Tag Archives: Feliciano and Uy

Cover photo of Chapter 8: Partial Differentiation of the book Differential and Integral Calculus by Feliciano and Uy

Chapter 8: Partial Differentiation

8.1 Partial Derivative

8.2 Geometric Interpretation of Partial Derivative

Exercise 8.1

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

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8.3 Partial Derivatives of Higher Order

Exercise 8.2

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

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8.4 Total Differentiation

Exercise 8.3

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

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8.5 Total Derivative

Exercise 8.4

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

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8.6 Partial Differentiation of Implicit Functions

Exercise 8.5

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

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8.7 Tangent Plane and Normal Line

Exercise 8.6

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

8.8 Maxima and Minima

Exercise 8.7

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

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Cover photo of Chapter 7: Derivatives from Parametric Equations, Radius and Center of Curvature of the book Differential and Integral Calculus by Feliciano and Uy

Chapter 7: Derivatives from Parametric Equations, Radius and Center of Curvature

7.1 Derivatives in Parametric Form

Exercise 7.1

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

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7.2 Differential of Arc Length

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7.3 Radius of Curvature

Exercise 7.2

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

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7.4 Center of Curvature

Exercise 7.2

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

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Cover photo of Chapter 6 The Differential of the book Differential and Integral Calculus by Feliciano and Uy

Chapter 6: The Differential

6.1 Differential: Definition and Interpretation

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6.2 Differential Formulas

Exercise 6.1

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

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6.3 Applications of the Differential

Exercise 6.2

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

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Cover photo of Chapter 5 The Indeterminate Forms of the book Differential and Integral Calculus by Feliciano and Uy

Chapter 5: The Indeterminate Forms

5.1 Rolle’s Theorem

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5.2 Mean Value Theorem

Exercise 5.1

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

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5.3 L’Hospital’s Rule

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5.4 The Indeterminate Forms \displaystyle \frac{0}{0} and \displaystyle \frac{\infty }{\infty }

Exercise 5.2

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

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5.5 The Indeterminate Forms \displaystyle 0\left( \pm \infty \right) and \displaystyle \infty -\infty

Exercise 5.3

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

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5.6 The Indeterminate Forms \displaystyle 0^0 , \displaystyle 1^\infty , and \displaystyle \infty ^0

Exercise 5.4

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

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Cover photo of Chapter 4 Differentiation of Transcendental Functions of the book Differential and Integral Calculus by Feliciano and Uy

Chapter 4: Differentiation of Transcendental Functions

4.1 The Function \displaystyle \frac{\sin u}{u}

Exercise 4.1

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

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4.2 Differentiation of Trigonometric Functions

Exercise 4.2

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

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4.3 Differentiation of Inverse Trigonometric Functions

Exercise 4.3

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

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4.4 The functions \displaystyle \left( 1+u \right)^{\frac{1}{u}}

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4.5 The Logarithmic and Exponential Functions

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4.6 Differentiation of Logarithmic Functions

Exercise 4.4

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

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4.7 Logarithmic Differentiations

Exercise 4.5

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

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4.8 Differentiation of Exponential Functions

Exercise 4.6

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

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4.9 The Hyperbolic Functions

Exercise 4.7

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

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4.10 Differentiation of Hyperbolic Functions

Exercise 4.8

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

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4.11 Differentiation of Inverse Hyperbolic Functions

Exercise 4.9

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

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Cover photo of Chapter 3 Some Applications of the Derivatives of the textbook Differential and Integral Calculus by Feliciano and Uy

Chapter 3: Some Applications of the Derivative

3.1 Equations of Tangents and Normals

Exercise 3.1

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

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3.2 Angle Between Two Curves

Exercise 3.2

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

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3.3 Increasing and Decreasing Functions

Exercise 3.3

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

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3.4 Maximum and Minimum Values of a Function

Exercise 3.4

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

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3.5 Significance of the Second Derivative

Exercise 3.5

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

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3.6 Applications of the Maxima and Minima

Exercise 3.6

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

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3.7 Related Rates

Exercise 3.7

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

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3.8 Rectilinear Motion

Exercise 3.7

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

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Cover photo for Chapter 2 Differentiation of Algebraic Functions of the textbook Differential and Integral Calculus by Feliciano and Uy

Chapter 2: Differentiation of Algebraic Functions

2.1 The Symbol Δ

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2.2 The Derivative of a Function

Exercise 2.1

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

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2.3 Geometric Significance of dy/dx

2.4 Rules for Differentiation

Exercise 2.2

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

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2.5 The Chain Rule

2.6 Differentiation of Inverse Functions

Exercise 2.3

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

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2.7 Higher Derivatives

Exercise 2.4

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

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2.8 Implicit Differentiation

Exercise 2.5

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

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Elementary Differential Equations by Dela Fuente, Feliciano and Uy Physical Application 2: Exponential Growth and Decay

A certain radioactive material follows the law of exponential change and has a half life of 38 hours. Find how long it takes for 90% of the radioactivity to be dissipated.

Solution:

Use the formula:

S=Ce^{-kt}

First, find the constant of proportionality. In the problem, after 38 hours, half of the radioactivity has been dissipated and a half has been retained. So we can assume that S = 0.5So when t = 38 hrs and C = So.

\left(0.5\right)So=\left(So\right)e^{-k\left(38\right)}

And then solve for k:

k=-0.018241

And then substitute k to the formula:

S=Ce^{-0.018241\left(t\right)}

Now we can solve for the time(t). According to the problem, 90% of the radioactivity is dissipated, so 10% is retained. So we can assume that S = 0.1So and change C = So.

\left(0.1\right)So=\left(So\right)e^{-0.018241\left(t\right)}

And then solve for time(t):

t\:=\:126.23\:hrs
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Elementary Differential Equations by Dela Fuente, Feliciano, and Uy Chapter 10 Problem 12 — Applications of Ordinary First-Ordered Differential Equations

A bacterial population follows the law of exponential growth. If between noon and 2 p.m. the population triples, at what time should the population become 100 times what it was at noon? At 10 a.m. what percentage was present?

SOLUTION:

First, we denote

P as the population of bacteria at anytime

Po as the original bacterial population

t = 0 (12 noon)

t = 2 (2 p.m.)

Let us determine the given and the required

GIVEN:

@12nn to 2p.m.; P= 3Po

REQUIRED:

  1. what time should the population become 100 times
  2. at noon
  3. percentage at 10 a.m.

Using the formula of Applications of Ordinary First-Ordered Differential Equations under Exponential Growth or Decay

\frac{dP}{dt}=kP \\
\int \:\frac{dP}{P}=\int \:kdt\\
e^{ln\:P}\:=\:e^{kt\:+\:C}\\
P=\:Ce^{kt}\:\:\:\:\:(Eq.1)\\
@t=0; P=P_o\\
P_o=Ce^{kt}\\
P_o=Ce^{k\left(0\right)}\\
P_o = C

Substituting to Eq.1., we get

P=\:P_{o\:}e^{kt}\:\:\:\:\:\:\:(Eq.2)

Then from the given condition, from 12 noon to 2 p.m., the population triples (using Eq.2), we will solve for the value of k

@t= 2\:;\:P= 3P_o\\
P=\:P_{o\:}e^{kt}\\
3P_{o\:}=\:P_{o\:}e^{k\left(2\right)}\\
k=0.54931

We will then come up with the working equation (WE), this will help us solve the required problems

P_{\:}=\:P_{o\:}e^{\left(0.54931\right)t}

1.) what time should the population become 100 times

Using WE,

t=?\:\:;\:\:P=100P_o\\
P_{\:}=\:P_{o\:}e^{\left(0.54931\right)t}\\
100P_{o\:}=\:P_{o\:}e^{\left(0.54931\right)t}\\
t=8.38\: hrs.\\
t= 8:22:48\: p.m. \; or\:8:23\:p.m.

2.) at noon 

P=P_o

3.) percentage at 10 a.m. 

@10 a.m.\:\:;\:\:t=-2\\
P_{\:}=\:P_{o\:}e^{\left(0.549\right)\left(-2\right)}\\
P_{\:}=\:P_{o\:}\left(0.33333\right)\\
\%=\frac{P}{P_o}{(100)}=\frac{P_o\left(0.33333\right)}{P_o}{(100)}\\
\%=\:33.33\%

Elementary Differential Equations by Dela Fuente, Feliciano, and Uy Chapter 10 Problem 2 — Applications of Ordinary First-Ordered Differential Equations

Find the equation of the curve so drawn that every point on it is equidistant from the origin and the intersection of the x-axis with the normal to the curve at the point.

Solution:

Plot points on the curve, 

A(x_{1},y_{1})

We all know that a Slope of a Tangent corresponds to m, and its negative reciprocal is equal to the Slope of a Normal. Thus, we use Point-Slope Formula.

y-y_{1}=-\frac{1}{m}\left(x-x_{1}\right)

As the normal intersects the x-axis, y = 0

Substituting to the previous equation, we get

\begin{align*}
-y_{1}&=-\frac{1}{m}\left(x-x_{1}\right)\\
-my_{1}&=-1\left(x-x_{1}\right)\\
-my_{1}&=-x+x_{1}\\
x&=x_{1}+my_{1}
\end{align*}

By using distance formula, from the origin (0,0), to point (x1+y1) = from intersection to (x1+y1) 

\begin{align*}
\sqrt{\left(x_{1}-0\right)^2+\left(y_{1}-0\right)^2}&=\sqrt{\left(x_{1}+my_{1}-x_{1}\right)^2+\left(0-y_{1}\right)^2}\\
\sqrt{x_{1}^2+y_{1}^2}&=\sqrt{m^2y_{1}^2+y_{1}^2}\\
x_{1}^2+y_{1}^2&=m^2y_{1}^2+y_{1}^2\\
x_{1}^2&=m^2y_{1}^2\\
x_{1}&=m_{1}y_{1}\:\:\:\:;m=\frac{dy}{dx}\\
x_{1}&=\frac{dy}{dx}y_{1}
\end{align*}

Change x1 and y1 to x and y,

\begin{align*}
x&=y\frac{dy}{dx}\\
xdx&=ydy
\end{align*}

By integrating,

\begin{align*}

\int \:ydx&=\int \:xdy\\
\frac{y^2}{2}&=\frac{x^2}{2}+C\\
y^2&={x^2}+2C\\
\end{align*}

We get,

y^2-x^2-=2C
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