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Math 11-Worksheet 08 - assignment

assignment
Course

Calculus I (MATH 011)

81 Documents
Students shared 81 documents in this course
Academic year: 2022/2023
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This document has been uploaded by a student, just like you, who decided to remain anonymous.
University of California, Merced

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Note that each of these worksheets are designed to offer plenty of practice material. You might not be able to finish the worksheet during the discussion section time. The strategy should be to look over all of the questions, discuss the basics of problem solving for each, and work in depth on those exercises that appear unfamiliar. You do not have to turn in your work for these worksheets, but one of the questions will be on the quiz during your discussion section.

1 Section 3.

  1. Compute the derivative

dy dx

for each of the following settings.

(a)

√

xy = cos− 1 (xy) (b) x 5 y 7 = tan(x) tan− 1 (y)

(c)

sin x cos y

=

x y (d) y = sin− 1 (y 2 ) +

√

1 − x

  1. The part of the graph of sin(x 2 + y) = x that contains the point (0, π) defines y as a function of x implicitly.

(a) Verify that (0, π) is a solution to the equation. (b) Is this graph increasing or decreasing near (0, π)?

  1. Find all points where the tangent line to the curve described by

y 3 = xy − 6

is either horizontal or vertical.

  1. Find an equation of the tangent line to the ellipse x 2 + 2y 2 = 1 at the point

(

1

√

2

,

1

2

)

.

  1. A bacterium follows a path that is represented by the equation, (x − y) 2 = − 2 x

√

y, for y > 0, with y = 0 one side wall of the petri dish. Here, y is an implicitly defined function of x.

(a) Find the velocity, dy dx

, of the bacterium.

(b) How fast is the bacterium travelling when it is at the location (− 1 , 1)? (c) Consider when y = 4. Are there any locations for which the bacterium has stopped? Give the coordinates of these locations. (d) Consider when y = k 2 , where k is a constant greater than 0. Are there locations for which the bacterium has stopped? Give the coordinates of these locations (in terms of k).

2 Section 4.

  1. For positive constants k and g, the velocity v of a particle of mass m at time t is given by

v(t) =

mg k

(

1 − e−kt/m

)

At what rate is the velocity changing at time t = 0? How about at time t = 1? What do your answers tell you about the motion?

  1. A certain quantity of gas occupies a volume of 20 cm 3 at a pressure of 1 atmosphere. The gas expands without the addition of heat, so for some constant k, its pressure P and volume V satisfy the relation

P V 1. 4 = k

The volume is increasing at 2 cm 3 /min when the volume is 30 cm 3. At that moment, is the pressure increasing or decreasing? How fast? Include the units.

  1. A voltage V across a resistance R generates a current

I =

V

R

If a constant voltage of 9 volts is put across a resistance that is increasing at a rate of 0 ohms per second when the resistance is 5 ohms, at what rate is the current changing?

  1. A water trough in the shape of an inverted triangle, as in the image below, is being filled with water. The width, W , is 1/3 the overall depth of the trough. Find the rate of change of the volume of water in the trough when the height of the water is 3 feet and is increasing at a rate of 2 ft/s for a 10 foot long trough. Note: the volume of a triangular prism such as this is V = 12 LW H.

  2. In an adiabatic process—one in which no heat transfer takes place—the pressure P and volume V of an ideal gas such as oxygen satisfy the equation P 5 V 7 = C, where C is a constant. Suppose that at a certain instant of time, the volume of the gas is 4 L, the pressure is 100 kPa, and the pressure is decreasing at the rate of 5 kPa/sec. Find the rate at which the volume is changing. 123

1 This was an exam question during Fall 2014. 2 Hint: isolate dV 3 dt after taking the derivatives with respect to t This is a variant of the ideal gas law; the exponents are indeed part of the math problem.

❼ Slope between endpoints is m =

s(3) − s(1) 3 − 1

= 16

2

= 8.

❼ Hence, there is some point, t = c, in [1, 3] where s′(t) = 8.

  1. Troubleshooting. Consder the same problem as above: Draw the conclusion given by the Mean Value Theorem for the function s(t) = 1 + 2t 2 over the interval [1, 3].

Explain what is wrong with the following set of statements. How should this have been done?

❼ Since s(t) is continuous on [1, 3], we can use MVT. ❼ s′(t) = 4t.

❼ Slope between endpoints is m =

s(3) − s(1) 3 − 1

= 16

2

= 8.

❼ Hence, 4t = 8, which indicates that t = c = 2.

  1. Verify that the function satisfies the two hypotheses of the Mean Value Theorem on the given interval, then find all numbers c that satisfy the conclusion of the Mean Value Theorem.

(a) f (x) = 2x 2 − 3 x + 1, [0, 2] (b) f (x) = x 3 + x − 1 , [0, 2] (c) f (x) = x x + 2

, [1, 4]

  1. Consider the function f (x) = 3

√

x over the interval [− 1 , 1].

(a) Show that this function does not satisfy the hypothesis of the Mean Value Theorem. (b) Compute the slope of the secant line between the two endpoints. (c) Show that, despite MVT not being applicable, there is exists a point x = c as in the conclusion of MVT.

Note: This is one justification that you cannot draw any conclusions, one way or the other, when the hypothesis of a theorem is not satisfied!

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Math 11-Worksheet 08 - assignment

Course: Calculus I (MATH 011)

81 Documents
Students shared 81 documents in this course

University: University of California, Merced

Was this document helpful?
Math 11 Worksheet 8 for Sections 3.8, 4.1, 4.2, and 4.4 Spring 2023
Note that each of these worksheets are designed to offer plenty of practice material. You might not be
able to finish the worksheet during the discussion section time. The strategy should be to look over all of the
questions, discuss the basics of problem solving for each, and work in depth on those exercises that appear
unfamiliar. You do not have to turn in your work for these worksheets, but one of the questions will be on
the quiz during your discussion section.
1 Section 3.8
1. Compute the derivative dy
dx for each of the following settings.
(a) √xy = cos−1(xy)
(b) x5y7= tan(x) tan−1(y)
(c) sin x
cos y=x
y
(d) y= sin−1(y2) + √1−x
2. The part of the graph of
sin(x2+y) = x
that contains the point (0, π) defines yas a function of ximplicitly.
(a) Verify that (0, π) is a solution to the equation.
(b) Is this graph increasing or decreasing near (0, π)?
3. Find all points where the tangent line to the curve described by
y3=xy −6
is either horizontal or vertical.
4. Find an equation of the tangent line to the ellipse x2+ 2y2= 1 at the point 1
√2,1
2.
5. A bacterium follows a path that is represented by the equation, (x−y)2=−2x√y, for y > 0, with
y= 0 one side wall of the petri dish. Here, yis an implicitly defined function of x.
(a) Find the velocity, dy
dx, of the bacterium.
(b) How fast is the bacterium travelling when it is at the location (−1,1)?
(c) Consider when y= 4. Are there any locations for which the bacterium has stopped? Give the
coordinates of these locations.
(d) Consider when y=k2, where kis a constant greater than 0. Are there locations for which the
bacterium has stopped? Give the coordinates of these locations (in terms of k).
2 Section 4.1
1. For positive constants kand g, the velocity vof a particle of mass mat time tis given by
v(t) = mg
k1−e−kt/m
At what rate is the velocity changing at time t= 0? How about at time t= 1? What do your answers
tell you about the motion?
1

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