High Velocity Calculus – Set A
Basic Concepts of Straight Line Motion
Position: A numeric value for the number of distance units after a designated zero position. Negative positions
mean positions below or to the left of position zero.
Time Point: A numeric value for the number of time units after a designated zero time. A negative time point
means indicates the number of time units before time zero.
Duration: The difference between two time points, equal to the second minus the first. The duration between
times –3seconds and 5 seconds is 8 seconds. Duration has direction. The duration between 5 seconds and –
3seconds is –8 seconds.
Motion: A change in position over a time interval. Straight line motion can be described by a function with time
as the independent variable and position as the dependent variable.
Displacement: When motion occurs during a time interval, displacement is the difference between the ending and
beginning positions. A negative displacement means that the second position was less than the first. For example,
if an object was at position 7 feet at time 2 minutes and at position 3 feet after 10 minutes, then the displacement
during the time interval 2 min. to 10 min. was –4 feet.
Distance Travelled: Divide a time interval into the largest subintervals in which the displacements are all positive
or all negative. For each subinterval, the distance travelled is the absolute value of the displacement over the whole
subinterval. The sum of all of these distances is the distance travelled over the whole time interval.
Average Velocity: The average velocity over a time interval is the displacement divided by the duration. Notice
that velocity is negative if and only if displacement is negative. Velocity is measured in position units divided by
time units; e.g., miles per hour.
Average Speed: The average speed over a time interval is the distance travelled divided by the duration. Notice
that average speed is never negative.
Velocity (or “instantaneous velocity”): For a motion function, the velocity at a point is the limit of average
velocities as the duration goes to zero around the point. Notice that the velocity could be negative if the average
velocities are all negative in some open interval around the point. Also, notice that velocity at a point might not
exists if there is no open interval around the point in which the average velocities go to a limit. For example, if
s(t ) | x  3 | is the position function of a motion, then look at its graph at the point (t , s(3))  (3,0) . The average
velocities over time intervals whose endpoints are twice as far to the right of 3 as to the left are all equal to 2 , no
3
matter how short the duration. But for the intervals whose endpoints are twice as far to the left as to the right of 3,
the average velocity is always  2 . So there is no limit as the duration goes to zero.
3
Speed: The speed at a point is the absolute value of the velocity at that point.
Average Acceleration: Average acceleration over a time interval is the change in velocity over the interval
divided by the duration of the interval. It can be positive or negative.
Acceleration (or instantaneous acceleration) : The limit of average accelerations around a point as the durations
of the intervals tends to zero. Like velocity, it may or may not exist at a given point and it may be positive or
negative.
Problem 1A: Let s (t ) in the graph at the right represent
the position in feet of an object at various times in
seconds. We can only see the position at whole
numbers of seconds.
1. What is a) s (2) ? b) s (4) ?
c) s (9) ?
2. For what t is s (t ) : a) a maximum? b) a minimum?
3. When is s (t )  0 ?
4. What is the average velocity for s (t ) between: a) t  0 and t  1? b) t  1and t  3 ?
c) t  0 and t  3 ? d) t  1and t  3 ? e) t  1and t  3 ? f) t  0 and t  5 ?
5. What is the slope of a line connecting: a) t  0 and t  1? b) t  1and t  3 ?
6. Explain why the answers to 4a and 4b are the same as the answers to 5a and 5b.
7. What is the mean value of s (t ) on t  {0,1, 2,...,9} ?
8. Suppose that the “true” s (t ) is continuous on the closed interval [0, 9], but we are only seeing
samples at each second mark. a) explain why we cannot estimate s (1.5) with any confidence
b) can we say where s (t ) is negative? why or why not? c) what is the max on s (t ) on [0, 9]?
9. (EVT) If s (t ) is continuous on [0, 9], a) explain why we can be sure it has a maximum on [0, 9]
b) if it is continuous on (0, 9) will it necessarily have a maximum on (0, 9)? why or why not?
c) if s (t ) exists for all t  [0, 9] but is not continuous, will it always have a max on [0, 9]?
10. (IVT) If s (t ) is continuous on [3, 4], a) is there is a point c  [3, 4] where s (t ) =0? Explain.
b) if s (t ) is continuous on [3, 4] is there is a point c  [3, 4] where s (t ) =0? Explain.
c) if s (t ) exists for all t  [3, 4] but is not continuous, is there necessarily a oint c  [3, 4] where
Explain.
11. If s (t ) is continuous on [6, 7], is there is necessarily a point c  [6,7] where s (t ) =0.6? Explain.
Problem 2A: (MVT) Let s (t ) be the same as in Problem 1A.
1. (MVT) Suppose s (t ) is really continuous on [0,1], but we only know the values at t  0 and
t  1. a) What is the average velocity for s (t ) between t  0 and t  1? b) Is there
necessarily a point c  [0,1] where the velocity of the particle is the same as the average
velocity between t  0 and t  1? Explain carefully. c) Suppose that s (t ) is not continuous on
[0, 1]. Would that change your answer to (b)? d) Suppose that s(t) is continuous only on
(0, 1)? Would that change your answer to (b)?
2. (Rolle’s) a) If s (t ) is continuous on [0, 2], is there necessarily a point c  [0,1] where the
velocity of the particle is 0? Explain. b) Would this answer change if s (t ) were only
continuous on (0, 2)? Why or why not?
3. What is the displacement between a) t  0 and t  1? b) t  1and t  3 ? c) t  1and t  2 ?
4. Can you find the distance travelled between t  0 and t  1? Explain why or why not.
5. Can you find the average speed between t  0 and t  1? Explain why or why not.
6. What is the duration between the first two times in the interval [0, 9] that s (t ) = 0?
7. Can we compute the average position between t  0 and t  2 ? Why or why not?
Problem 3A: Let x(t ) in the graph at the
right represent the position of a particle
moving along the x-axis at time t.
1. Estimate: a) x(1) b) x (2) c) x (3)
d) x (3.5) e) x (4)
2. When is the particle furthest: a) to the right?
b) to the left?
s (t ) =0?
3. When does the particle change directions?
4. When is the velocity: a) positive? b) negative? c) zero? d) undefined?
5. Can you think of a graph of x(t ) where the particle changes direction where the velocity is not
equal to zero or undefined? Explain.
6. On what time interval is the particle: to the left of the origin? b) to the right of the origin?
7. What was the displacement of the particle between: a) t=0 and t=1? b) t=0 and t=2?
c) t=0 and t=3? d) t=0 and t=5? e) t=0 and t=5?
8. What was the distance travelled between: : a) t=0 and t=1? b) t=0 and t=2?
c) t=0 and t=3? d) t=0 and t=5? e) t=0 and t=5?
9. What was the average velocity between: : a) t=0 and t=1? b) t=0 and t=2?
c) t=0 and t=3? d) t=0 and t=5? e) t=0 and t=5?
10. What was the average speed between: : a) t=0 and t=1? b) t=0 and t=2?
c) t=0 and t=3? d) t=0 and t=5? e) t=0 and t=5?
11. What was the velocity at: a) 2.5 seconds? b) 4.5 seconds? c) 1 second?
12. What does the velocity appear to be at: a) 0 seconds? b) 2 seconds?
13. If the position x(0.5)  0.8 cms, what is the average velocity between: a) 0 secs and 0.5 secs?
b) 0.5 secs and 1.0 secs?
14. From the graph, is the average velocity positive for all intervals between t=0 and t=1? Why?
15. From the graph, does it appear that the instantaneous velocity positive for all points between
t=0 and t=1? Why?
16. What is the average rate of change of the velocity between 0 secs and 1 secs? (include units)
17. What is the average acceleration of the particle between: a) t=0 and t=1? b) t=1 and t=2?
c) t=0 and t=2? d) t=2 and t=3?
Problem 4A: Let s(t) represents the position of a
car on I-10 in miles west of Houston.
1. Where is the car at: a) hour 2? b) hour 9?
2. What does s(2) mean? b) what are its units?
3. What is the displacement between:
a) hr. 1 and hr. 3? b) hr. 1 and hr. 5?
4. What is the distance travelled between:
a) hr. 1 and hr. 3? b) hr. 1 and hr. 5?
5. What was the average speed of the car between: a) hr. 0 and hr. 1? b) hr. 1 and hr. 2?
c) hr. 2 and hr. 3? d) hr. 1 and hr. 3? e) hr. 1 and hr. 5?
6. In what direction was the car moving at: a) t=1.5? b) t=2.5? c) t=4? d) t=1?
7. What was the average velocity of the car between: a) hr. 0 and hr. 1? b) hr. 1 and hr. 2?
c) hr. 2 and hr. 3? d) hr. 1 and hr. 3? e) hr. 1 and hr. 5?
8. On what intervals was v(t): a) positive? b) negative? c) zero?
9. Did the velocity of the car change between t = 0.1 and t = 0.9? Explain.
10. If you have a 78 average basket weaving and you grade on every assignment was the same,
what else can you conclude about the grades on the individual assignments?
11. What was the velocity at 0.5 seconds; i.e., v(1.5) = ?
Problem 5A: Suppose a car is travelling on I-10 through Texas. Let v(t )  40 miles per hour (mph) be the velocity
of the car for t=0 hours to t=10 hours. There are mile posts every mile that mark the distance west of Houston and
s(t) represents the position of the car at time t.
1. What exactly does 40 miles per hour mean?
2. a) What is v(5)? b) what are the units? c) what exactly does v(5) mean?
3. How far does the car travel during the interval t = 3 to t = 10? (always include units)
4. Draw a graph of v(t), for t  [0,10] .
5. What is the area under the graph for the interval t = 3 to t = 10? (i.e., the area between the
graph of v(t) and the t-axis between the lines t = 3 and t = 10?)
6. What is the distance travelled between t = 0 hours and t = 10 hours?
7. What is the area under the curve between t = 0 and t = 10?
8. Explain why the answers to 6 and 7 are the same.
9. Using only the answers to 3 and 6, what is the area travelled between t = 0 and t = 3? Explain.
10. Using only the answers to 5 and 7, what is the area under the curve between t = 0 and t = 3?
11. What was the average velocity between t = 2 and t = 4?
12. How long will it take to travel: a) 40 miles? b) 60 miles? c) 20 miles?
13. Why can you not say where the car is after hour 1?
14. Suppose the car was at mile post 20 at t=0. Where is it at: a) t=2? b) t=10? c) t=x?
15. Draw a graph of s(t) between t=0 and t=10, given that s(0)=20.
16. What is the slope of the graph of s(t)?
17. In general, if v(t) is constant, what is the slope of s(t)?
18. Suppose v(t)=40 for all t  [0,10] and s(5)=180. a) What is s(0)? b) What is s(1.3)? )?
c) What does it mean that these numbers are negative? d
19. What is the average acceleration of the car between t = 0 hours and t = 10 hours?
Problem 6A: Suppose x(t) is the x coordinate of a particle moving along the x-axis and v (t ) is its velocity in
centimeters per second. Suppose v (t ) is constant at v cps.
1. If x (5)  83 , what units go with 83?
2. If x (5)  83 and x(10)  163 : a) what is v? b) what is x (12) ?
c) what is the slope of the graph of x(t)? d) what is the vertical intercept of the graph of x(t)?
3. If x (5)  83 and x(6)  113 : a) what is the vertical intercept of the graph of x(t)?
b) What does it mean that the vertical intercept is a negative number? c) x(0) = ? d) x(1) = ?
4. Does the particle ever reverse directions? Explain why or why not.
5. Suppose v = 7 and x(1)  10 : a) x(0) = ? b) x(60) = ? c) x(0)  60v  ? d) slope of x(t) = ?
e) Explain why the answers to (b) and (c) are the same?
6. In any straight line motion problem, if v(t) is a constant, explain why x(t )  x(0)  v  t for all t.
7. Draw a graph of v (t ) if v (t ) =v. a) what is the slope? b) what is the vertical intercept?
c) what is a formula for the area under the curve between t = 2 and t = 5? ( that is, the area
between the graph of v (t ) and the t-axis and between the vertical lines t = 2 and t = 5?)
d) what is a formula for the displacement between t = 2 and t = 5?
8. In this problem, let v = –5 cps: a) What is the displacement between t = 2 and t = 6?
b) why does the answer to (a) not depend on where the particle started at time 0?
c) what is the distance travelled between t = 2 and t = 6?
d) what is the speed between t = 2 and t = 6?
e) what is the average acceleration between t = 2 and t = 6?
f) when the velocity is a negative constant, what is the relation between velocity and speed?
g) if x(2)  24 , what would be x(20) ?
Problem 7A: Suppose a car is driving on a road with velocity given by v(t) miles per hour and position by s(t),
 40mph if 0  t <4

where t is time in hours. Suppose v(t )   60mph if 4  t <8
 50mph if 8  t <10

1.
2.
3.
4.
5.
6.
7.
8.


.


Draw a graph of v(t) from t = 0 to t = 10.
What are: a) v(3)? b) v(4)? c) v(5)? d) v(8)? e) v(8.3)?
What is the average velocity between: a) t = 1 and t = 3? b) t = 0 and t = 6?
What is the area under the v(t) curve between: a) t = 4 and t = 8? b) t = 0 and t = 6?
What is the distance travelled between: a) t = 4 and t = 8? b) t = 0 and t = 6?
What is the displacement between: a) t = 4 and t = 8? b) t = 0 and t = 6?
Based on (6), what was the average velocity between: a) t = 4 and t = 8? b) t = 0 and t = 6?
Suppose s(1) = 60 miles. a) Draw a graph of s between t = 0 and t = 10?
What are: a) s(0)? b) s(3)? c) s(4)? d) s(5)? e) s(8)? f) s(10)?
9. a) What is the distance travelled between t = 0 and t = 10? b) Did the car ever reverse
direction between t = 0 and t = 10? c) s(10) – s(6) = ? d) Does (c) depend on the fact that
s(1) = 60? e) Where was the car at time 0?
10. What is the area under the curve between t = 0 and t = 10? Explain exactly how and why (9a)
and (10) are related.
11. Let Ab ( x) be the area under the velocity curve between b and x on the graph of v from (1),
where b is fixed. What are: a) A1 (2) ?
b) A2 (6) ?
c) A4 (10) ?
d) Is it true that A5 (6)  A6 (8)  A5 (8) ? e) Explain why using geometry.
f) If b  c  d and v(t) > 0, would it always be true that Ab (c)  Ac (d )  Ab (d ) . Why?
g) In this problem, what is a formula for A0 ( x) : a) if x < 4? b) if 4 < x < 8?
h) If v(t) remained the same at all time except at t = 4, and now v(4) = 100, would this change
answer to (g). Why or why not?
i) Is Ab ( x) a function of x? Why or why not?
the
j) Draw a graph of A0 (t ) for t = 0 to t = 10. k) Is the graph continuous?
l) What is the displacement of the car between t = 1 and t = 2? Relate this to (11a).
m) What is the displacement of the car between t = 2 and t = 6? Relate this to (11b).
12. (FTC) Suppose s(1) = 60. a) calculate s(1) + A1 (5) b) calculate s(5) c) compare (a) & (b)
d) Explain why (c) turns out the way it does. e) Explain why, in general, it will turn out
that if b < c then s(b) + Ab (c) = s(c).
13. In the graph in (8), what is the slope of the line between (3, s (3)) and (9, s (9)) .
14. What is the average acceleration between: a) t = 1 and t = 2? b) t = 2 and t = 6?
c) Why are the answers to 14(b) and (13) the same?
15. Let a(t) be the acceleration of the car at time t. What are: a) a(1)? b) a(5)? c) a(4)?
16. Why does question (14) say “between” but question (15) says “at”?
17. Why is a(8) undefined?
21. Calculate
A1 (3)
, where Ab ( x) is defined in (11).
3 1
22. What is the average height of the graph of v between t = 1 and t = 3?
23. What is the average velocity between t = 1 and t = 3?
21. Calculate
A2 (7)
72
22. What is the average height of the graph of v between t = 2 and t = 7?
23. What is the average velocity between t = 2 and t = 7?
Problem 8A: Suppose a particle travels in a straight line with its position in meters being given by the function of
time, t, in minutes:
 5  3t if 0  t  2

s(t )   11  2t if 2  t  5
 7  4t if 5  t  10






1. Carefully draw a graph of s(t).
2. Is the graph continuous on [0, 10]?
3. If you has not drawn the graph, could we tell from the formula for s(t) whether it is continuous?
ds
represent the slope of the graph of s(t) when t = x; that is, it represents the change in s (rise)
dt x
ds
ds
divided by the change in t (run) over a very small interval around x. Then what are: a)
? b)
?
dt 3
dt 1
ds
ds
c)
?
d)
?
dt 6
dt 2
4. Use the symbol
5. What is the average velocity between: a) t = .5 and t = 1.5?
6. What is the relation between the answers to (4) and (5)?
b) t = 3 and t = 4?
7. Between t = 1 and t = 9, what are: a) the displacement b) the duration c)the average velocity?
8. What is the average velocity between: a) t = 1.5 and t = 1.6? b) t = 3.9 and t = 4.0?
c) t = 7.1and t = 7.2?
d) t = 7.14 and t = 7.15?
9. Explain the relation between average velocities in (8c) and the slope in (4c).
10. Let v(t) be the (instantaneous) velocity at time t. a) where is v(t) undefined? b) what would
you estimate v(3) to be. Why?
11. Give a good argument why the slope of the graph of s is the same as v; i.e., that
ds
= v(x).
dt x
Draw a graph of v(t), for t  (0,10) . Put in open circles wherever v(t) is undefined..
Write the formula for v(t) as a piecewise function.
Make a rule: whenever the graph of s(t) is connected line segments, the graph of v(t) is…
What is the area under the curve of v(t) between t = 1 and t = 9?
What is s(9) – s(1).
How and why are (15) and (16) related?
Use the graph of v(t) to find the average velocity between t = 1 and t = 9.
Use the formula for s(t) to find the average velocity between t = 1 and t = 9.
Explain why we get the same answer for (18) and (19).
Let be the acceleration of the particle at time t. What are: a) a(1)? b) a(3)?
c) a(6)? d) a(1)? e) Where is a(t) undefined?
22. What is the average acceleration between: a) t = 1 and t = 1.5? b) t = 1 and t = 3?
c) t = 1 and t = 4?
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
Problem 9A: Let s(t) be the position, in miles west of Houston, of a car on I-10, where t is the time in hours. Let
the velocity of the car be given by the formula v(t )  8  10t .
1. Draw a graph of v from t = 0 to t = 12.
2. On the same graph, draw v*(t), an approximation of v, given by:
 8mph if 0  t <4

v *(t )   42mph if 4  t <8
 74mph if 8  t <12
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
3.
4.
5.
6.
What are: a) v(4)? b) v*(4)? c) v(3)? b) v*(3)?
What is the area under the graph of v*(t) between: a) t = 1 to t = 3? b) t = 0 to t = 12 ?
Approximate, using v*(t), the distance travelled between time 0 hours and 12 hours.
Let’s use a better approximation for v by dividing [0, 12] into 6 equal intervals.
Graph v and this new v*(t) on the same coordinates.
7. Explain why the new approximation in more accurate.
8. Using this new v*, what is the approximate distance travelled between t = 0 and t = 12?
9. What is the area under v* between t = 0 and t = 12?
10. Explain why we would get a more accurate estimate still of the distance travelled if we
divided the time interval into 120 equal subintervals.
11. Problem for later: what is the exact number we would get by adding the 120 distances?
12. As we divide into more and more intervals, the area under the approximation curve is
becoming closer and closer to the area of what figure?
13. Why would we say that the true (exact) distance travelled between t = 0 and t = 12 is the area
under the line v(t)?
14. Using elementary geometry, what is the area under v(t) between t = 0 and t = 12?
15. Exactly how far did the car travel between t = 0 and t = 12?
16. If the car were at mile 17 at time 0, where was it at time 12?
17. What is the formula for A0 (5) , the area under the velocity curve between t = 0 and t = 5?
18. What is the formula for A0 (t ) , the area under the velocity curve between t = 0 and any time t?
19. (FTC) If s(2) = 32, what would be If s(7)?
20. (FTC) Explain why, in general, a the equation s(c)  s(b)  Ab (c) .
21. (FTC) Rewrite the formula in 20 to show the displacement: s (c)  s (b)  ?
Problem 10A: Suppose a particle travels along the y-axis with its position given by y(t), where t is time measured
in minutes. Let the velocity of the particle be given by v(t) = 10 – 2t inches/min.
Suppose y(0) = 2.
1. (a) Draw a graph of v(t) for 0  t  8 . (b) What is the t intercept? (c) …the vertical intercept?
2. What are the values of: (a) v(0)? (b) v(5)? (c) v(8)?
3. (a) When is v(t) positive? (b) In what intervals is the particle moving upward?
4. At what time in [0, ) is the particle furthest up? … down?
5. How far does the particle travel: (a) upward? (b) downward?
6. How far does the particle travel between: (a) t=5 and t=8? (b) t=0 and t=5? (c) t=0 and t= 8?
7. What is the displacement between: (a) t=5 and t=8? (b) t= 0 and t=5? (c) t=0 and t= 8?
8. What is the average velocity between: (a) t=5 and t=8? (b) t= 0 and t=5? (c) t=0 and t= 8?
9. What is: (a) A0 (2) (a trapezoid?)
(b) y(2)?
10. Evaluate (using area of a trapezoid formula): (a) A2 (5)
(b) A5 (8)
11. What is the displacement between: (a) t=2 and t=5? (b) t=5 and t=8?
12. What are: (a) y(2)?
(b) y(5)?
(c) y(8)?
Use the following notation for the rest of the questions.
c
If b  c and f ( x)  0 for all x  (b, c) , then

b
f ( x)dx  Ab (c) and
b
 f ( x)dx   A (c) .
b
c
c
If b  c and f ( x)  0 for all x  (b, c) , then
 f ( x)dx   A (c) .
b
b
In other words, if the lower limit of the integral of a positive function is less than the upper limit, then the
definite integral of f(x) is the area under f between the limits.
5
10. Evaluate: (a)
2
 y(t )dt
(b)
 y(t )dt
7
(c)
5
2
 y(t )dt
5
(d)
5
 y(t )dt
7
5
11. Evaluate: (a) y(0) +
 y(t )dt
(b) y(5)
0
9
12. Evaluate: (a) y(6) +
 y(t )dt
(b) y(9)
6
x
13. Evaluate: (a)
 y(t )dt , if 0 < x < 5.
(Use formula for area of a trapezoid.)
2
x
(b) y(2) +
 y(t )dt
(b) y(x)
6
14. Using the formula for y(x) from (13b), draw a graph of the equation of y(t) from t=0 to t= 8.
15. What is the shape of this graph?
x
16) Explain why the Fundamental Theorem of Calculus, y(b) +
 y(t )dt
= y(x), is true.
b
17) What is the speed of the particle at: (a) t = 4? (b) t = 8?
18) Draw a graph of the speed of the particle over [0 min, 8 min].
7
19) Evaluate
 y(t ) dt .
0
20) How and why are (6c) and (19) related?
Problem 11A: Suppose a particle travels along the y-axis with its position given by y(t), where t is time measured
in minutes. Let the velocity of the particle be given by v(t) = 20 – 4t inches/min.
Suppose y(0) = 2.
1. What is the average acceleration between: (a) t=5 and t=8? (b) t=0 and t=5? (c) t=0 and t= 8?
2. What is the average acceleration between: (a) t=6and t=7? (b) t=6.4 and t=6.5?
3. What is the acceleration at t = 6.45?
4. Draw a graph of the acceleration function over t  [0,8] .
5. When does v change from positive to negative? When is the particle highest over [0, 6] ?
6. When is it lowest on [4,6] ?
Problem 12A: Suppose a particle travels along the x-axis with its position at t seconds given by x(t ) and it
10  2t if t  6 
 . Suppose x(0) = 40 feet.
-8+t if 6  t 
velocity by v (t ) , where v(t )  
1.
2.
3.
4.
What are the units for v?
Draw a graph of v (t ) .
When is v (t ) : (a) increasing? (b) decreasing? (c) positive? (d) negative?
Explain what it means for the motion of the particle for the velocity to be positive versus for
the velocity to be increasing.
5. When is the particle’s speed: (a) increasing? (b) decreasing? (c) zero?
6. Use a dotted line to draw a graph of the speed on the same coordinates as the velocity.
3
7. Evaluate:
(a)
5
 v(t )dt
(b)
0
(e) for x > 6,

x
6
 v(t )dt
0
v(t )dt
(c)

6
0
v(t )dt
(f) for x > 6,
8
(d)

x
0
 v(t )dt
0
v(t )dt
8. What is: (a) the displacement between t=0 and t=3? ( Use (7) to answer this. )
9. Use the FTC to find: (a) x(3)?
(b) x(6)?
(c) x(7)? (d) x(8)? (e) for t >6, x(t)?
10. Draw a graph of x(t) for t  [0,12] on a new set of axis.
11. On the interval [0,10], when it the particle furthest: (a) to the right? (b) to the left?
12. When does the particle change directions?
13. What is the maximum value of x(t) for t  [0,12] ?
14. A rule for finding the maximum value of x on a closed interval is to check the value of x at
both endpoints of the interval and at any point where the velocity changes from positive to
negative. Explain why this makes sense.
15. In (14), why do we not need to check points where the velocity turns from negative to
16. Prove that v(t) is continuous without referring to the graph of v.
17. When does x(t) reach a minimum on the interval: (a) [5,7]? (b) [5,9]? (c) [2,9]?
18. Make up a rule like in (14) for finding the minimum of x on a closed interval.
19. Suppose x(t) was not a continuous function on an interval. Would your rule in (18)
necessarily work? Why or why not?
positive?