Calculus without Limits: Trigonometry the Theory The derivative - PowerPoint PPT Presentation

Calculus without Interpolation Limits C. K. Raju Outline Trigonometry The derivative ◮ The next idea is that in the process of linear Fundamental theorem of calculus interpolation Functions ◮ we naturally run into the derivative = difference Zeroism quotient = slope of a chord Conclusions Appendix: report ◮ We also run into this if we use the elementary of an experiment arithmetic rule of 3 The experiment ◮ or similar triangles. Results

��� ��� ��� � ��� � � ��� ��� ��� ��� � ��� ��� ��� Calculus without Linear interpolation: continued Limits C. K. Raju Outline ◮ From the graph for any x value Trigonometry The derivative Fundamental theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results Figure: Interpolating sine values graphically

��� ��� ��� ��� � � � ��� ��� ��� ��� � ��� ��� ��� Calculus without Linear interpolation: continued Limits C. K. Raju Outline ◮ From the graph for any x value Trigonometry ◮ we can read off the corresponding y value. The derivative Fundamental theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results Figure: Interpolating sine values graphically

Calculus without Method 2: Linear interpolation using similar Limits triangles C. K. Raju Outline Trigonometry The derivative Fundamental theorem of calculus ◮ The process of reading off from the graph may involve Functions errors. Zeroism Conclusions Appendix: report of an experiment The experiment Results

Calculus without Method 2: Linear interpolation using similar Limits triangles C. K. Raju Outline Trigonometry The derivative Fundamental theorem of calculus ◮ The process of reading off from the graph may involve Functions errors. Zeroism Conclusions ◮ We can instead work numerically. Appendix: report of an experiment The experiment Results

Calculus without Method 2: Linear interpolation using similar Limits triangles C. K. Raju Outline Trigonometry The derivative Fundamental theorem of calculus ◮ The process of reading off from the graph may involve Functions errors. Zeroism Conclusions ◮ We can instead work numerically. Appendix: report ◮ Method 2: To calculate sin( x ) for a given x . of an experiment The experiment Results

Calculus without Method 2: Linear interpolation using similar Limits triangles C. K. Raju Outline Trigonometry The derivative Fundamental theorem of calculus ◮ The process of reading off from the graph may involve Functions errors. Zeroism Conclusions ◮ We can instead work numerically. Appendix: report ◮ Method 2: To calculate sin( x ) for a given x . of an experiment The experiment ◮ In the table we first locate x 1 and x 2 between which x Results lies.

Calculus without Limits C. K. Raju ◮ To refer to the entries in the table, let us rewrite the Outline Trigonometry table as follows. The derivative Fundamental sin( x ) x theorem of calculus Functions x 1 0 0 y 1 Zeroism x 2 0.2617 0.2588 y 2 Conclusions 0.5235 0.5 x 3 y 3 Appendix: report x 4 0.7853 0.7071 y 4 of an experiment 1.0471 0.8660 x 5 y 5 The experiment x 6 1.3089 0.9659 y 6 Results 1.5707 1 x 7 y 7

��� � � ��� ��� � ��� ��� ��� ��� ��� � ��� ��� � � � � � � � � �� � � � � ��� Calculus without Slope Limits C. K. Raju ◮ If x lies between x 1 and x 2 , say. Outline Trigonometry The derivative Fundamental theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results Figure: Derivative and slope

��� � � ��� � � ��� ��� ��� ��� ��� ��� ��� ��� � � � � � � � � �� � � � � ��� Calculus without Slope Limits C. K. Raju ◮ If x lies between x 1 and x 2 , say. ◮ Let y 1 = sin( x 1 ), and y 2 = sin( x 2 ) Outline Trigonometry The derivative Fundamental theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results Figure: Derivative and slope

��� � ��� ��� � � ��� ��� ��� ��� ��� ��� ��� ��� � � � � � � � � �� � � � � � Calculus without Slope Limits C. K. Raju ◮ If x lies between x 1 and x 2 , say. ◮ Let y 1 = sin( x 1 ), and y 2 = sin( x 2 ) Outline ◮ Let Trigonometry The derivative ∆ y = y 2 − y 1 , ∆ x = x 2 − x 1 Fundamental theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results Figure: Derivative and slope

��� � ��� ��� � � ��� ��� ��� ��� ��� ��� ��� ��� � � � � � � � � �� � � � � � Calculus without Slope (continued) Limits C. K. Raju ◮ The quantity ∆ x = y 2 − y 1 ∆ y Outline = tan a Trigonometry x 2 − x 1 The derivative Fundamental theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results Figure: Derivative and slope

� � ��� ��� � � ��� ��� ��� ��� ��� ��� ��� ��� � � � � � � � � �� � � � � ��� Calculus without Slope (continued) Limits C. K. Raju ◮ The quantity ∆ x = y 2 − y 1 ∆ y Outline = tan a Trigonometry x 2 − x 1 The derivative ◮ is just the slope of the chord joining the point ( x 1 , y 1 ) Fundamental to ( x 2 , y 2 ). theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results Figure: Derivative and slope

� � �� � �� � � � � � � � � � � � � � � � � � � � � � � �� � � � � � ��� �� Calculus without Using slope for linear interpolation Limits C. K. Raju ◮ From the figure it is clear that Outline y − y 1 = tan a = slope Trigonometry x − x 1 The derivative Fundamental theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results Figure: Using the slope to interpolate

� � � � �� � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � ��� �� Calculus without Using slope for linear interpolation Limits C. K. Raju ◮ From the figure it is clear that Outline y − y 1 = tan a = slope Trigonometry x − x 1 The derivative ◮ or Fundamental y − y 1 = ( x − x 1 ) × slope theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results Figure: Using the slope to interpolate

� � � � � ��� �� � � �� � � � � � �� � � � � � � � � � � � � � � � � � � � Calculus without Using slope for linear interpolation (contd) Limits C. K. Raju ◮ Since x lies between x 1 and x 2 . Outline Trigonometry The derivative Fundamental theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results Figure: Using the slope to interpolate

� � � � �� � � � � � � � � � � � � � � � � � �� � � � � �� � � � � � ��� �� Calculus without Using slope for linear interpolation (contd) Limits C. K. Raju ◮ Since x lies between x 1 and x 2 . ◮ we already know the slope = ∆ y Outline ∆ x , so Trigonometry The derivative Fundamental theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results Figure: Using the slope to interpolate

� �� � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � ��� �� Calculus without Using slope for linear interpolation (contd) Limits C. K. Raju ◮ Since x lies between x 1 and x 2 . ◮ we already know the slope = ∆ y Outline ∆ x , so Trigonometry ◮ The derivative y − y 1 = ( x − x 1 ) × ∆ y Fundamental ∆ x theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results Figure: Using the slope to interpolate

Calculus without Using slope for linear interpolation Limits C. K. Raju Outline Trigonometry The derivative ◮ From Fundamental theorem of calculus y − y 1 = ( x − x 1 ) × ∆ y Functions ∆ x Zeroism Conclusions Appendix: report of an experiment The experiment Results

Calculus without Using slope for linear interpolation Limits C. K. Raju Outline Trigonometry The derivative ◮ From Fundamental theorem of calculus y − y 1 = ( x − x 1 ) × ∆ y Functions ∆ x Zeroism ◮ we can immediately calculate the desired y value: Conclusions Appendix: report y = y 1 + ( x − x 1 ) × ∆ y of an experiment ∆ x The experiment Results

Calculus without Method 3: Linear interpolation by rule of 3 Limits C. K. Raju Outline Trigonometry The derivative ◮ x lies between x 1 and x 2 . Fundamental theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results

Calculus without Method 3: Linear interpolation by rule of 3 Limits C. K. Raju Outline Trigonometry The derivative ◮ x lies between x 1 and x 2 . Fundamental theorem of calculus ◮ Let y 1 = sin( x 1 ), and y 2 = sin( x 2 ) Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results

Calculus without Method 3: Linear interpolation by rule of 3 Limits C. K. Raju Outline Trigonometry The derivative ◮ x lies between x 1 and x 2 . Fundamental theorem of calculus ◮ Let y 1 = sin( x 1 ), and y 2 = sin( x 2 ) Functions ◮ Change in sine value = y 2 − y 1 = ∆ y Zeroism Conclusions Appendix: report of an experiment The experiment Results

Calculus without Method 3: Linear interpolation by rule of 3 Limits C. K. Raju Outline Trigonometry The derivative ◮ x lies between x 1 and x 2 . Fundamental theorem of calculus ◮ Let y 1 = sin( x 1 ), and y 2 = sin( x 2 ) Functions ◮ Change in sine value = y 2 − y 1 = ∆ y Zeroism ◮ This change takes place over a distance x 2 − x 1 = ∆ x . Conclusions Appendix: report of an experiment The experiment Results

Calculus without Method 3: Linear interpolation by rule of 3 Limits C. K. Raju Outline Trigonometry The derivative ◮ x lies between x 1 and x 2 . Fundamental theorem of calculus ◮ Let y 1 = sin( x 1 ), and y 2 = sin( x 2 ) Functions ◮ Change in sine value = y 2 − y 1 = ∆ y Zeroism ◮ This change takes place over a distance x 2 − x 1 = ∆ x . Conclusions Appendix: report ◮ ∴ of an experiment = ∆ y unit rate of change = y 2 − y 1 The experiment ∆ x x 2 − x 1 Results

Calculus without Linear interpolation by rule of 3 Limits C. K. Raju Outline Trigonometry The derivative ◮ unit rate of change = y 2 − y 1 x 2 − x 1 = ∆ y Fundamental ∆ x theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results

Calculus without Linear interpolation by rule of 3 Limits C. K. Raju Outline Trigonometry The derivative ◮ unit rate of change = y 2 − y 1 x 2 − x 1 = ∆ y Fundamental ∆ x theorem of calculus ◮ ∴ change y − y 1 over the distance x − x 1 , is Functions Zeroism y − y 1 = ∆ y Conclusions ∆ x ( x − x 1 ) Appendix: report of an experiment change = unit rate of change × distance The experiment Results

Example: calculating sin 1 ◦ Calculus without Limits C. K. Raju Outline ◮ 1 ◦ = 180 = 0 . 01745 radians. π Trigonometry The derivative Fundamental theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results

Example: calculating sin 1 ◦ Calculus without Limits C. K. Raju Outline ◮ 1 ◦ = 180 = 0 . 01745 radians. π Trigonometry ◮ We can calculate sin 1 ◦ as follows The derivative Fundamental sin 0 . 01745 = sin 0+ sin 15 ◦ − sin 0 ◦ theorem of calculus 15 × 0 . 01745 − 0 × (1 × 0 . 01745 − 0) Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results

Example: calculating sin 1 ◦ Calculus without Limits C. K. Raju Outline ◮ 1 ◦ = 180 = 0 . 01745 radians. π Trigonometry ◮ We can calculate sin 1 ◦ as follows The derivative Fundamental sin 0 . 01745 = sin 0+ sin 15 ◦ − sin 0 ◦ theorem of calculus 15 × 0 . 01745 − 0 × (1 × 0 . 01745 − 0) Functions Zeroism ◮ From the table, we read off: Conclusions 15 ◦ = 15 × 0 . 01745 = 0 . 2617 radians, and Appendix: report of an experiment sin 15 ◦ = 0 . 2588. The experiment Results

Example: calculating sin 1 ◦ Calculus without Limits C. K. Raju Outline ◮ 1 ◦ = 180 = 0 . 01745 radians. π Trigonometry ◮ We can calculate sin 1 ◦ as follows The derivative Fundamental sin 0 . 01745 = sin 0+ sin 15 ◦ − sin 0 ◦ theorem of calculus 15 × 0 . 01745 − 0 × (1 × 0 . 01745 − 0) Functions Zeroism ◮ From the table, we read off: Conclusions 15 ◦ = 15 × 0 . 01745 = 0 . 2617 radians, and Appendix: report of an experiment sin 15 ◦ = 0 . 2588. The experiment ◮ Hence, Results sin 1 ◦ = sin 0 . 01745 = 0 . 2588 0 . 2617 × 0 . 01745 = 0 . 01725

Example: calculating sin 1 ◦ Calculus without Limits C. K. Raju Outline ◮ 1 ◦ = 180 = 0 . 01745 radians. π Trigonometry ◮ We can calculate sin 1 ◦ as follows The derivative Fundamental sin 0 . 01745 = sin 0+ sin 15 ◦ − sin 0 ◦ theorem of calculus 15 × 0 . 01745 − 0 × (1 × 0 . 01745 − 0) Functions Zeroism ◮ From the table, we read off: Conclusions 15 ◦ = 15 × 0 . 01745 = 0 . 2617 radians, and Appendix: report of an experiment sin 15 ◦ = 0 . 2588. The experiment ◮ Hence, Results sin 1 ◦ = sin 0 . 01745 = 0 . 2588 0 . 2617 × 0 . 01745 = 0 . 01725 ◮ We can compare this with the value of sin 1 ◦ from a calculator, which comes out to be 0.01745.

Calculus without More accurate sine values Limits C. K. Raju Outline Trigonometry ◮ We have approximated a curved line by a straight line. The derivative Fundamental theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results

Calculus without More accurate sine values Limits C. K. Raju Outline Trigonometry ◮ We have approximated a curved line by a straight line. The derivative Fundamental ◮ Smaller parts of a curved line are better approximated theorem of calculus by a straight line. Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results

Calculus without More accurate sine values Limits C. K. Raju Outline Trigonometry ◮ We have approximated a curved line by a straight line. The derivative Fundamental ◮ Smaller parts of a curved line are better approximated theorem of calculus by a straight line. Functions Zeroism ◮ So, to get more accurate sine values, we must have Conclusions more values in our table. Appendix: report of an experiment The experiment Results

Calculus without More accurate sine values Limits C. K. Raju Outline Trigonometry ◮ We have approximated a curved line by a straight line. The derivative Fundamental ◮ Smaller parts of a curved line are better approximated theorem of calculus by a straight line. Functions Zeroism ◮ So, to get more accurate sine values, we must have Conclusions more values in our table. Appendix: report ◮ The earth is round, but because we see only a small of an experiment The experiment part of it, it appears flat. Results

Calculus without More accurate sine values Limits C. K. Raju Outline Trigonometry ◮ We have approximated a curved line by a straight line. The derivative Fundamental ◮ Smaller parts of a curved line are better approximated theorem of calculus by a straight line. Functions Zeroism ◮ So, to get more accurate sine values, we must have Conclusions more values in our table. Appendix: report ◮ The earth is round, but because we see only a small of an experiment The experiment part of it, it appears flat. Results ◮ 8th c. mathematician Lalla: “Mathematicians say 1 100 th part of the earth is flat.”

Calculus without Difference quotient vs derivative Limits C. K. Raju Outline Trigonometry ◮ The quantity ∆ y The derivative ∆ x = slope = unit rate of change is Fundamental usually called the difference quotient. theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results

Calculus without Difference quotient vs derivative Limits C. K. Raju Outline Trigonometry ◮ The quantity ∆ y The derivative ∆ x = slope = unit rate of change is Fundamental usually called the difference quotient. theorem of calculus ◮ and is distinguished from the derivative dy Functions dx which Zeroism involves limits. Conclusions Appendix: report of an experiment The experiment Results

Calculus without Difference quotient vs derivative Limits C. K. Raju Outline Trigonometry ◮ The quantity ∆ y The derivative ∆ x = slope = unit rate of change is Fundamental usually called the difference quotient. theorem of calculus ◮ and is distinguished from the derivative dy Functions dx which Zeroism involves limits. Conclusions ◮ If we take an infinite number of values in our table, Appendix: report which are all only an infinitesimal distance apart the of an experiment difference quotient will agree with the derivative. The experiment Results

Calculus without Difference quotient vs derivative Limits C. K. Raju Outline Trigonometry ◮ The quantity ∆ y The derivative ∆ x = slope = unit rate of change is Fundamental usually called the difference quotient. theorem of calculus ◮ and is distinguished from the derivative dy Functions dx which Zeroism involves limits. Conclusions ◮ If we take an infinite number of values in our table, Appendix: report which are all only an infinitesimal distance apart the of an experiment difference quotient will agree with the derivative. The experiment Results ◮ However, there is no way to build a table with an infinite number of entries.

Calculus without Difference quotient vs derivative Limits C. K. Raju Outline ◮ Therefore, in calculus without limits we will treat the Trigonometry difference quotient ∆ y ∆ x as the same as the derivative dy The derivative dx Fundamental theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results

Calculus without Difference quotient vs derivative Limits C. K. Raju Outline ◮ Therefore, in calculus without limits we will treat the Trigonometry difference quotient ∆ y ∆ x as the same as the derivative dy The derivative dx ◮ The understanding is that, like the number π , we may Fundamental theorem of calculus not be able to write down an exact value for dy dx . Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results

Calculus without Difference quotient vs derivative Limits C. K. Raju Outline ◮ Therefore, in calculus without limits we will treat the Trigonometry difference quotient ∆ y ∆ x as the same as the derivative dy The derivative dx ◮ The understanding is that, like the number π , we may Fundamental theorem of calculus not be able to write down an exact value for dy dx . Functions ◮ We use different values for π , such as 22 355 7 , 113 3.1415 Zeroism etc. depending upon the exact accuracy we want. Conclusions Appendix: report of an experiment The experiment Results

Calculus without Difference quotient vs derivative Limits C. K. Raju Outline ◮ Therefore, in calculus without limits we will treat the Trigonometry difference quotient ∆ y ∆ x as the same as the derivative dy The derivative dx ◮ The understanding is that, like the number π , we may Fundamental theorem of calculus not be able to write down an exact value for dy dx . Functions ◮ We use different values for π , such as 22 355 7 , 113 3.1415 Zeroism etc. depending upon the exact accuracy we want. Conclusions ◮ Likewise, for dy Appendix: report dx we have a value to a certain accuracy of an experiment as estimated by ∆ y ∆ x . The experiment Results

Calculus without Difference quotient vs derivative Limits C. K. Raju Outline ◮ Therefore, in calculus without limits we will treat the Trigonometry difference quotient ∆ y ∆ x as the same as the derivative dy The derivative dx ◮ The understanding is that, like the number π , we may Fundamental theorem of calculus not be able to write down an exact value for dy dx . Functions ◮ We use different values for π , such as 22 355 7 , 113 3.1415 Zeroism etc. depending upon the exact accuracy we want. Conclusions ◮ Likewise, for dy Appendix: report dx we have a value to a certain accuracy of an experiment as estimated by ∆ y ∆ x . The experiment Results ◮ A more accurate value of dy dx can usually be obtained by taking the points x 1 and x 2 closer to each other.

Calculus without Difference quotient vs derivative Limits C. K. Raju Outline ◮ Therefore, in calculus without limits we will treat the Trigonometry difference quotient ∆ y ∆ x as the same as the derivative dy The derivative dx ◮ The understanding is that, like the number π , we may Fundamental theorem of calculus not be able to write down an exact value for dy dx . Functions ◮ We use different values for π , such as 22 355 7 , 113 3.1415 Zeroism etc. depending upon the exact accuracy we want. Conclusions ◮ Likewise, for dy Appendix: report dx we have a value to a certain accuracy of an experiment as estimated by ∆ y ∆ x . The experiment Results ◮ A more accurate value of dy dx can usually be obtained by taking the points x 1 and x 2 closer to each other. ◮ (However, a more accurate interpolation is usually obtained by taking higher derivatives.)

Calculus without Summing successive differences Limits C. K. Raju Outline Trigonometry The derivative Fundamental ◮ If successive differences are summed, theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results

Calculus without Summing successive differences Limits C. K. Raju Outline Trigonometry The derivative Fundamental ◮ If successive differences are summed, theorem of calculus Functions ◮ the result is the difference between the first and last Zeroism value. . Conclusions Appendix: report of an experiment The experiment Results

Calculus without Summing successive differences Limits C. K. Raju Outline Trigonometry The derivative Fundamental ◮ If successive differences are summed, theorem of calculus Functions ◮ the result is the difference between the first and last Zeroism value. . Conclusions ◮ ( y 3 − y 2 ) + ( y 2 − y 1 ) = y 3 − y 1 , or Appendix: report of an experiment The experiment Results

Calculus without Summing successive differences Limits C. K. Raju Outline Trigonometry The derivative Fundamental ◮ If successive differences are summed, theorem of calculus Functions ◮ the result is the difference between the first and last Zeroism value. . Conclusions ◮ ( y 3 − y 2 ) + ( y 2 − y 1 ) = y 3 − y 1 , or Appendix: report of an experiment ◮ Similarly, ( y 4 − y 3 ) + ( y 3 − y 2 ) + ( y 2 − y 1 ) = y 4 − y 1 , The experiment and so on Results

Calculus without The “fundamental theorem of calculus” Limits C. K. Raju Outline Trigonometry ◮ This process extends to any number of terms. The derivative Fundamental theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results

Calculus without The “fundamental theorem of calculus” Limits C. K. Raju Outline Trigonometry ◮ This process extends to any number of terms. The derivative Fundamental ◮ ( y n − y n − 1 ) + ( y n − 1 − y n − 2 ) + . . . + ( y 2 − y 1 ) = y n − y 1 . theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results

Calculus without The “fundamental theorem of calculus” Limits C. K. Raju Outline Trigonometry ◮ This process extends to any number of terms. The derivative Fundamental ◮ ( y n − y n − 1 ) + ( y n − 1 − y n − 2 ) + . . . + ( y 2 − y 1 ) = y n − y 1 . theorem of calculus ◮ This is written as Functions Zeroism n Conclusions � ∆ y i = y n − y 1 Appendix: report of an experiment i =1 The experiment Results

Calculus without The “fundamental theorem of calculus” Limits C. K. Raju Outline Trigonometry ◮ This process extends to any number of terms. The derivative Fundamental ◮ ( y n − y n − 1 ) + ( y n − 1 − y n − 2 ) + . . . + ( y 2 − y 1 ) = y n − y 1 . theorem of calculus ◮ This is written as Functions Zeroism n Conclusions � ∆ y i = y n − y 1 Appendix: report of an experiment i =1 The experiment ◮ Here Σ is a Greek letter used for the “S” of “Sum” Results (just as the Greek letter ∆ was used for the “D” of “Difference”).

Calculus without The “fundamental theorem of calculus” Limits C. K. Raju Outline Trigonometry ◮ Historically, instead of Σ, some people used an The derivative � elongated S, like this . Fundamental theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results

Calculus without The “fundamental theorem of calculus” Limits C. K. Raju Outline Trigonometry ◮ Historically, instead of Σ, some people used an The derivative � elongated S, like this . Fundamental theorem of calculus ◮ This has now come to be known as the integral sign. Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results

Calculus without The “fundamental theorem of calculus” Limits C. K. Raju Outline Trigonometry ◮ Historically, instead of Σ, some people used an The derivative � elongated S, like this . Fundamental theorem of calculus ◮ This has now come to be known as the integral sign. Functions ◮ If we use just “d” for “difference” we can rewrite the Zeroism above Conclusions � x n Appendix: report dy ( x ) = y ( x n ) − y ( x 1 ) of an experiment x 1 The experiment Results

Calculus without The “fundamental theorem of calculus” Limits C. K. Raju Outline Trigonometry ◮ Historically, instead of Σ, some people used an The derivative � elongated S, like this . Fundamental theorem of calculus ◮ This has now come to be known as the integral sign. Functions ◮ If we use just “d” for “difference” we can rewrite the Zeroism above Conclusions � x n Appendix: report dy ( x ) = y ( x n ) − y ( x 1 ) of an experiment x 1 The experiment ◮ a statement often called the “fundamental theorem of Results calculus”,

Calculus without The “fundamental theorem of calculus” Limits C. K. Raju Outline Trigonometry ◮ Historically, instead of Σ, some people used an The derivative � elongated S, like this . Fundamental theorem of calculus ◮ This has now come to be known as the integral sign. Functions ◮ If we use just “d” for “difference” we can rewrite the Zeroism above Conclusions � x n Appendix: report dy ( x ) = y ( x n ) − y ( x 1 ) of an experiment x 1 The experiment ◮ a statement often called the “fundamental theorem of Results calculus”, ◮ that summation ( � ) is the inverse of the difference ( d ).

Calculus without Fundamental theorem of calculus Limits C. K. Raju Outline Trigonometry ◮ This leads naturally to the problem of numerical The derivative Fundamental solution of ODE. theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results

Calculus without Fundamental theorem of calculus Limits C. K. Raju Outline Trigonometry ◮ This leads naturally to the problem of numerical The derivative Fundamental solution of ODE. theorem of calculus ◮ Given the initial value y (0), and the value of the Functions derivative/difference quotient at any point y ′ ( x ) = f ( x ) Zeroism Conclusions Appendix: report of an experiment The experiment Results

Calculus without Fundamental theorem of calculus Limits C. K. Raju Outline Trigonometry ◮ This leads naturally to the problem of numerical The derivative Fundamental solution of ODE. theorem of calculus ◮ Given the initial value y (0), and the value of the Functions derivative/difference quotient at any point y ′ ( x ) = f ( x ) Zeroism Conclusions ◮ How to determine the value of the function? Appendix: report of an experiment The experiment Results

Calculus without Fundamental theorem of calculus Limits C. K. Raju Outline Trigonometry ◮ This leads naturally to the problem of numerical The derivative Fundamental solution of ODE. theorem of calculus ◮ Given the initial value y (0), and the value of the Functions derivative/difference quotient at any point y ′ ( x ) = f ( x ) Zeroism Conclusions ◮ How to determine the value of the function? Appendix: report ◮ If we are dealing with finite differences, it is very easy to of an experiment The experiment calculate the answer by summing successive differences. Results

Calculus without Fundamental theorem of calculus Limits C. K. Raju Outline Trigonometry ◮ This leads naturally to the problem of numerical The derivative Fundamental solution of ODE. theorem of calculus ◮ Given the initial value y (0), and the value of the Functions derivative/difference quotient at any point y ′ ( x ) = f ( x ) Zeroism Conclusions ◮ How to determine the value of the function? Appendix: report ◮ If we are dealing with finite differences, it is very easy to of an experiment The experiment calculate the answer by summing successive differences. Results ◮ In the implicit case, y ′ ( x ) = f ( x , y ) we use the elementary technique today called an “Euler” solver.

Calculus without Fundamental theorem of calculus Limits C. K. Raju Outline ◮ To look at matters in way. Trigonometry The derivative Fundamental theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results

Calculus without Fundamental theorem of calculus Limits C. K. Raju Outline ◮ To look at matters in way. Trigonometry The derivative ◮ if we want to teach calculus for purposes of calculations Fundamental in physics and engineering theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results

Calculus without Fundamental theorem of calculus Limits C. K. Raju Outline ◮ To look at matters in way. Trigonometry The derivative ◮ if we want to teach calculus for purposes of calculations Fundamental in physics and engineering theorem of calculus Functions ◮ students should learn how to apply Newton’s laws of Zeroism motion. Conclusions Appendix: report of an experiment The experiment Results

Calculus without Fundamental theorem of calculus Limits C. K. Raju Outline ◮ To look at matters in way. Trigonometry The derivative ◮ if we want to teach calculus for purposes of calculations Fundamental in physics and engineering theorem of calculus Functions ◮ students should learn how to apply Newton’s laws of Zeroism motion. Conclusions ◮ This requires the ability to calculate the solution of Appendix: report of an experiment ordinary differential equations. The experiment Results

Calculus without Fundamental theorem of calculus Limits C. K. Raju Outline ◮ To look at matters in way. Trigonometry The derivative ◮ if we want to teach calculus for purposes of calculations Fundamental in physics and engineering theorem of calculus Functions ◮ students should learn how to apply Newton’s laws of Zeroism motion. Conclusions ◮ This requires the ability to calculate the solution of Appendix: report of an experiment ordinary differential equations. The experiment ◮ Note: the operative term is calculate not prove. Results

Calculus without Fundamental theorem of calculus Limits C. K. Raju Outline ◮ To look at matters in way. Trigonometry The derivative ◮ if we want to teach calculus for purposes of calculations Fundamental in physics and engineering theorem of calculus Functions ◮ students should learn how to apply Newton’s laws of Zeroism motion. Conclusions ◮ This requires the ability to calculate the solution of Appendix: report of an experiment ordinary differential equations. The experiment ◮ Note: the operative term is calculate not prove. Results ◮ If the object to send a rocket to the moon, what is required is the ability to calculate the solution, and not prove its existence and uniqueness.

Calculus without Fundamental theory of calculus Limits contd C. K. Raju Outline Trigonometry ◮ An Euler solver simply uses the above interpolation The derivative procedure to extrapolate. Fundamental theorem of calculus Functions Zeroism Conclusions Appendix: report of an experiment The experiment Results

Calculus without Fundamental theory of calculus Limits contd C. K. Raju Outline Trigonometry ◮ An Euler solver simply uses the above interpolation The derivative procedure to extrapolate. Fundamental theorem of calculus ◮ This is a good way to solve ODEs, though higher Functions precision can be obtained by taking higher derivatives Zeroism (i.e., higher-order difference quotients), as in my Conclusions package calcode (which also visualises the solution). Appendix: report of an experiment The experiment Results

Calculus without Fundamental theory of calculus Limits contd C. K. Raju Outline Trigonometry ◮ An Euler solver simply uses the above interpolation The derivative procedure to extrapolate. Fundamental theorem of calculus ◮ This is a good way to solve ODEs, though higher Functions precision can be obtained by taking higher derivatives Zeroism (i.e., higher-order difference quotients), as in my Conclusions package calcode (which also visualises the solution). Appendix: report of an experiment ◮ Numerical solution of an ODE is a superior substitute The experiment for the fundamental theorem of calculus. Results

Calculus without Fundamental theory of calculus Limits contd C. K. Raju Outline Trigonometry ◮ An Euler solver simply uses the above interpolation The derivative procedure to extrapolate. Fundamental theorem of calculus ◮ This is a good way to solve ODEs, though higher Functions precision can be obtained by taking higher derivatives Zeroism (i.e., higher-order difference quotients), as in my Conclusions package calcode (which also visualises the solution). Appendix: report of an experiment ◮ Numerical solution of an ODE is a superior substitute The experiment for the fundamental theorem of calculus. Results ◮ From a practical point of view there is no doubt that this is a better approach, and it also allows the solution (and visualisation) of a wide variety of problems.