Particle Motion
Ex. A particle moves along a horizontal line so that its position at any time t  0 is given by
s  t   2t 3  7t 2  4t  5 , where s is measured in meters and t in seconds.
(a) Find the velocity at time t and at t = 1 second.
(b) When is the particle at rest? Moving left? Moving right? Justify your answers.
(c) Find the acceleration at time t and at t = 1 seconds.
(d) Find the displacement of the particle between t = 0 and t = 3 seconds. Explain the meaning of
your answer.
(e) Find the distance traveled by the particle between t = 0 and t = 3 seconds.
Ex. (continued)
A particle moves along a horizontal line so that its position at any time t  0 is given
by s  t   2t 3  7t 2  4t  5 , where s is measured in meters and t in seconds.
(f) When is the particle speeding up? Slowing down? Justify your answer.
(Hint: Since speed is the absolute value of velocity, the particle is:
1) Speeding up when the velocity and acceleration have the same sign (both pos. or both neg.)
2) Slowing down when the velocity and acceleration have opposite signs (one pos. and one neg.)
Homework: Worksheet 1 on Particle Motion
Particle Motion, Day 2
Ex. A particle moves along a horizontal line so that its position at any time t  0 is given by
s  t   t 3  5t 2  2t  3.
(a) When is the particle moving right? moving left? Justify your answer.
(b) Find the distance traveled by the particle from t = 0 to t = 5. Justify your answer.
(c) Find the intervals where the speed is increasing. Justify your answer.
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Ex. (Multiple Choice) A particle moves along the x-axis so that at any time t  0 , its velocity is
given by v  t   3  4.1cos (0.9t ) . What is the acceleration of the particle at time t = 4?
(A) – 2.016
(B) – 0.677
(C) 1.633
(D) 1.814
(E) 2.978
Homework: Worksheet 2 on Particle Motion
3.9 Linear Approximations and Differentials
The tangent line to a curve gives us a useful way to approximate a value on the curve. We usually write our
tangent line equations in point-slope form: y  y1  m  x  x1  or for a particular value of x1 named a, we could
write y  f  a   f   a  x  a  .
When we are using a tangent line to find an approximation of a value on the curve, we might want to rearrange
the terms of the tangent line equation to write: y  f  a   f   a  x  a  .
Note: y represents the y-coordinate of a on the tangent line. This gives us an approximation of f  a  .
The y-coordinate of a on the function f  x  is represented by f  a  .
f  a  is the actual value.
2
Ex. Given f  x   x .
(a) Write the equation of the tangent line to f at x = 1
(b) Use your answer to (a) to find an approximation for f 1.01 .
(c) Find the actual value of f 1.01 .
(d) Find the amount of error of your approximation.
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Ex. Use a tangent line approximation to approximate the value of 3 127. To do this, let f  x   3 x , and then
choose a perfect cube close to 127, and build your tangent line at that x-value.
(From Calculus by Finney, Demana, Waits, Kennedy)
dy
Leibniz used the notation
to represent the derivative of y with respect to x. The notation looks like a
dx
quotient of real numbers, but it is really a limit of quotients in which both numerator and denominator go to zero
(without actually equaling zero). Leibniz did most of his calculus using dy and dx as separate entities, but he
never quite settled the issue of what they were. To him, they were “infinitesimals” ---nonzero numbers but
infinitesimally small.
It is tricky to define dy and dx as separate entities, but since we really only need to define dy and dx as
formal variables, textbooks define them in terms of each other so that their quotient must be the derivative.
Definition: Let y  f  x  be a differentiable function. The differential dx is an independent variable. The
differential dy is dy  f   x  dx. The variable dy is always a dependent variable; it depends on both x and
dx .
Ex. Find the differential dy, given y  5 x3 .
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Ex. The radius r of a circle increases from 10 m to 10.1 m.
(a) Use the differential dA to estimate the increase in the circle’s area A.
(b) Compare the estimate you found in (a) with the true change A , and find the error in your approximation.
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Ex. Suppose the earth were a perfect sphere and we determined its radius to be 3959  0.1 miles. What
effect would the tolerance of 0.1 miles have on our estimate of the earth’s
Homework: Worksheet on Linear Approximation and Differentials
3.8 Newton’s Method of Finding Zeros
Suppose f is a differentiable function on (a, b). If f  a 
and f  b  differ in sign, then f has at least one zero in (a, b).
Newton made a guess that this zero occurs at some x1 in (a, b),
and he discovered that the zero of the tangent line was close to the
zero of the function.
Write the equation of the tangent line to f at
 x1 , f  x1   , and solve for x, the x-intercept of the tangent line.
Each successive approximation is called an iteration.
Formula for Newton’s Method:
xn 1  xn 
f  xn 
f   xn 
Formula for Newton’s Method:
xn 1  xn 
f  xn 
f   xn 
TI-89 steps: Let y1  function and y2  derivative
On the home screen, type the x1 you were given and press Enter.
Then type: Answer 
y1 Answer 
y 2  Answer 
Enter Enter Enter ...
The result after the first “Enter” is x2 .
The result after the second “Enter” is x3 .
The result after the third “Enter” is x4 .
Keep pressing Enter until your iterations give you as accurate an answer as
the directions ask you for.
Ex. Use Newton’s Method to approximate the zero of f  x   2 x3  x2  x  1. Continue the
iterations until two successive approximations differ by less than 0.001. Let x1  1.2 .
f  x   2 x3  x 2  x  1 so let y1  2 x3  x 2  x  1
f  x 
so let y 2 
Then go to the home screen and type – 1.2 Enter and then
Answer 
y1 Answer 
y 2  Answer 
Enter Enter Enter ...
x2 
x3 
x4 
By Newton’s Method, the zero is approximately _____________.
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Ex. Use Newton’s Method to estimate the x-coordinate of the positive point of intersection
of y  tan x and y  2 x. Let x1  1.25. Continue the iterations until two successive
approximations differ by less than 0.001. After you have estimated the x-coordinate,
find the y-coordinate to go with it.
Homework: Worksheet on Newton’s Method and Tangent Line Approximations