter 2 Test Review #1
AP Calculus AB Chapter Test Review # Open-Ended Practice Problems:. Nicole just loves drinking chocolate milk out of her special cone cup which has a radius of inches and a height of 8 inches. Nicole pours
x f(x)
1. Name three different reasons that a function can fail to be differential at a point. Give an example for each reason, and explain why your examples are valid. 2. Given the following table of values,
p144 Section 2.6: Related Rates Find a related rate Use related rates to solve real life problems
1 2 p144 Section 2.6: Related Rates Find a related rate Use related rates to solve real life problems Finding Related Rates We have used the chain rule to find dy/dx implicitly, but you can also use the
p144 Section 2.6: Related Rates Find a related rate Use related rates to solve real life problems
p144 Section 2.6: Related Rates Find a related rate Use related rates to solve real life problems Finding Related Rates We have used the chain rule to find dy/dx implicitly, but you can also use the chain
Implicit Differentiation
Week 6. Implicit Differentiation Let s say we want to differentiate the equation of a circle: y 2 + x 2 =9 Using the techniques we know so far, we need to write the equation as a function of one variable
3.8 Exponential Growth and Decay
3.8 Exponential Growth and Decay Suppose the rate of change of y with respect to t is proportional to y itself. So there is some constant k such that dy dt = ky The only solution to this equation is an
MCV4U1 Worksheet 4.7. dh / dt if neither r nor h is constant?
MCV4U1 Worksheet 4.7 This worksheet serves as an additional exercise to complement the lesson and the examples given. Worksheets may take more than one day to complete. If you are stuck, read again the
Welcome to Math 104. D. DeTurck. January 16, University of Pennsylvania. D. DeTurck Math A: Welcome 1 / 44
Welcome to Math 104 D. DeTurck University of Pennsylvania January 16, 2018 D. DeTurck Math 104 002 2018A: Welcome 1 / 44 Welcome to the course Math 104 Calculus I Topics: Quick review of Math 103 topics,
Review: A Cross Section of the Midterm. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Review: A Cross Section of the Midterm Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the limit, if it eists. 4 + ) lim - - ) A) - B) -
Derivatives and Rates of Change
Sec.1 Derivatives and Rates of Change A. Slope of Secant Functions rise Recall: Slope = m = = run Slope of the Secant Line to a Function: Examples: y y = y1. From this we are able to derive: x x x1 m y
Sample Questions Exam II, FS2009 Paulette Saab Calculators are neither needed nor allowed.
Sample Questions Exam II, FS2009 Paulette Saab Calculators are neither needed nor allowed. Part A: (SHORT ANSWER QUESTIONS) Do the following problems. Write the answer in the space provided. Only the answers
MATH1910Chapter2TestReview
Class: Date: MATH1910Chapter2TestReview Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the slope m of the line tangent to the graph of the function
Math 241 Homework 6 Solutions
Math 241 Homework 6 s Section 3.7 (Pages 161-163) Problem 2. Suppose that the radius r and surface area S = 4πr 2 of a sphere are differentiable functions of t. Write an equation that relates ds/ to /.
Related Rates Problems. of h.
Basic Related Rates Problems 1. If V is the volume of a cube and x the length of an edge. Express dv What is dv in terms of dx. when x is 5 and dx = 2? 2. If V is the volume of a sphere and r is the radius.
Due: Wed Oct :30 AM MDT. Question Instructions Make sure you have easy access to all three of these documents.
Related Rates II: Guided (10862409) Due: Wed Oct 4 2017 07:30 AM MDT Question 1 2 3 4 5 6 Instructions Make sure you have easy access to all three of these documents. Today's Notes and Learning Goals Tips
DIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES If we are pumping air into a balloon, both the volume and the radius of the balloon are increasing and their rates of increase are related to each other. However,
Name Date Period. AP Calculus AB/BC Practice TEST: Curve Sketch, Optimization, & Related Rates. 1. If f is the function whose graph is given at right
Name Date Period AP Calculus AB/BC Practice TEST: Curve Sketch, Optimization, & Related Rates. If f is the function whose graph is given at right Which of the following properties does f NOT have? (A)
4.1 Implicit Differentiation
4.1 Implicit Differentiation Learning Objectives A student will be able to: Find the derivative of variety of functions by using the technique of implicit differentiation. Consider the equation We want
d dx x = d dx (10,000 - p2 ) 1/2 dx [10,000 - p2 ] p' = dv = 0 dl dv V + n Things to remember: dt dt ; dy dt = 3
![d dx x = d dx (10,000 - p2 ) 1/2 dx [10,000 - p2 ] p' = dv = 0 dl dv V + n Things to remember: dt dt ; dy dt = 3](/thumbs/77/74907890.jpg "d dx x = d dx (10,000 - p2 ) 1/2 dx [10,000 - p2 ] p' = dv = 0 dl dv V + n Things to remember: dt dt ; dy dt = 3")
45. x 0,000 " p 2 (0,000 - p 2 ) /2 d x d (0,000 - p2 ) /2 2 (0,000 - p2 ) -/2 d [0,000 - p2 ] 2(0,000 " p 2 ) 2 (!2pp') p' "p p # 0,000 " p 2 " 0,000 " p2 p 47. (L + m)(v + n) k (L + m)() + (V + n) dl
5. Find the intercepts of the following equations. Also determine whether the equations are symmetric with respect to the y-axis or the origin.
MATHEMATICS 1571 Final Examination Review Problems 1. For the function f defined by f(x) = 2x 2 5x find the following: a) f(a + b) b) f(2x) 2f(x) 2. Find the domain of g if a) g(x) = x 2 3x 4 b) g(x) =
Section 3.8 Related Rates
Section 3.8 Related Rates Read and re-read the problem until you understand it. Draw and label a picture which gives the relevant information (if possible). Introduce notation. Assign a symbol to every
Trigonometry Math 076
Trigonometry Math 076 133 Right ngle Trigonometry Trigonometry provides us with a way to relate the length of sides of a triangle to the measure of its angles. There are three important trigonometric functions
Calculus 437 Semester 1 Review Chapters 1, 2, and 3 January 2016
Name: Class: Date: Calculus 437 Semester 1 Review Chapters 1, 2, and 3 January 2016 Short Answer 1. Decide whether the following problem can be solved using precalculus, or whether calculus is required.
Review Sheet for Second Midterm Mathematics 1300, Calculus 1
Review Sheet for Second Midterm Mathematics 300, Calculus. For what values of is the graph of y = 5 5 both increasing and concave up? >. 2. Where does the tangent line to y = 2 through (0, ) intersect
1 + x 2 d dx (sec 1 x) =
Page This exam has: 8 multiple choice questions worth 4 points each. hand graded questions worth 4 points each. Important: No graphing calculators! Any non-graphing, non-differentiating, non-integrating
Section 4.1: Related Rates
1 Section 4.1: Related Rates Practice HW from Stewart Textbook (not to hand in) p. 67 # 1-19 odd, 3, 5, 9 In a related rates problem, we want to compute the rate of change of one quantity in terms of the
( f + g ) (3) = ( fg ) (3) = g(x) = x 7 cos x. s = 200t 10t 2. sin x cos x cos2x. lim. f (x) = 7 x 5. y = 1+ 4sin x, (0,1) f (x) = x 2 g(x)
Stewart - Calculus ET 6e Chapter Form A 1. If f ( ) =, g() =, f () =, g () = 6, find the following numbers. ( f + g ) () = ( fg ) () = ( f / g) () = f f g ( ) =. Find the points on the curve y = + 1 +
Apply & Practice 3.5 Set 1: P #3-18 (mult. of 3); 19 #21 write explicit #27-33 (mult. of 3) point #39-40 eqn tang line from graph
Ch 0 Homework Complete Solutions V Part : S. Stirling Calculus: Earl Transcendental Functions, 4e Larson WATCH for the product rule and the chain rule. If the order that our terms are in differ from the
PRACTICE PROBLEMS CH 8 and Proofs
GEOM PRACTICE PROBLEMS CH 8 and Proofs Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the length of the missing side. The triangle is not drawn to
PLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 1. (a) (b) (c) (d) (e) 2. (a) (b) (c) (d) (e) (a) (b) (c) (d) (e) 4. (a) (b) (c) (d) (e)...
Math 0550, Exam October, 0 The Honor Code is in effect for this examination. All work is to be your own. No calculators. The exam lasts for hour and 5 min. Be sure that your name is on every page in case
Days 3 & 4 Notes: Related Rates
AP Calculus Unit 4 Applications of the Derivative Part 1 Days 3 & 4 Notes: Related Rates Implicitly differentiate the following formulas with respect to time. State what each rate in the differential equation
Chapter 8: Radical Functions
Chapter 8: Radical Functions Chapter 8 Overview: Types and Traits of Radical Functions Vocabulary:. Radical (Irrational) Function an epression whose general equation contains a root of a variable and possibly
dx dt = x 2 x = 120
Solutions to Review Questions, Exam. A child is flying a kite. If the kite is 90 feet above the child s hand level and the wind is blowing it on a horizontal course at 5 feet per second, how fast is the
TMTA Calculus and Advanced Topics Test 2010
. Evaluate lim Does not eist - - 0 TMTA Calculus and Advanced Topics Test 00. Find the period of A 6D B B y Acos 4B 6D, where A 0, B 0, D 0. Solve the given equation for : ln = ln 4 4 ln { } {-} {0} {}
U of U Math Online. Young-Seon Lee. WeBWorK set 1. due 1/21/03 at 11:00 AM. 6 4 and is perpendicular to the line 5x 3y 4 can
U of U Math 0-6 Online WeBWorK set. due //03 at :00 AM. The main purpose of this WeBWorK set is to familiarize yourself with WeBWorK. Here are some hints on how to use WeBWorK effectively: After first
AP Calculus. Applications of Derivatives. Table of Contents
AP Calculus 2015 11 03 www.njctl.org Table of Contents click on the topic to go to that section Related Rates Linear Motion Linear Approximation & Differentials L'Hopital's Rule Horizontal Tangents 1 Related
Use the following problem to answer popper questions 25 and 26.
Example 9: A 6-foot man is walking towards a 5 foot lamp post at the rate of 10 ft/sec. How fast is the length of his shadow changing when he is 0 feet from the lamp post? Hint: This example will use similar
4.6 Related Rates Notes RELATED RATES PROBLEMS --- IT S AS EASY AS 1 2-3!
4.6 Related Rates Notes RELATED RATES PROBLEMS --- IT S AS EASY AS 1 2-3! 1) Draw a picture. Label all variables and constant values. Identify the given rate of change, the rate to be found, and when to
North Carolina Math 2 Transition Edition Unit 5 Assessment: Trigonometry
Name: Class: _ Date: _ North Carolina Math 2 Transition Edition Unit 5 Assessment: Trigonometry Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the
LESSON 11 PRACTICE PROBLEMS
LESSON 11 PRACTICE PROBLEMS 1. a. Determine the volume of each of the figures shown below. Round your answers to the nearest integer and include appropriate units of b. Determine the volume of each of
dy dx dx dx as a BC Calculus 1 The Chain Rule is notation for a which says that we have the
2.4 2.6 BC Calculus 1 The Chain Rule dy is notation for a which says that we have the for an expression set equal to (the dependent variable), where the variable is x. This is read dee why, dee or the
Geometry Review- Chapter Find e, and express your answer in simplest radical form.
Name: Date: Period: Geometry Review- Chapter 10 1. The diagonal of a rectangle measures 15 cm long, and the width is 10. Find the height of the rectangle and epress your answer in simplest radical form.
Almost all of the questions involving Related Rates will require one of the following relationships to link together the various changing rates:
Related Rates All quantities that we meet in every-day life change with time, this is especially true in scientific investigations. Related Rate problems are those in which an equation epresses some relationship
Quiz 4A Solutions. Math 150 (62493) Spring Name: Instructor: C. Panza
Math 150 (62493) Spring 2019 Quiz 4A Solutions Instructor: C. Panza Quiz 4A Solutions: (20 points) Neatly show your work in the space provided, clearly mark and label your answers. Show proper equality,
APPLICATIONS OF DERIVATIVES UNIT PROBLEM SETS
APPLICATIONS OF DERIVATIVES UNIT PROBLEM SETS PROBLEM SET #1 Related Rates ***Calculators Allowed*** 1. An oil tanker spills oil that spreads in a circular pattern whose radius increases at the rate of
Math 147 Exam II Practice Problems
Math 147 Exam II Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture, all homework problems, all lab
V = π 3 r2 h. dv dt = π [ r 2dh dt r2. dv 3 dt +2rhdr dt
9 Related Rates Related rates is the phrase used to describe the situation when two or more related variables are changing with respect to time. The rate of change, as mentioned earlier, is another expression
Name Date Period. Multiple Choice
Name Date Period Worksheet 3.8 Related Rates Show all work. Calculator permitted. Show all set-ups and analysis. Report all answers to 3 decimals and avoid intermediate rounding error. Multiple Choice
AP Calculus AB Semester 2 Practice Final
lass: ate: I: P alculus Semester Practice Final Multiple hoice Identify the choice that best completes the statement or answers the question. Find the constants a and b such that the function f( x) = Ï
Chapter 2: Differentiation 1. Find the slope of the tangent line to the graph of the function below at the given point.
Chapter : Differentiation 1. Find the slope of the tangent line to the graph of the function below at the given point. f( ) 10, (, ) 10 1 E) none of the above. Find the slope of the tangent line to the
Name Date Class. Logarithmic/Exponential Differentiation and Related Rates Review AP Calculus. Find dy. dx. 1. y 4 x. y 6. 3e x.
Name Date Class Find dy d. Logarithmic/Eponential Differentiation and Related Rates Review AP Calculus 1. y 4. 1 y ln. y ln 1 4. y log9 1 5. e y 6. y log 7. y e 8. e y e 4 1 1 9. y e e 10. 1 y ln 1 e 11.
1. The cost (in dollars) of producing x units of a certain commodity is C(x) = x x 2.
APPM 1350 Review #2 Summer 2014 1. The cost (in dollars) of producing units of a certain commodity is C() 5000 + 10 + 0.05 2. (a) Find the average rate of change of C with respect to when the production
Solve the equation for the specified variable. Use the distributive property to factor as necessary. 2) -9s + 8p = tp - 8 for p
College Readiness Math Final Eam Review Sheet Solve the equation. 1) r + 6 = r + 8 3 6 Solve the equation for the specified variable. Use the distributive propert to factor as necessar. 2) -9s + 8p = tp
DRAFT - Math 101 Lecture Note - Dr. Said Algarni
3 Differentiation Rules 3.1 The Derivative of Polynomial and Exponential Functions In this section we learn how to differentiate constant functions, power functions, polynomials, and exponential functions.
1 Antiderivatives graphically and numerically
Math B - Calculus by Hughes-Hallett, et al. Chapter 6 - Constructing antiderivatives Prepared by Jason Gaddis Antiderivatives graphically and numerically Definition.. The antiderivative of a function f
Guidelines for implicit differentiation
Guidelines for implicit differentiation Given an equation with x s and y s scattered, to differentiate we use implicit differentiation. Some informal guidelines to differentiate an equation containing
a right triangle), we see that x 200 or equivalently x = 200 tan θ. The rate at which the ray of light moves along the shore is
Example 1: A revolving beacon in a lighthouse makes one revolution every 15 seconds. The beacon is 00 ft from the nearest point P on a straight shoreline. Find the rate at which a ray from the light moves
Review Questions, Exam 3
Review Questions, Exam. A child is flying a kite. If the kite is 90 feet above the child s hand level and the wind is blowing it on a horizontal course at 5 feet per second, how fast is the child paying
Radicals and Pythagorean Theorem Date: Per:
Math 2 Unit 7 Worksheet 1 Name: Radicals and Pythagorean Theorem Date: Per: [1-12] Simplify each radical expression. 1. 75 2. 24. 7 2 4. 10 12 5. 2 6 6. 2 15 20 7. 11 2 8. 9 2 9. 2 2 10. 5 2 11. 7 5 2
Name: Date: Period: Calculus Honors: 4-2 The Product Rule
Name: Date: Period: Calculus Honors: 4- The Product Rule Warm Up: 1. Factor and simplify. 9 10 0 5 5 10 5 5. Find ' f if f How did you go about finding the derivative? Let s Eplore how to differentiate
Find the slope of the curve at the given point P and an equation of the tangent line at P. 1) y = x2 + 11x - 15, P(1, -3)
Final Exam Review AP Calculus AB Find the slope of the curve at the given point P and an equation of the tangent line at P. 1) y = x2 + 11x - 15, P(1, -3) Use the graph to evaluate the limit. 2) lim x
1 x 3 3x. x 1 2x 1. 2x 1 2x 1. 2x 1 2x 1. x 2 +4x 1 j. lim. x3 +2x x 5. x2 9
MATHEMATICS 57 Final Eamination Review Problems. Let f 5. Find each of the following. a. fa+b b. f f. Find the domain of each function. a. f b. g +. The graph of f + is the same as the graph of g ecept
WW Prob Lib1 Math course-section, semester year
Young-Seon Lee WW Prob Lib Math course-section, semester year WeBWorK assignment due /4/03 at :00 PM..( pt) Give the rational number whose decimal form is: 0 7333333 Answer:.( pt) Solve the following inequality:
Extra Practice Recovering C
Etra Practice Recovering C 1 Given the second derivative of a function, integrate to get the first derivative, then again to find the equation of the original function. Use the given initial conditions
Calculus I - Lecture 14 - Related Rates
Calculus I - Lecture 14 - Related Rates Lecture Notes: http://www.math.ksu.edu/ gerald/math220d/ Course Syllabus: http://www.math.ksu.edu/math220/spring-2014/indexs14.html Gerald Hoehn (based on notes
Final Exam Review / AP Calculus AB
Chapter : Final Eam Review / AP Calculus AB Use the graph to find each limit. 1) lim f(), lim f(), and lim π - π + π f 5 4 1 y - - -1 - - -4-5 ) lim f(), - lim f(), and + lim f 8 6 4 y -4 - - -1-1 4 5-4
I. Horizontal and Vertical Tangent Lines
How to find them: You need to work with f " x Horizontal tangent lines: set f " x Vertical tangent lines: find values of x where f " x I. Horizontal and Vertical Tangent Lines ( ), the derivative of function
Maths GCSE Langdon Park Foundation Calculator pack A
Maths GCSE Langdon Park Foundation Calculator pack A Name: Class: Date: Time: 96 minutes Marks: 89 marks Comments: Q1. The table shows how 25 students travel to school. Walk Bus Car Taxi 9 8 7 1 Draw a
Pre-Calculus Vectors
Slide 1 / 159 Slide 2 / 159 Pre-Calculus Vectors 2015-03-24 www.njctl.org Slide 3 / 159 Table of Contents Intro to Vectors Converting Rectangular and Polar Forms Operations with Vectors Scalar Multiples
Math 0200 Final Exam Review Questions
Math 000 Final Eam Review Questions 1. Simplif: 4 8i + 8 ( 7). Simplif: 11 ( 9) + 6(10 4) + 4. Simplif ( 5 + 7) ( ) 8 6 4. Simplif: (4 ) 9 i 5. Simplif: 4 7 6. Evaluate 4 + 5 when = and = Write each of
Math 370 Exam 3 Review Name
Math 370 Exam 3 Review Name The following problems will give you an idea of the concepts covered on the exam. Note that the review questions may not be formatted like those on the exam. You should complete
e x for x 0. Find the coordinates of the point of inflexion and justify that it is a point of inflexion. (Total 7 marks)
Chapter 0 Application of differential calculus 014 GDC required 1. Consider the curve with equation f () = e for 0. Find the coordinates of the point of infleion and justify that it is a point of infleion.
8.6 Inverse Trigonometric Ratios
www.ck12.org Chapter 8. Right Triangle Trigonometry 8.6 Inverse Trigonometric Ratios Learning Objectives Use the inverse trigonometric ratios to find an angle in a right triangle. Solve a right triangle.
Math Exam 02 Review
Math 10350 Exam 02 Review 1. A differentiable function g(t) is such that g(2) = 2, g (2) = 1, g (2) = 1/2. (a) If p(t) = g(t)e t2 find p (2) and p (2). (Ans: p (2) = 7e 4 ; p (2) = 28.5e 4 ) (b) If f(t)
The Primary Trigonometric Ratios Word Problems
The Primary Trigonometric Ratios Word Problems A. Determining the measures of the sides and angles of right triangles using the primary ratios When we want to measure the height of an inaccessible object
Section MWF 12 1pm SR 117
Math 1431 Section 12485 MWF 12 1pm SR 117 Dr. Melahat Almus almus@math.uh.edu http://www.math.uh.edu/~almus COURSE WEBSITE: http://www.math.uh.edu/~almus/1431_sp16.html Visit my website regularly for announcements
MATH 130 FINAL REVIEW
MATH 130 FINAL REVIEW Problems 1 5 refer to triangle ABC, with C=90º. Solve for the missing information. 1. A = 40, c = 36m. B = 53 30', b = 75mm 3. a = 91 ft, b = 85 ft 4. B = 1, c = 4. ft 5. A = 66 54',
Trigonometric ratios:
0 Trigonometric ratios: The six trigonometric ratios of A are: Sine Cosine Tangent sin A = opposite leg hypotenuse adjacent leg cos A = hypotenuse tan A = opposite adjacent leg leg and their inverses:
; approximate b to the nearest tenth and B or β to the nearest minute. Hint: Draw a triangle. B = = B. b cos 49.7 = 215.
M 1500 am Summer 009 1) Given with 90, c 15.1, and α 9 ; approimate b to the nearest tenth and or β to the nearest minute. Hint: raw a triangle. b 18., 0 18 90 9 0 18 b 19.9, 0 58 b b 1.0, 0 18 cos 9.7
MAC 2311 Review
Name: Class: Date: MAC 2311 Review 2.6-2.9 Numeric Response 1. Calculate y. xy 4 +x 2 y =2x +3y 2. Calculate y. cos xy =x 6 y 3. The position function of a particle is given by s =t 3 10.5t 2 2t,t 0 When
Exam Review Sheets Combined
Exam Review Sheets Combined Fall 2008 1 Fall 2007 Exam 1 1. For each part, if the statement is always true, circle the printed capital T. If the statement is sometimes false, circle the printed capital
TMTA CONTEST CALCULUS AND ADVANCED TOPICS. d) t 1 b. a) 0 b) 2/15 c) 2/15 d) 4/15 e)-4/15
TMTA CONTEST CALCULUS AND ADVANCED TOPICS 1. Suppose Q te bt, with b a positive constant. For which values of increase? t will Q a) t b b) t b c) t 1 b d) t 1 b e) t b 2 2. Find the finite area bounded
Unit 3 Right Triangle Trigonometry - Classwork
Unit 3 Right Triangle Trigonometry - Classwork We have spent time learning the definitions of trig functions and finding the trig functions of both quadrant and special angles. But what about other angles?
CHAPTER 3 Applications of Differentiation
CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. 0 Section. Rolle s Theorem and the Mean Value Theorem. 07 Section. Increasing and Decreasing Functions and the First
BC VECTOR PROBLEMS. 13. Find the area of the parallelogram having AB and AC as adjacent sides: A(2,1,3), B(1,4,2), C( 3,2,7) 14.
For problems 9 use: u (,3) v (3, 4) s (, 7). w =. 3u v = 3. t = 4. 7u = u w (,3,5) 5. wt = t (,, 4) 6. Find the measure of the angle between w and t to the nearest degree. 7. Find the unit vector having
AP Calculus. Slide 1 / 101. Slide 2 / 101. Slide 3 / 101. Applications of Derivatives. Table of Contents
Slide 1 / 101 Slide 2 / 101 AP Calculus Applications of Derivatives 2015-11-03 www.njctl.org Table of Contents click on the topic to go to that section Slide 3 / 101 Related Rates Linear Motion Linear
Math 1303 Part II. The opening of one of 360 equal central angles of a circle is what we chose to represent 1 degree
Math 1303 Part II We have discussed two ways of measuring angles; degrees and radians The opening of one of 360 equal central angles of a circle is what we chose to represent 1 degree We defined a radian
4.1 Related Rates Goal: 1. Understand how variables change with respect to time. 2. Understand "with respect to".
Goal: 1. Understand how variables change with respect to time. 2. Understand "with respect to". Study 4.1, # 1 7, 11, 17, 19, 23, 25, 29, 35 water drop ripples pour cement high rise @ 1:30 water drop ripples
Math 1431 Final Exam Review. 1. Find the following limits (if they exist): lim. lim. lim. lim. sin. lim. cos. lim. lim. lim. n n.
. Find the following its (if they eist: sin 7 a. 0 9 5 b. 0 tan( 8 c. 4 d. e. f. sin h0 h h cos h0 h h Math 4 Final Eam Review g. h. i. j. k. cos 0 n nn e 0 n arctan( 0 4 l. 0 sin(4 m. cot 0 = n. = o.