Históricamente la idea de integral se halla unida al cálculo de áreas a través del teorema fundamental del cálculo. Ampliamente puede decirse que la integral contiene información de tipo general mientras que la derivada la contiene de tipo local.
El concepto operativo de integral se basa en una operación contraria a la derivada a tal razón se debe su nombre de: antiderivada.
Las reglas de la derivación son la base que de cada operación de integral indefinida o antiderivada.
Históricamente la idea de integral se halla unida al cálculo de áreas a través del teorema fundamental del cálculo. Ampliamente puede decirse que la integral contiene información de tipo general mientras que la derivada la contiene de tipo local.
El concepto operativo de integral se basa en una operación contraria a la derivada a tal razón se debe su nombre de: antiderivada.
Las reglas de la derivación son la base que de cada operación de integral indefinida o antiderivada.
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Dalam materi ini, kita membahas tentang Himpunan penyelesaian dari persamaan trigonometri
ada dua cara dalam menyelesaikan persamaan trigonometri yaitu:
dengan gambar
dengan rumus
The trapezoidal rule is used to approximate the area under a curve by dividing it into trapezoids. It takes the average of the function values at the beginning and end of each sub-interval multiplied by the sub-interval width. The general formula sums these values over all sub-intervals divided by the number of intervals. An example calculates the area under y=1+x^3 from 0 to 1 using n=4 sub-intervals and gets an approximate value of 1.26953125.
Integration by parts is a technique for evaluating integrals of the form ∫udv, where u and v are differentiable functions. It works by expressing the integral as uv - ∫vdu. Some examples of integrals solved using integration by parts include ∫xe^xdx, ∫lnxdx, and ∫xe^-xdx. The technique can also be used repeatedly and for definite integrals between limits a and b using the formula ∫abudv = uv|_a^b - ∫avdu.
Đây chỉ là bản mình dùng để làm demo trên web. Để tải bản đầy đủ bạn vui lòng truy cập vào website tuituhoc.com nhé, chúc bạn tìm được nhiều tài liệu hay
Dalam materi ini, kita membahas tentang Himpunan penyelesaian dari persamaan trigonometri
ada dua cara dalam menyelesaikan persamaan trigonometri yaitu:
dengan gambar
dengan rumus
The trapezoidal rule is used to approximate the area under a curve by dividing it into trapezoids. It takes the average of the function values at the beginning and end of each sub-interval multiplied by the sub-interval width. The general formula sums these values over all sub-intervals divided by the number of intervals. An example calculates the area under y=1+x^3 from 0 to 1 using n=4 sub-intervals and gets an approximate value of 1.26953125.
Integration by parts is a technique for evaluating integrals of the form ∫udv, where u and v are differentiable functions. It works by expressing the integral as uv - ∫vdu. Some examples of integrals solved using integration by parts include ∫xe^xdx, ∫lnxdx, and ∫xe^-xdx. The technique can also be used repeatedly and for definite integrals between limits a and b using the formula ∫abudv = uv|_a^b - ∫avdu.
This document discusses calculating the volume of solids of revolution formed by rotating an area bounded by graphs around an axis. It provides the formula for finding the volume of a cylindrical shell as well as the formula for finding the total volume of a solid of revolution by summing the volumes of infinitely thin cylindrical shells. It includes two example problems demonstrating how to set up and solve the integrals to find the volume of solids of revolution.
7.2 volumes by slicing disks and washersdicosmo178
This document discusses different methods for calculating the volumes of solids of revolution: the disk method and washer method. It provides step-by-step explanations of how to set up and evaluate the definite integrals needed to calculate these volumes, whether the region is revolved about an axis that forms a border or not. Examples are given to illustrate each method. The key steps are to divide the solid into slices, approximate the volume of each slice, add the slice volumes using a limit of a Riemann sum, and evaluate the resulting definite integral.
The document discusses calculating the area between two curves. It explains that this area is defined as the limit of sums of the areas of rectangles between the curves as the number of rectangles approaches infinity, which is represented by a definite integral. It provides examples of finding the area between curves defined by various functions through setting up and evaluating the appropriate definite integrals.
This document discusses integration by substitution. It provides an example of recognizing a composite function and rewriting the integral in terms of the inside and outside functions. Specifically, it shows rewriting the integral of (x2 +1)2x dx as the integral of the outside function (x2 + 1) with the inside function (x) plugged in, plus a constant. It then provides additional practice problems applying the technique of substitution to rewrite integrals in terms of u-substitutions.
This document discusses the indefinite integral and antiderivatives. It defines an antiderivative as a function whose derivative is the original function, and notes that there are infinitely many antiderivatives that differ by a constant. The process of finding antiderivatives is called indefinite integration or antidifferentiation. Initial conditions can be used to determine a unique particular solution by solving for the constant of integration.
6.1 & 6.4 an overview of the area problem areadicosmo178
The document discusses different methods for approximating the area under a curve:
- Lower estimate (LAM) uses the left endpoints of intervals
- Upper estimate (RAM) uses the right endpoints
- Average estimate (MAM) uses the midpoints
Formulas are provided for calculating the area using each method by summing the areas of rectangles. Examples are shown for finding the area under y=x^2 from 0 to 2 using each method. Finally, the document introduces using the antiderivative method to find the exact area under a curve by calculating the antiderivative and evaluating it over the bounds.
This document discusses rectilinear motion and concepts related to position, velocity, speed, and acceleration for objects moving along a straight line. It defines velocity as the rate of change of position with respect to time and speed as the magnitude of velocity. Acceleration is defined as the rate of change of velocity with respect to time. Examples are given to show how to calculate position, velocity, speed, and acceleration functions from a given position function. The document also analyzes position versus time graphs to determine characteristics of the particle's motion at different points in time.
This document discusses Rolle's theorem and the mean value theorem. It provides the definitions and formulas for each theorem. It then gives examples of applying each theorem to find values of c where a derivative is equal to zero or a tangent line is parallel to a secant line. Rolle's theorem examples find values of c where the derivative of a function over an interval is zero. The mean value theorem examples find values of c where the slope of a tangent line equals the slope of a secant line over an interval.
1. Optimization problems involve finding the maximum or minimum value of a function subject to certain constraints.
2. To solve maximum/minimum problems: draw a figure, write the primary equation relating quantities, reduce to one variable if needed, take the derivative(s) to find critical points, and check solutions in the domain.
3. Examples show applying this process to find the dimensions that maximize volume of an open box, minimize cost of laying pipe between points, maximize area of two corrals with a fixed fence length, and find the largest volume cylinder that can fit in a cone.
The document provides information on finding absolute maximum and minimum values (absolute extrema) of functions on different interval types. It discusses determining absolute extrema on closed, infinite, and open intervals. Examples are provided finding the absolute extrema of specific functions on given intervals, including finding any critical points and limits to determine if absolute extrema exist. Practice problems are also provided at the end to find the absolute extrema of additional functions on specified intervals.
This document provides guidance on sketching graphs of functions by considering key features such as symmetries, intercepts, extrema, asymptotes, concavity, and inflection points. It then works through an example of sketching the graph of the function f(x) = (2x^2 - 8)/(x^2 - 16). Key steps include finding vertical and horizontal asymptotes, critical points and inflection points, intervals of increase/decrease, and finally sketching the graph.
This document discusses methods for finding relative extrema of functions:
1. The First Derivative Test (FDT) states that a critical point is a relative maximum if the derivative changes from positive to negative, and a relative minimum if the derivative changes from negative to positive.
2. The Second Derivative Test (SDT) states that a critical point is a relative maximum if the second derivative is negative, and a relative minimum if the second derivative is positive.
3. Examples are provided to demonstrate using the FDT and SDT to find relative extrema of functions.
This document discusses increasing and decreasing functions, concavity of functions, and finding intervals where functions are increasing, decreasing, concave up, or concave down. It provides examples of finding the intervals for the functions f(x)=x-4x^2+3 and f(x)=x-5x^4+9x showing the steps to determine where the functions are increasing or decreasing and where they are concave up or concave down. It also discusses inflection points and provides an example of finding intervals of increase, decrease, concavity and the inflection point for the function f(x)=x^3-3x^2+1.
4.3 derivatives of inv erse trig. functionsdicosmo178
This document discusses derivatives of inverse trigonometric functions and differentiability of inverse functions. It provides examples of finding the derivative of inverse trig functions like sin^-1(x^3) and sec^-1(e^x). It also explains that if a function f(x) is differentiable on an interval I, its inverse f^-1(x) will also be differentiable if f'(x) is not equal to 0. It gives the formula for the derivative of the inverse function and an example confirming this formula. It also discusses monotonic functions and how if f'(x) is always greater than 0 or less than 0, f(x) is one-to-one and its inverse will be different
Integration by parts is a technique for evaluating integrals of the form ∫udv, where u and v are differentiable functions. It works by expressing the integral as uv - ∫vdu. Some examples of integrals solved using integration by parts include ∫xe^xdx, ∫lnxdx, and ∫xe^-xdx. Repeated integration by parts may be necessary when the integral ∫vdu generated cannot be directly evaluated. Integration by parts also applies to definite integrals between limits a and b using the formula ∫_a^budv = uv|_a^b - ∫_a^bvdu.
The trapezoidal rule is used to approximate the area under a curve by dividing it into trapezoids. It takes the average of the function values at the beginning and end of each sub-interval. The area is calculated as the sum of the areas of each trapezoid multiplied by the width of the sub-interval. An example calculates the area under y=1+x^3 from 0 to 1 using n=4 sub-intervals, giving an approximate result of 1.26953125. The document also provides an example of using the trapezoidal rule with n=8 sub-intervals to estimate the area under the curve of the function y=x from 0 to 3.
2. This method used the sum of
the area of intervals under a
curve- called Reimann Sums
3. The limit of the sums of intervals is the same as
a definite integral over the same interval.
4.
5.
6. b
A (x)
• A’ (x) = f (x)
• A (a) = 0 and F (x) = A (x) + C
• A (b) = A
F b
( )-F a
( )= A b
( )+C
é
ë ù
û- A a
( )+C
é
ë ù
û=
A b
( )- A a
( )= A-0 = A
11. Practice Time !!!
Find the total area between the curve y = 1 – x2
and the x-axis over the interval [0, 2].
A = 1- x2
dx
0
2
ò = 1- x2
( )dx
0
1
ò + - 1- x2
( )dx
1
2
ò =
x -
x3
3
æ
è
ç
ö
ø
÷
ù
û
ú
0
1
- x -
x3
3
æ
è
ç
ö
ø
÷
ù
û
ú
1
2
= 1-
1
3
-0
æ
è
ç
ö
ø
÷- 2-
8
3
æ
è
ç
ö
ø
÷- 1-
1
3
æ
è
ç
ö
ø
÷
é
ë
ê
ù
û
ú=
2
3
- -
4
3
æ
è
ç
ö
ø
÷ =
6
3
= 2
12. The Mean Value Theorem for Integrals:
Over any interval, there exists an x value which creates a y value
that is the height of a rectangle which will equal the area under the
curve.
The Average Value:
The function value, f(c), found by the
Mean Value Theorem
Þ f c
( ) =
1
b-a
f x
( )
a
b
ò
13. Example
faverage =
1
2-
1
2
x2
+1
x2
æ
è
ç
ö
ø
÷
1
2
2
ò dx =
2
3
1+
1
x2
æ
è
ç
ö
ø
÷
1
2
2
ò dx =
2
3
x -
1
x
æ
è
ç
ö
ø
÷
ù
û
ú
1
2
2
=
2
3
2-
1
2
æ
è
ç
ö
ø
÷-
1
2
-2
æ
è
ç
ö
ø
÷
é
ë
ê
ù
û
ú=
2
3
1
1
2
æ
è
ç
ö
ø
÷- -1
1
2
æ
è
ç
ö
ø
÷
é
ë
ê
ù
û
ú=
2
3
×3= 2
14. In analyzing the graph of F(x) we would look
at the derivative:
d
dx
F x
( ) =???
F x
( )= sint
( )
a
x
ò dt =-cost]a
x
=-cosx - -cosa
( )=
d
dx
F x
( ) =
d
dx
-cosx +cosa
[ ] =
-cosx+cosa
sin x
f (x)
