Chapter 3 – Part 1: THE DERIVATIVE



Unit 2 Packet

[pic]

[pic]

Calculus Homework NAME _________________________________

Derivatives and Limit Review

1. [pic]

2. If [pic], which of the following are true statements?

I. [pic] is continuous at x = 0.

II. [pic] is differentiable at x = 0.

III. [pic]

3. Let b > 0 be a fixed constant. Then [pic]

*JUST TELL WHAT #3 means, YOU WILL NOT BE ABLE TO “DO” IT*

4. Let [pic]be defined as follows, where [pic]:

[pic]

Which of the following are true about f?

[pic][pic]

[pic]I. [pic]exists.

[pic]II. [pic]exists.

[pic]III. [pic]is continuous at [pic].

A) I only B) II only C) I and II only D) All three are true E) None are true

5. [pic]

A) 0 B) [pic] C) [pic] D) [pic] E) [pic]

6. [pic]

A) 0 B)[pic] C) [pic] D) [pic] E) Does not exist

7. [pic]

A) 1 B) 6 C) 0 D) –1 E) Does not exist

8. Let [pic] Find a value of a so that h is continuous at [pic].

A) [pic] B) [pic] C) [pic] D) [pic]

E) It is not possible to find a value of a so that [pic]is continuous at [pic]

9. If [pic] is continuous at [pic], then [pic]

A) 3 B) –3 C) 0 D) 6 E) –6

10. [pic]

A) 0 B) 1 C) –1 D) [pic] E) [pic]

11. The graph of which function has [pic] as an asymptote?

A) [pic] B) [pic] C) [pic] D) [pic] E) [pic]

12. [pic]

13. The [pic]at x = 3 is:

A) 0 B) 1 C) 3 D) –1 E) nonexistent

14. If y = 7 is a horizontal asymptote of a rational function, f, then which of the following must be true?

[pic]

15. [pic]is:

[pic]

Position, Velocity, and Acceleration

1. A balloonist drops a sandbag from a balloon 160 feet above the ground. After t seconds, the sandbag is 160-16t2 feet above the ground.

a) Find the velocity of the sandbag at t=1.

b) With what velocity does the sandbag hit the ground?

2. A projectile is fired directly upward from the ground with an initial velocity of 112ft/sec. Its distance above the ground after t seconds is 112t-16t2 feet.

a) Find the velocity of the projectile at t = 2, t=3, and t=4.

b) When does the projectile hit the ground?

c) Find the velocity at the instant it hits the ground.

3. A projectile is shot upward from the surface of the earth with and initial velocity of 384 ft/sec. If s(t)= 384t-16t2, what it its velocity after 5 seconds? After 10 seconds?

4. A pebble is dropped from a height of 600 feet. If s(t)=600-16t2, find the pebble’s velocity when it hits the ground.

5. A ball is thrown straight down from the top of a 220 foot building with an initial velocity of –22 ft/sec. If s(t)=220-22t-16t2,

a) What is the velocity after 3 seconds?

b) What is its velocity after falling 108 feet?

6. A ball is dropped from a height of 100 feet and s(t) = 100-16t2 ft/sec. One second later another ball is dropped from a height of 75 feet and s(t)=75-16t2 ft/sec. Which one hits the ground first?

7. An object is dropped from a balloon which is stationary at 1600 ft. Express its height above the ground as a function of t. (Hint: Look at the pattern of the equations in 1-6). How long does it take the object to reach the ground?

8. A ball is thrown vertically upward from the ground with an initial velocity of 60 ft/sec. How high will the ball go?

9. A ball is thrown upward with an initial velocity of 64ft/sec from a height of 80 ft.

a) Find the position function for this motion.

b) When will the ball hit the ground?

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