Math 1210-001 Lab 3
Spring 2016
Due Date: 02/04/2016
uID:
Name:
Instructions: This lab relates to sections 2.1-2.2, introduction to derivatives; and 2.3, rules
for finding derivatives. Please show all of your work as partial credit will be given where
appropriate, and there may be no credit for problems where there is no work shown. All answers should be completely simplified, unless otherwise stated. No calculators or electronics
of any kind are allowed, unless explicitly mentioned. This assignment is due at the start of
your lab meeting on Thursday February 4.
1. (18 points) Tangent Lines
The parabola shown below is the graph y = f (x), for the quadratic function f (x) =
x2 − 3x + 1.
(a) Find and label the coordinates of the three points on the parabola for which x =
−2, x = 0, x = 2. Then sketch the three tangent lines to the parabola passing
through those points. (You can make a nice straight-edge by folding a piece of
paper.)
(b) Estimate the slopes of the three tangent lines, like we did in class.
(c) Use the limit definition of tangent line slope, as a limit of secant line slopes (i.e. the
definition of derivative), to algebraically find the slopes you estimated in part (b).
(Note: You can do this with one limit computation.) Are the exact slope values
close to your estimates in part (b)?
(d) Write the equations for the three tangent lines you sketched in part (a), using pointslope form and the exact slopes you computed in part (c). Simplify these equations
so that they are in slope-intercept form. Discuss how the computed y−intercepts of
the tangent lines compare with the y−intercepts of the tangent lines you sketched
in part (a).
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2. (14 points) Estimation
The following graph shows an estimation of the number of people in the Union on
Thursday at each hour of the day.
(a) What is the most number of people in the Union during Thursday, and when does
this happen?
(b) What is the approximate rate of change in the number of people, in units of people
per hour, at 2 PM?
(c) What is the approximate rate of change at 9 AM?
(d) What is the approximate average rate of change between 10 AM and 1 PM?
(e) Name the times where the rate of change is equal to zero.
(f) At which time is the rate of change the greatest?
(g) At which time is the rate of change the lowest?
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3. (12 points) Falling Objects
A falling object will fall approximately 10t2 meters in t seconds under the influence
of gravity, neglecting air resistance and assuming the object is dropped with no initial
velocity.
(a) How far will an object fall between t = 0 and t = 1 second?
(b) How far will it fall between t = 2 and t = 3 (assuming it doesn’t hit the ground)?
(c) What is its average velocity on the interval 2 ≤ t ≤ 3?
(d) What is its average velocity on the interval 2.9 ≤ t ≤ 3?
(e) Find its instantaneous velocity at t = 3 seconds.
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4. (10 points) Electric current
The rate of change of electric charge with respect to time is called current. Suppose that
1 3
t + t coulombs of charge flow through a wire in the first t seconds after the electric
3
line is switched on. Find the current in amperes (coulombs per second) after 3 seconds.
At what time will a 50-ampere fuse in the line blow?
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5. (10 points) Use the limit definition of derivative to find the derivatives of the given
functions.
1
(a) (5 points) f (t) =
t+3
(b) (5 points) f (x) = √
4
1−x
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uID:
Name:
6. (12 points)The given limit is a derivative, but of what function and at what point?
Note: You are not asked to actually compute the limit value. This problem is practice
in recognizing the equivalent limit notations for derivatives.
√
4
16 + h − 2
(a) (3 points) lim
h→0
h
2x − 32
x→5 x − 5
(b) (3 points) lim
cos(π + h) + 1
h→0
h
(c) (3 points)lim
t4 + t − 2
t→1
t−1
(d) (3 points) lim
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7. (9 points) The graph of f is given. State, with reasons(corner, discontinuity, or vertical
tangent) , the numbers at which f is not differentiable.
(Hint: See Figure 4 on page 103 of the text)
(a) (3 points)
(b) (3 points)
(c) (3 points0
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8. (15 points) Use section 2.3 ”shortcuts” to find the derivatives f 0 (x), for the given functions
(a) (3 points) f (x) = πx4 + 2x2 − 5x + 100
(b) (4 points) f (x) =
3x2 − x + 10
2x + 1
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(c) (4 points) f (x) =
2π
9
−4
−
x
+
x3
x7
1
(d) (4 points) f (x) = (5x3 + 1)(x4 − 2x2 − x)
2
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