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Contents
Contents
Temporalización: 11 horas
Rate of change (tasa de cambio)
Derivative rules (reglas de derivación).
Applications of differentiation (aplicaciones de las derivadas).
Introductory activity
Introductory activity
The five minutes test.
Introduction
Introduction
Let's watch the following videos:
Main concepts of calculus
Main concepts of calculus
Write down the ones we haven't seen yet.
Video based on Bohemian Rhapsody, by Queen.
Derivative as a concept
Derivative as a concept
Introducción a las derivadas
Introducción a las derivadas
Este vídeo contiene un error básico. ¿Puedes encontrarlo?
Main concepts of calculus (in order of appearance)
Main concepts of calculus (in order of appearance)
Continuous, limit, one-side limits (implicit), function, asymptotes, graph, differentiable, calculus, derivatives, integrals (also integrate, integration), area, curves, y', derivative of y, dy/dx, product rules, chain rule, area, exponent, reciprocal
Average Rate of Change
Average Rate of Change
En español: Tasa de Variación Media. Check how it works in desmos.
You can move the orange points by clicking on this link, and see how the rate changes.
Formal definition of derivative
Formal definition of derivative
Given f(x), a real function, then
f'(a) = the slope of the tangent line to the function f at x = a.
This is the graph of x²
We have its formula, x², and want to find out the slope of the tangent to the curve at x=1
That is to find out the slope of the tangent line at point A
In other words, I want to find the slope of that red line.
To find it out precisely we apply the formal definition.
Notice that x → 1 because we want to find f'(1)
We solve this limit as usual (see Unit 10). The result is 2, as it was expected.
Instead of x=1 we can consider x=a, so we have a general expression for f'(a)=2a
That expresion, written usually as f'(x)=2x is called the derivative function of f(x)=x²
Activities
Activities
Derivative rules
Derivative rules
It would be tedious to calculate derivatives using limits every time we need them, so they have prepared a table with the derivative functions of common functions (see Unit 10) and some general rules that can be easily proved by using limits.
Derivative of common functions, by Math is Fun.
Derivative rules, by Math is fun.
Find a more complete table at symbolab.
Homework Ex. 2 to 8 from page 310
Chain rule ❤️❤️ [f∘g(x)]’ = f[g(x)]’ = f’[g(x)]·g’(x) ❤️❤️
Chain rule ❤️❤️ [f∘g(x)]’ = f[g(x)]’ = f’[g(x)]·g’(x) ❤️❤️
Regla de la cadena [f∘g(x)]’ = f’(g(x))·g’(x)
Remember: (x+1)³ = 1·x³·1⁰ + 3·x²·1¹ + 3·x¹·1² + 1·x⁰·1³ = x³+3x²+3x+1
So, if we want to derive (x+1)³, we can derive x³+3x²+3x+1, what give us 3x²+6x+3
What if we want to derive (4x-5)⁷? It's hard to expand. But it's also the composition of two functions: g(x) = 4x-5 and f(x) = x⁷.
[f∘g(x)] = f(4x-5) = (4x-5)⁷ ⇒ D[(4x-5)⁷] = 7(4x-5)⁶·4 = 28(4x-5)⁶
Other example: f(x)=(3x² -7x + 9)⁵ ⇒ f'(x) = 5(3x²-7x+9)⁴·(3·2x-7) = 5(3x²-7x+9)⁴·(6x-7). We can leave it that way, there's no need to expand it by now.
More examples:
f(x) = sin(3x) ⇒ f'(x) = 3·cos (3x)
← To the left, the chain rule explained by NancyPi, with some examples. She uses a change of variable.
These exercises about chain rule comes from Eddie Woo worksheets.
More exercises: 9, 10, page 326.
applications of derivatives
applications of derivatives
Tangent line at x = a
Tangent line at x = a
Use this form: y - f(a) = f'(a)·( x - a)
Find the equation of the tangent line to the function f(x) at the given point.
f(x)=x²−3x+5 at x=2
f(x)=1/x at x=1
f(x)=sqrt(x+4) at x=0
f(x)=x³+2x at x=−1
f(x)=sin(x) at x=π/4
f(x)=(x+1)/(x−1) at x=2
f(x)=e^x⋅cos(x) at x=0
Us desmos to check if the line founded is tangent to the curve at the given point.
L'Hôpital's rule
L'Hôpital's rule
Let's apply that rule to the following limit. Before knowing L'Hôpital's rule, it was impossible for us.
Be careful! Not apply the quotient rule of derivatives. It's just a derivative over another derivative.
"Special" points of a function
"Special" points of a function
We call "special" points to those with horizontal tangent, that is, the solutions of f'(x)=0.
They can be maxima, minima or inflection points.
There are other "special" points, such as, intersection with the axis, but those have nothing to do with derivatives.
Drawing or sketching functions
Drawing or sketching functions
Follow this steps:
Infinite branches: limits as x→∞
Infinite branches: vertical asymptotes:
Polynomials don't have them;
Rational functions: candidates x=a, being a a zero of the denominator.
Special points: those with horizontal tangent, solutions of f'(x)=0. They can be maxima, minima or inflection points.
Intersection of the function with axis:
OX: solutions of f(x)=0
OY: f(0)
Get all pieces of information together.
Example: let's draw y=x³-3x²+4
1st: infinite branches. The limit as x→-∞ is -∞, so "it comes from down below"
the limit as x→+∞ is +∞, so "it goes up above"
2nd: asymptotes.
It has not, it's a polynomial.
3rd: special points.
Those in which f'(x)=0
They are 0 and 1. There, after calculating f(0) and f(1), we know there is an horizontal tangent line.
4th: joint the information together.
Sometimes it's good to know the points where the curve touches the axis.
Practise; draw the following functions:
y = x³ - 3x + 2
y = x³ - 9x² + 24x - 20
y = x⁵ - 5x³ - 1
y = x/(1-x²)
y = x⁴ - 8x² + 2
y = x²/(x²+x-2)
y = x²/(1-x²)
y=(x-4)/(2x-8)
y=(x-4)/(x²-2x-8)
y=(2x³-4x²-6x)/(x²-2x-3)
Tangent line
Tangent line
Be it f, the function given by f(x) = ln x². Find out the tangent line to its graph when x=1.
This is the graph of the function. f(1) = 0, so the tangencial point is A(1,0)
We have the point, we need something more (i.e. the slope of the line). See unit 8.
According to the definition of the derivative, m = f'(x1)
Then, x=1 ⇒ y' = 2
Applying the point-slope equation, y - 0 = 2(x - 1), what gives us y = 2x -2 as the solution of our problem.
Graphic solution (not necessary).
Increasing and decreasing intervals of a function
Increasing and decreasing intervals of a function
Since f'(a) is the slope (m) of the tangent line at x=a, and the sign of m indicates if that line is increasing or decreasing, the derivative of a function at a allows you to tell if the function is increasing or decreasing at a. To remember this concepts, have a look a this section.
Optimization problems
Optimization problems
An eccentric millionaire offers you the following game:
"I will give you €12 for every cm³ that fits inside a box made from an A4 sized piece of cardboard."
Give an example of a box you could build and the amount of money you would earn from it.
Create a table with different values for the cut length (x) in the first column, the corresponding volume of the resulting box (V) in the second column, and the money earned (€) in the last column.
Which cut would earn you the most money?
Find the formula for the function that gives the volume (V) of the box depending on the cut length (x).
Plot the graph of this function using the website desmos.com.
Observe the different boxes that can be built on this website. Answer the questions you find there in your notebook.
Word problems on optimization
Word problems on optimization
A pizza company wants to design a rectangular box (without a top) with a square base to hold one pizza. If they have 600 cm² of cardboard, what dimensions will maximize the volume of the box?
You're organizing a music concert and need to build a rectangular stage with an area of 10000 m². If the back wall is more expensive than the sides and front, what dimensions will minimize the total cost of the stage?. Data: price of back wall 3 €/m; price of sides and front walls: 1.8 €/m
A phone manufacturer wants to design a screen with a fixed area of 120 cm² but minimal perimeter (to reduce frame material). What dimensions (length and width) should the screen have?
Your school has 200 meters of fencing to enclose a rectangular sports field. One side borders a wall and doesn’t need fencing. What dimensions give the largest area?
A company wants to make cylindrical cans that hold 330 ml. What radius and height minimize the amount of aluminum used for the can?
You're designing a rectangular poster with a printed area of 800 cm². The margins at the top and bottom are 5 cm, and the side margins are 3 cm. What are the dimensions of the poster that use the least paper?
You’re making a playlist for a party. Each fast song is 3 minutes, and each slow song is 5 minutes. If people get tired after 60 minutes of dancing, how many of each type should you include to maximize the number of songs?
A student walks from home to school, which lies across a rectangular park. Should they cut across the park or follow the sidewalks along the edges to minimize walking time?
Links
Links
Answer key to some activities above
Answer key to some activities above
Main concepts of calculus (in order of appearance), initial video
Continuous, limit, one-side limits (implicit), function, asymptotes, graph, differentiable, calculus, derivatives, integrals (also integrate, integration), area, curves, y', derivative of y, dy/dx, product rules, chain rule, area, exponent, reciprocal
The approximated tangent lines to the red curve at certain points: x=14, the maximum, x=42, x=70.
They allow us to measure the rate of growth of the curve by knowing the slopes of those lines.
As higher is the slope, as higher is the growth; if the slope is negative, it indicates a decreasing. Slope equal to 0 shows the maximum.
Answers to the chain-rule activities.
Soluciones al ejercicio 10 del libro (p. 326)
Soluciones al ejercicio 9 del libro (p. 326)
Saberes básicos implicados en la unidad
Saberes básicos implicados en la unidad
B. Sentido de la medida.
MATE.1.B.2. Cambio.
MATE.1.B.2.3 Derivada de una función: definición a partir del estudio del cambio en diferentes contextos. Derivación de funciones polinómicas, racionales, irracionales, exponenciales, logarítmicas y trigonométricas. Reglas de derivación de las operaciones elementales con funciones y regla de la cadena. Aplicaciones de las derivadas: ecuación de la recta tangente a una curva en un punto de la misma; obtención de extremos relativos e intervalos de crecimiento y decrecimiento de una función. Cálculo de derivadas sencillas por definición.
D. Sentido algebraico.
MATE.1.D.2. Modelo matemático.
MATE.1.D.2.1 Relaciones cuantitativas en situaciones sencillas: estrategias de identificación y determinación de la clase o clases de funciones que pueden modelizarlas.
MATE.1.D.4. Relaciones y funciones.
MATE.1.D.4.1 Análisis, representación gráfica e interpretación de relaciones mediante herramientas tecnológicas. Concepto de función real de variables real: expresión analítica y gráfica. Cálculo gráfico y analítico del dominio de una función.
MATE.1.D.4.2 Propiedades de las distintas clases de funciones, incluyendo, polinómicas, exponenciales, irracionales, racionales sencillas, logarítmicas, trigonométricas y a trozos: comprensión y comparación. Estudio y representación gráfica de funciones polinómicas y racionales a partir de sus propiedades globales y locales obtenidas empleando las herramientas del análisis matemático (límites y derivadas).
MATE.1.D.4.3 Álgebra simbólica en la representación y explicación de relaciones matemáticas de la ciencia y la tecnología.
Criterios de evaluación
Criterios de evaluación
3.2. Emplear herramientas tecnológicas adecuadas en la formulación o investigación de conjeturas o problemas.
5.1. Manifestar una visión matemática integrada, investigando y conectando las diferentes ideas matemáticas.
5.2. Resolver problemas en contextos matemáticos, estableciendo y aplicando conexiones entre las diferentes ideas matemáticas y usando enfoques diferentes.
6.1. Resolver problemas en situaciones diversas utilizando procesos matemáticos, estableciendo y aplicando conexiones entre el mundo real, otras áreas del conocimiento y las matemáticas.
6.2. Analizar la aportación de las matemáticas al progreso de la humanidad, reflexionando sobre su contribución en la propuesta de soluciones a situaciones complejas: consumo responsable, medio sostenibilidad, ambiente, etc., y a los retos científicos y tecnológicos que se plantean en la sociedad.
7.1. Representar ideas matemáticas, estructurando diferentes razonamientos matemáticos y seleccionando las tecnologías más adecuadas.
7.2. Seleccionar y utilizar diversas formas de representación, valorando su utilidad para compartir información.
8.1. Mostrar organización al comunicar las ideas matemáticas, empleando el soporte, la terminología y el rigor apropiados.
8.2. Reconocer y emplear el lenguaje matemático en diferentes contextos, comunicando la información con precisión y rigor.
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