Download Calculus-Mathmatics-Lecture Handoit and more Lecture notes Mathematics in PDF only on Docsity!
CALCULUS
Areas of Focus:
- Differentiation
- Maximization and minimization
- Partial derivatives
- Integration
- Integration over a line
- Double integrals
- Integration over an area
- Centroid of an area
- Integrating differential equations
Differentiation:
The derivative of a function is a measure of how the function changes as a result of a change in the value of its argument. Given the function f ( x ), the derivative of f with
respect to x is written as or as , and is defined by
As shown in the figure, the derivative of the function f ( x ) at point x gives the slope of the function at x in terms of the ratio of the rise divided by the run for the line AB that is tangent to the curve at point x.
The derivative of the function f ( x ) is also sometimes written as f ' ( x ). As shown in the figure, one can also write the definition of the derivative as
The basic rules of differentiation are
Consider a function such as f ( x )= g ( x ) h ( x ) that is the product of two functions. The product rule can be used to calculate the derivative of f with respect to x. The product rule states that
For example, To take the derivative of f ( x )=(2 x +3)(4 x +5) 2 one can follow these steps
The chain rule for derivatives:
Consider the function f ( U ), where U is a function of x. One can calculate the derivative of f with respect to x by using the chain rule given by
For example, to calculate the derivative of f ( U ) = Un^ with respect to x , where U = x 2 +a , one can follow these steps
Maximization and minimization:
A point on a smooth function where the derivative is zero is a local maximum, a local minimum, or an inflection point of the function. This can be clearly seen in the figure, where the function has three points at which the tangent to the curve is horizontal (the slope is zero). This function has a local maximum at A , a local minimum at B , and an inflection point at C.
The global maximum or minimum of a smooth function in a specific interval of its argument occurs either at the limits of the interval or at a point inside the interval where the function has a derivative of zero. As can be seen in the figure, the function shown has a global maximum at point A on the left boundary of the interval under consideration, a global minimum at B , a local maximum at C, and a local minimum at D.
Partial derivative:
The derivative of a function of several variable with respect to only one of its variables is called a partial derivative. Given a function f ( x,y ), its partial derivative with respect to its
first argument is denoted by and defined by
Since all other variables are kept constant during the partial derivative, it represents the slope of the curve one obtains when varying only the designated argument of the function.
For example, the partial derivative of the function with respect to x is evaluated by treating y as a constant so that one gets
The chain rule:
Consider a function f [ U ( x ), V ( x )] of two arguments U and V , each a function of x. The chain rule can be used to find the derivative of f with respect to x by the rule
For example, consider the function f = UV^2 where U =2x+3 and V=4x 2. The derivative of f with respect to x is given by
This approximation approaches the value of the integral as the width of the strips approaches zero.
Indefinite Integrals:
A function F ( x ) is the indefinite integral of the function f ( x ) if
The indefinite integral is also know as the anti-derivative. Since the derivative of a constant is zero, the indefinite integral of a function can only be evaluated up to the addition of a constant. Therefore, given a function F ( x ) to be an anti-derivative of f ( x ), the function F ( x ) + C , where C is any constant, is also an anti-derivative of f ( x ). This constant is known as the constant of integration and may be determined only if one has additional information about the integral. Normally, a known value of the integral at a specified point is used to calculate the constant of integration.
The basic rules of integration are
The indefinite integral of commonly used functions:
The following is a list of indefinite integrals of commonly used functions,
up to a constant of integration [ ]:
For example, given the function we can write this
function as where U = ax+b and. Therefore, the integral of f can be evaluated by using the following steps.
Change of variables for a definite integral is similar with an additional change in the limits of integration. The resulting equation is
For example, given the function , we can write this function
as where U = ax and. Therefore, the integral of f can be evaluated by using the following steps.
Integration by parts:
Given the functions U ( x ) and V ( x ), one can use integration by parts to integrate the following integral using the relation
For example, to evaluate the integral
one can take and so that and
. Using integration by parts we get
Integration over a line:
For example, consider integrating the function f ( x,y ) = xy^2 over the straight line defined in the figure from point A to point B. Direct integration, using the relations
would yield
Double integral:
The double integral of function f ( x,y ) first integrating over x and then integrating over y is given by
The notation implies that the inner integral over x is done first, treating y as a constant.
Unlike the example, the limits of integration need not be constants. There will be no problem as long as the inner integral is conducted fist and the limits are substituted into the resulting expression before the outer integral is evaluated. For example, consider the following integral.
Integration over an area:
The double integral F of the function over the area A is written as
This integral is the sum of fdA over the area A. Each fdA is the volume of the column with base dA and height f. Thus, the integral gives the volume under the surface
. To accomplish the summation that is represented by the integral, one needs to section the domain of integration A into small parts (i.e., into many small dAs). As shown in the figure, the domain of integration can be sections into rectangular
sections, each with an area. At the limit of small element sizes, the sum of these area elements adds up to the original domain of integration.
