Math 1310 Lab 11.
Name/Unid:
Lab section:
Note: no credit is given for answers without any explanation.
1. (Motion)
A world-class sprinter lines up at the starting line of a 100-meter race. After the gun
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sounds, the sprinter’uares acceleration at t seconds is given by a(t) =
meters per
t+1
second square.
(a) Write a function v(t) which describes the velocity of the sprinter at t seconds.
(Hint: To determine the integration constant, remember that the sprinter starts at
a velocity of 0 meters per second.) (2 pts)
(b) Write a function p(t) which describes the position of the sprinter at t seconds.(Hint:
To determine the integration constant, remember that the sprinter starts at a position of 0 meters.) (2 pts)
(c) Will this sprinter beat the world record of 9.58 seconds held by Usain Bolt?(Explain
and show your work.) (2 pts)
2. (Distance v.s. Displacement)
Suppose a car moves in a straight line with the velocity v(t) = tsin(t) meters per second.
(a) Graph the function v(t) = tsin(t) from 0 to 2π. (1 pt)
(b) What’s the displacement of the car after 2π seconds? (Hint: You may need to
integrate by parts.) (2 pts)
(c) What’s the distance that the car have travelled after 2π seconds? (Hint: Since
sin(t) will change sign, the velocity will change direction. Break up the integral
Rc
Rb
Rc
using the property that a f dx = a f dx + b f dx) (3 pts)
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3. (Differentiation
Rx
d
v.s Integration a )
dx
d
In class, we learned the following theorem: If f is a continuous function, then
is an
dx
Rx
d Rx
inverse of a in the sense that
f (t)dt = f (x).
dx a
d Rx 2
(t + 1)dt = x2 + 1 by direct calculation. (2 pts)
(a) Verify
dx 0
(b) Is it still true if the order of the differentiation and integration is changed? Or
R x df (t)
equivalently, if f is a differentiable function, is a
dt = f (x) true? Prove it or
dt
give a differentiable function f such that the identity in question is false and explain
why the identity is false by showing your calculation. Hint: Try the function in
part(a) and let a = 0 in the domain of the integral. (2 pts)
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2 Rx 2
4. (The Error function) The error function erf (x) = √ 0 e−t dt is important in probπ
ability, statistics and engineering.
√
R b −t2
π
(a) Show that a e dt =
[erf (b) − erf (a)] (2 pts)
2
2 Rx t 2
(b) Suppose x is positive. Follow the inequality erf (x) ≥ √ 0 e−t dt to give a lower
x
π
bound of erf (x). In other words, simplify the integral into a function of x that does
not involve t. (3 pts)
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5. (Integration techniques) Solve the following integrations.
R 2π
(a) 0 cos(x)ex dx (1 pt)
(b)
R2
(c)
R1
1
0
ln(x)dx ( 1pt)
x
dx (1 pt)
1 + x2
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(d)
R4 x+3
dx (1 pt)
2
x2 − 1
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