Technical Calculus I Homework. Instructions

Size: px

Start display at page:

Download "Technical Calculus I Homework. Instructions"

Adam Howard
5 years ago
Views:

1 Technical Calculus I Homework Instructions 1. Each assignment is to be done on one or more pieces of regular-sized notebook paper. 2. Your name and the assignment number should appear at the top of the first sheet. 3. Please do not use a red pen or red pencil to do the homework. 4. Please do not circle answers. Answer to a problem is not a single epression or number, it is the entire solution. 5. All relevant work is required. Problems are graded on the quality and correctness of the presented work. 6. Work through the homework problems referring to your notes and the lesson notes when necessary. 7. Redo the homework problems before an eam without referring to any other materials. It is best to do this more than once. 1. Limits Homework Problems 1. The graph of y = f() is shown below. y A. Find lim 2 f() if it eists. 1

2 B. Find f(2) if it eists. C. Find lim f() if it eists. 1 D. Find f( 1) if it eists. 2. The graph of y = f() is shown below. y A. Find lim f() if it eists. 1 B. Find f( 1) if it eists. C. Find lim 0 f() if it eists. D. Find f(0) if it eists. E. Find lim 2 f() if it eists. F. Find f(2) if it eists. 3. Let y = f() = sin(). Copy the following table, fill it out, and use it to determine sin() lim. (Make sure your calculator is in radian mode when evaluating sin(). In 0 calculus, angles must always be in radians for computations.) 2

3 y y Let y = f() = Copy the following table, fill it out, and use it to determine lim. 2 2 ( 5. Compute lim ). 3 ( 6. Compute lim 2 2 ) ( ) Compute lim 2/ Compute lim Compute lim (2 + h) Compute lim. h 0 h 11. Compute lim h h 1 3 h. y y Tangent Lines Homework Problems 1. A. Find the slope of the line tangent to the graph of f() = when = 2. B. Find the slope-intercept form of the tangent line. C. Sketch a graph which includes both the function and the tangent line. Be sure to include the point the two have in common. 3

4 2. A. Find the slope of the line tangent to the graph of g() = 3 5 when = 4. B. Find the slope-intercept form of the tangent line. C. Are there any differences between g() and the tangent line? Eplain your answer. 3. A. Find the slope of the line tangent to the graph of h() = 3 when = 0. B. Find the slope-intercept form of the tangent line. 4. A. Find the slope of the line tangent to the graph of f() = for all. B. Find the slope of the tangent line when = 3. C. Find the slope-intercept form of the tangent line. 5. A. Find the slope of the line tangent to the graph of h() = 3 for all. B. Find the slope of the tangent line when = 1/2. C. Find the slope-intercept form of the tangent line. 6. A. Find the slope of the line tangent to the graph of g() = 1 for all. 2 B. Find the slope of the tangent line when = 1/4. C. Find the slope-intercept form of the tangent line. 3. Velocity Homework Problems In the following problems, s = f(t) is the position s at time t of an object in linear motion. 1. Suppose s = 4t + 1. A. Find v from t = 0 to t = 3. B. Find v at time t. C. What does part B say about v? 2. Suppose s = t 2 + 3t. A. Find v from t = 2 to t = 3. B. Find v at time t. C. What is v when t = 2? D. What is v when t = 3? E. Why is the v found in part A the average of the v found in part C and the v found in part D? 4

5 3. Suppose s = 1 t. A. What is s when t = 1? B. What is s when t = 2? C. What is v from t = 1 to t = 2? D. Why is the above v negative? E. Compute v for time t. F. Why is v negative when t > 0? 1. Suppose y = The Derivative Homework Problems A. Use the definition of the derivative to compute dy d. B. Find the slope of the line tangent to the graph of y when = 1. C. Find the slope-intercept form of the tangent line. 2. A. Use the definition of the derivative to compute f () if f() = 1. B. What is f (5)? 3. Suppose y = 4 3. Eplain why y is a constant without computing it. 4. If s = 4t 2 2t + 1 is the position s at time t of an object in linear motion, then v = s = 8t 2 and v = 8. What does v represent? Eplain your answer. 5. Polynomial Differentiation Homework Problems 1. Find y if y = Find dy d if y = Find f () if f() = Find g () if g() = π. 5. Find dy d if y = ( ) ( ). = 1 5

6 6. Find y (0) if y = ( ) ( ). 7. Find h (1) if h() = ( ) Find y if y = ( 2) Find dy dt if y = 7t (t 7) Find f (z) if f(z) = 16z3 4z 2 + 3z Find d dy if = 13 ( 5 y 20) + 6y A. Find the slope of the tangent line of y = when = 2. B. Find the slope-intercept form of the tangent line. 13. Let s = 2t 3 + 6t 2 1 be the position function of an object in linear motion. Find v when t = Product and Quotient Rules Homework Problems 1. Use the product rule to find y if y = ( + 1)(3 4). Simplify the answer. 2. Use the product rule to find dy d if y = (2 5) ( ). Simplify the answer. 3. Use the product rule to find f () if f() = 2 ( ). Simplify the answer. 4. Use the product rule to find y if y = ( ) 2. Simplify the answer. 5. Use the product rule to find dy d if y = ( + 1)( + 2). Simplify the answer. 6. Use the quotient rule to find y if y = 2. Simplify the answer Use the quotient rule to find dy d if y = 8. Use the quotient rule to find f () if f() =. Simplify the answer Simplify the answer A. Find the slope of the tangent line of y = 2 4 when = B. Find the slope-intercept form of the tangent line. 10. If s = 1 t + 1 is the position function of an object in linear motion, find v when t = 8. 6

7 7. Chain Rule Homework Problems 1. Use the general power rule to find y in each of the following parts. A. y = (8 1) 4 B. y = 7 ( ) 3 C. y = D. y = E. y = (3 4) 2 F. y = Find f () in each of the following parts. A. f() = 3 2 (5 + 2) 3 Factor f (). 2 B. f() = (3 2) 2 Simplify f (). 3. Suppose s = (2t + 1) 3 is the position s at time t of an object in linear motion. Find v when t = Find the slope of the line tangent to the graph of f() = 3(1 ) 5 when = Implicit Differentiation Homework Problems 1. Use implicit differentiation to find dy d A y 2 = 1 B y = 8 C y 4y 2 = 2 D. 4 y 2 = 1 in each of the following parts. 2. The graph of 2 + y 2 = 5 is a circle with center (0, 0) and radius 5. A. What two points are on the graph of 2 + y 2 = 5 when y = 1? 7

8 B. Find the slopes of the lines tangent to the graph of 2 + y 2 = 5 when y = 1. C. Find the slope-intercept forms of the lines tangent to the graph of 2 + y 2 = 5 when y = 1. D. Sketch a graph which shows the circle and the two tangent lines. 9. Higher Derivatives Homework Problems 1. Find f () if f() = Find y (4) if y = Find d 3 y d 3 if y = Find g () if g() = Suppose that s = 3t 3 2t 2 + 7t 1 is the location at time t of an object in linear motion. Find a when t = Etrema and Concavity Homework Problems Complete each part for the functions y = f() listed below. A. Find all intercepts of y. B. Find all the critical values of y. C. Where is y increasing? Where is y decreasing? D. Find all the critical points of y. E. What are the relative minimum points of y? What are the relative maimum points of y? F. Find all possible inflection points of y. G. Where is y concave up? Where is y concave down? H. What are the inflection points of y? I. Sketch the graph of y. Include all intercepts, relative etreme points, and inflection points. 8

9 Problems 1. y = Note: y = (3 + 5)(2 3), y = , and y = y = Note: y = 2 2 ( + 4), y = 2(3 + 8), and y = 4(3 + 4). 3. y = Note: y = ( 1) 3, y = ( 1) 2 (4 1) and y = 6( 1)(2 1). 4. y = 25 ( 2 1 ) 3 Note: y = 25( 1) 3 ( + 1) 3, y = 150( 1) 2 ( + 1) 2, and y = 150( 1)( + 1) ( ). 5. y = Note: y = 6 ( + 1), y = 5 (7 + 6), and y = 6 4 (7 + 5). 11. Optimization Homework Problems 1. Find two positive numbers such that their sum is 40 and their product is as large as possible. 2. Find the dimensions of the rectangle which has area 9 and the smallest possible perimeter. 3. A long rectangular sheet of aluminum that is 12 wide is to be made into a gutter as shown in the following diagram. 12 Determine the value of such that the gutter can hold the largest possible volume of water. 4. Find the line which is tangent to y = and which has the largest possible slope. 5. Let s(t) = t 3 + 3t t be the position function of an object in linear motion when 0 t 4. When does the object attain its maimum velocity? What is the maimum velocity of the object? What is the position of the object when it attains its maimum velocity? What is the acceleration of the object when it attains its maimum velocity? 6. Z can swim at a rate of 1 m/s and run at a rate of 2 m/s. Z plans to move from point A to point B by first swimming in a straight line across the pool and then running along the edge of the pool. If the quickest route is taken, how long will Z take? 9

10 B Pool 20 m 50 m A 12. Special Function Derivatives Homework Problems 1. Find y in each of the following parts. A. y = 3 sin(6) B. y = sin(4 3) C. y = 6 sin ( 2 2 ) D. y = 2 cos ( 3 2) E. y = 7 cos ( ) F. y = cos ( ( + 2) 3) G. y = 4 sin 2 ( 2 2) H. y = sin(9) cos(5) I. y = cos (sin(2)) J. y = 4 tan ( ) K. y = sec(2 8) cot(1 2) L. y = 2 M. y = csc ( 4 2) N. y = tan ( 2 1 ) O. y = 4 sec ( 3) P. y = cot ( 4 5) Q. y = 8 csc (sin()) R. y = sec 2 (2 + 4) ( (3 S. y = 7 tan ) ) 4 10

11 T. y = 2 sin 2 (3) + 2 cos 2 (3) 2. Find the slope-intercept form of the tangent line of y = 2 cos(3) when = π Find y in each of the following parts. A. y = 4 sin 1 (2) B. y = cos 1 ( 3 2) C. y = 2 tan 1 ( 2 1 ) D. y = 2 sin 1 (3) 4 cos 1 (3) E. y = tan 1 ( 3 2 ) ( ) 4. Find the slope-intercept form of the tangent line of y = 12 tan 1 when = Find y in each of the following parts. A. y = ln(3) B. y = 5 ln(2 3) C. y = ln ( ) ( ( D. y = ln 2 1 ) ) 3 E. y = ln (cos(2)) ( ) 5 F. y = 4 ln G. y = sin (ln()) 6. Use logarithmic differentiation to find dy d. A. y = 32 B. y = 4 (2 3) 2 ( + 1) 3 7. Find y in each of the following parts. A. y = e 43 B. y = e 1/ C. y = 2 e 7 4 e 5 11

12 D. y = 9 e E. y = 3 e sin() F. y = cos ( e 2) G. y = 6 ln ( e 2) H. y = e 3 ln(4) 13. Indefinite Integrals Homework Problems Determine each of the following indefinite integrals d d d d 2 d 33/ d (2 + 1)( 2) d ( + 3) 2 d ( ) 5 4 d 10. (2 2 + ) 2 2 d 12

13 14. Indefinite Integral Applications Homework Problems 1. Find y = f() if y = 3 and the graph of y passes through the point (1, 4) Find y = f() if y = and the graph of y passes through the point ( 2, 48). 3. Find the position function s of an object in linear motion if a = 3t + 2, v(0) = 2, and s(0) = 1/2. 4. How long does it take for an object dropped from a height of 1000 meters to hit the ground? Assume air resistance is negligible. 5. An object is thrown straight up into the air from a height of 3 meters with an initial velocity of 20 m/s. What is the maimum height of the object? When does the object hit the ground? 6. What is the initial velocity of an object that reaches a maimum height of 50 meters when it is thrown straight up into the air from the ground? 7. An object is thrown straight up into the air from a height of one meter. The object hits the ground 3 seconds after it was thrown. What was the maimum height of the object? 1. Evaluate 2. Evaluate 3. Evaluate 15. Definite Integrals Homework Problems Find the area of the region bounded by the curves y = 1, = 1, = 4, and y = 0. A 2 sketch of the region is required. 5. Find the area of the region bounded by the curves y = 3 + 1, = 1, and y = 0. A sketch of the region is required. 6. Find the area of the region bounded by the curves y =, y = 2 2, and y = 0 when > 0. A sketch of the region is required. 13

14 16. Integration by Substitution Homework Problems 1. Evaluate (3 5) Evaluate (8 7) Evaluate 3 4. Evaluate 5. Evaluate 6. Evaluate ( 3 2 ) 4 7 (4 2 5) 2 ( 7. Evaluate 2 1 ) ( ) Evaluate ( 2 2 4) 4 (3 2 ln()) 2 9. Evaluate Evaluate 2(ln(2)) Evaluate cos(4)(sin(4) + 1) Evaluate 13. Evaluate 14. Evaluate 15. Evaluate sec 4 (5) tan(5) tan( + 1) sec 2 ( + 1) tan 1 (6) ( 1 e 42 ) 3 e 42 14

15 16. Evaluate ( 17. Evaluate ) Evaluate 19. Evaluate 20. Evaluate (2 1) Special Function Integrals Homework Problems 1. Evaluate Evaluate Evaluate cos(2) 3 sin(2) Evaluate e e5 5. Evaluate 6 e Evaluate e2 7. Evaluate 3 e Evaluate 9. Evaluate 10. Evaluate cos(7) ( ) sin 2 sec ( 2) 15

16 11. Evaluate 12. Evaluate 13. Evaluate 14 sec(3) tan(3) e 3 sin ( 4 e 3) cos (tan(2)) sec 2 (2) ( 14. Evaluate ) tan 3/2 sec (ln ( 2 )) 15. Evaluate 16

Solutions Exam 4 (Applications of Differentiation) 1. a. Applying the Quotient Rule we compute the derivative function of f as follows:

Solutions Exam 4 (Applications of Differentiation) 1. a. Applying the Quotient Rule we compute the derivative function of f as follows: MAT 4 Solutions Eam 4 (Applications of Differentiation) a Applying the Quotient Rule we compute the derivative function of f as follows: f () = 43 e 4 e (e ) = 43 4 e = 3 (4 ) e Hence f '( ) 0 for = 0