Advanced Calculus - Math - 12th Grade by Slidesgo.pptx

Calculo matematico

1. Emily Torres Calculo Integral Biomedicina

2. Contenido de los Temas Integrales Inmediatas directas Integrales Compuestas (Cambio de variable) Integrales Racionales (División de polinomios) Integrales tipo Arcotangente Integrales por partes Integrales por cambio de variable complicado SECCION 1 SECCION 2 SECCION 3 SECCION 4 SECCION 5 SECCION 6

3. Introducción al cálculo integral ¿Qué es? Es una rama fundamental de las matemáticas ¿Que aplicación tiene? Diversos campos, como la física, la ingeniería, la economía y la biología, entre otros. ¿Qué estudia? L a acumulación y la variación de las cantidades a lo largo de un intervalo. Aplicación en Biomedicina Modelar sistemas biomédicos Analizar imágenes biomédicas Fenómenos fisiológicos

4. PROBLEMAS Y RESPUESTAS EJERCICIOS Instrucciones : Resolver las siguientes integrales Integral Inmediatas directas Compuestas Racionales Problema ∫ (2x^3 - 5x + 1) dx ∫ (x^2 + 1)^3 dx ∫ (x^2 + 3x + 2) / (x + 1) dx Desarrollo = (2x^4 / 4) - (5x^2 / 2) + x + C = (1/2) ∫ u^3 du = (1/2) (u^4 / 4) + C = (1/8) u^4 + C =x – 1 x^2 – x =x^3 - 2x^2 + 5x – 3 x^3 - x Solución = x^4 - 2.5x^2 + x + C = (1/8) (x^2 + 1)^4 + C ∫ (x^2 + 3x + 2) / (x + 1) dx = x + 2 + C

5. PROBLEMAS Y RESPUESTAS EJERCICIOS Instrucciones : Resolver las siguientes integrales Integral Tipo arco tangente Por partes Por cambio de variable Problema ∫ (1 / (1 + x^2)) dx ∫ x * e^x dx Desarrollo Solución = arctan (x) + C

6. Es importante destacar que el cálculo integral es una disciplina en constante evolución, con nuevas aplicaciones y desarrollos surgiendo continuamente Conclusiones y consideraciones finales El Cálculo Integral tiene un papel crucial en el desarrollo de nuevas tecnologías en diversos campos. En resumen, el cálculo integral es una herramienta poderosa y versátil que tiene numerosas aplicaciones en diversos campos.

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8. Mercury is the closest planet to the Sun and the smallest in the entire Solar System Venus has a beautiful name and is the second planet from the Sun. It’s terribly hot Earth is the third planet from the Sun and the only one that harbors life in the Solar System The concept of functions Linear function Quadratic function Exponential function

9. Limits and continuity Venus has a beautiful name and is the second from the Sun Earth is the only planet that harbors life in the Solar System Despite being red, Mars is actually a cold place Jupiter is a gas giant and the biggest planet of them all Removable Jump discontinuity Infinite Discussive

10. ∫ (2x^3 - 5x + 1) dx SOLUCION ∫ (2x^3 - 5x + 1) dx = (2x^4 / 4) - (5x^2 / 2) + x + C = x^4 - 2.5x^2 + x + C Venus is the second planet from the Sun Despite being red, Mars is actually a cold place Jupiter is the biggest planet of them all Earth is the only planet known to harbor life Ejemplos resueltos 1. Integrales Inmediatas Directas: Product Quotient Sum

11. Mercury is the smallest planet of them all Venus is the second planet from the Sun Despite being red, Mars is actually a cold place Jupiter is the biggest planet of them all Earth is the only planet known to harbor life Saturn is a gas giant and has several rings Applications of derivatives A rate of change The tangent lines Optimization problems Newton’s method Many related rates Curve sketching

12. Awesome words

13. Images reveal large amounts of data, so remember: use an image instead of a long text. Your audience will appreciate it A picture reinforces the concept

14. 4,498,300 Big numbers catch your audience’s attention

15. You can replace the image on the screen with your own work. Just right-click on it and select “Replace image” Desktop mockup

16. You can replace the image on the screen with your own work. Just right-click on it and select “Replace image” Tablet mockup

17. You can replace the image on the screen with your own work. Just right-click on it and select “Replace image” Phone mockup

18. Calculus competitions Competition C Pluto is now classified as a dwarf planet Competition A The Sun is the closest star to our planet Competition B The Moon is Earth’s only natural satellite

19. Key concepts in calculus Concepts Definition Application Derivative Rate of change of a function Finding instant rates of change Integral Accumulation of a function Finding areas under total curves Limit Behavior of a function as x approaches a value Evaluation function behavior with continuity Fundamental theorem of calculus Relates differentiation and integration Evaluating definite integrals and antiderivatives

20. You can use this graph Finding x Mercury is the closest planet to the Sun About the graph Venus has a beautiful name and is the second planet from the Sun. It’s terribly hot, even hotter than Mercury, and its atmosphere is extremely poisonous Follow the link in the graph to modify its data and then paste the new one here. For more info, click here

21. Exercises You can do a brief review of the topics covered before proceeding to the exercises. It’s best to be prepared, as it will maximize the chances of understanding things right

22. Stepping it up Exercise ∫ √xdz Solution: ∫ √xdz = ∫x 4 dx = ⅔x ⅔ + C = ⅔x √x + C Instructions: analyze the following and compare your results with the solutions 1) ∫ 3e 2 dx Solution: ∫ 3e2dx = 3 ∫e 2 dx = 3e 2 + C 2) ∫ (3x2 - √5x + 2)dx Solution: = x 3 - ⅔x √5x * x * ⅔x + 2x + C 3) ∫ ½x - 2/x 2 + 3/√x Solution: = ½ln |x| -2 * (-1)x -1 + 3 * 2x ⅔ + C 4)

23. The last exercise for you Exercise Instructions: solve the following functions Exercise Solution Solve the equation: 3x^2 - 5x + 2 = 0. x = 1 and x = \frac{2}{3} Factor the expression: x^2 + 7x + 10. (x + 5)(x + 2) Find the vertex of the quadratic function: f(x) = 2x^2 - 4x - 3. Vertex: (1, -7) Calculate the value of f(2) for f(x) = 3x^2 + 2x - 1. f(2) = 15 Graph the quadratic function: g(x) = -x^2 + 4x - 5. -------------------