MATH1013 Calculus I, 2013-14 Fall

Week 12 Eleventh tutorial Worksheet: Mean Value theorem, L’Hopital’s rule,Newton’s method, Anti-derivatives (L10, T11A)

Name: ID No.: Tutorial Section:

Complete at least TWO questions from the following questions!

(Solution of this worksheet will be available at the course website the week after.)

1. (Demonstration) (p.288, Q. 27) Without evaluating derivatives, which of the functions g(x) = 2x10,h(x) = x10 + 2, and p(x) = x10 − ln 2 have the same derivative as f(x) = x10?

2. (Demonstration) (page 289, Q. 31) A state patrol officer saw a car start from rest at a highwayon-ramp. She radioed ahead to a patrol officer 30 miles along the highway. When the car reached thelocation of the second officer 28 mins later, it was clocked going 60 mile/hour. The car driver was given aticket for exceeding the 60 mile/hour speed limit. Why can the officer conclude that the driver exceededthe speed limit?

3. (Demonstration) (p.300, Q. 18) Using L’Hopital’s rule to evaluate limx→14 arctanx− π

x− 1.

4. (Demonstration) (p.309, Q. 11) Use Newton’s method to approximate the root of f(x) = sinx+x− 1with the initial approximation x0 = 1.5. Calculate up to seven iterations.

5. (Demonstration) (page 321, Q.92) Given the acceleration a(t) = 2 cos t of an object moving along aline, find the position function with the initial velocity and position: v(0) = 1, and s(0) = 0.

6. (Demonstration) (page 321, Q. 97) A softball is popped up vertically from the ground level with avelocity of 30m/s. We assume that the acceleration due to gravitation is 9.8m/s2.

• Find the velocity of the object as a function of time t;

• Find the position of the object as a function of time t;

• Find when the object reaches the maximum height and at find this height;

• Find the time when the object strikes the ground.

7. (Class work) (page 300, Q. 46, 52) Evaluate limx→1−(1− x) tan(πx

2

)and limx→∞(x−

√x2 + 1)

Answer

8. (Class work) (p.288, Q. 27) Without evaluating derivatives, which of the functions f(x) = lnx, g(x) =ln 2x, and h(x) = lnx2 and p(x) = ln(10x2) have the same derivative as f(x) = x10?

Answer

9. (Class work) (page 289, Q. 32) A state patrol officer saw a car start from rest at a highway on-ramp.She radioed ahead to a patrol officer 30 miles along the highway. When the car reached the location ofthe second officer 30 mins later, it was clocked going 60 mile/hour. The car driver was given a ticket

1

for exceeding the 60 mile/hour speed limit. Can the officer conclude that the driver exceeded the speedlimit?

Answer

10. (Class work) (p.300, Q. 17, 20) Using L’Hopital’s rule to evaluate limx→1lnx− 1

x− eand limx→1

x sinx+ x2 − 4π2

x− 2π.

Answer

11. (Class work)(page 300, Q. 47, 54) Evaluate limx→0(csc 6x sin 7x) and limx→∞(x−√x2 + 4x)

Answer

12. (Class work) (p.309, Q. 13) Use Newton’s method to approximate the root of f(x) = tanx− 2x withthe initial approximation x0 = 1.5. Calculate up to seven iterations.

Answer

13. (Class work) (page 320, Q.49, 51) Find primitives of (i)

∫6√

25− x2dx, (ii)

∫dx

x√x2 − 100

.

Answer

14. (Class work) (page 320, Q.49, 51) Find primitives of (i)

∫6√

25− x2dx, (ii)

∫dx

x√x2 − 100

.

2

Answer

15. (Class work) (page 321, Q.93) Suppose a(t) = 3 sin 2t is the acceleration of an object moving along aline, find the position function with the given initial velocity and position: v(0) = 1, and s(0) = 10.

Answer

16. (Class work) (page 321, Q. 100) A payload is released at an elevation of 400m from a hot-air balloonthat is rising at a rate of 10 m/s. We assume that the acceleration due to gravitation is 9.8m/s2.

• Find the velocity of the object as a function of time t;

• Find the position of the object as a function of time t;

• Find when the object reaches the maximum height and at find this height;

• Find the time when the object strikes the ground.

Answer

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