Explain in your own words - Assignment Example

Mathematics assignment; Calculus Mathematics assignment; Calculus Question The fundamental theorem of calculus explains the relationship that exists between integration and differentiation, or rather integrals and derivatives, which are reverse processes of each other. The theorem is divided into two parts; the first part provides the reversibility condition of indefinite integral through differentiation. That is; given f(x) a continuous function moving from a to b (fixed intervals [a, b], suppose its indefinite integral is F, the definite integral is F (b) – F (a).

Thus; The second part of this theorem clarifies that, no matter which indefinite integral or antiderivative is used to calculate a definite integral, the solution remains unchanged. Note that, the indefinite integrals of a function only differ by a constant (Dawkins 2014). Intuitively, this second part, opens the first part of the theorem to include the entire interval, not limited by (a,b).It holds that if F (x)and f(x) are functions such that , F(x) is an integral of f(x), then F’(x) = f(x)I.e. if Taking a point f(x) anywhere between a and b, and using the first theory then By differentiating the right hand side, we obtain, F’(g(x))g’(x) – F’(a) which is equal to Assuming that a,is constant, then f (a) is zero, and if f(x) is x, then g’(x) =1, hence the whole term becomes f (g(x).

Therefore, the definite integral is only influenced by the derivative, with the constant in the indefinite integral being a dummy. Question 2L’Hospital’s rule is a principle that helps to determine the limits of indeterminate forms. There are two most common indeterminate forms I.e. 0/0 and ∞/-∞ in which L’hospital rule majorly occurs. Note that these indeterminate forms create a controversy in the expected results. For instance, if we divide zero by zero, the answer may be 1(any number divided by itself is one), ∞ (any number divided by 0 = ∞) or 0 (zero divided by any number is zero) and so does the second case.

therefore, there is need to manipulate indeterminate functions in order to obtain limits. Secondly, factoring out functions could be so tedious, and since derivatives reduce the highest power, then L’hospital rule can also be used (Ikenaga, 2005). L’hospital rule states that; In times when then it true that For example, to evaluate Note that as x approaches infinity then the numerator also approaches infinity while the denominator approaches negative infinity, and thus (∞/-∞), moreover, if we x takes the value 2, then the function becomes (0/0), both of which are indeterminate forms.

Obtaining the first derivative, which also gives (∞/-∞)But also, (taking the second derivative)Therefore; Question 3The Taylor series is an expression of a given function as the summation of infinite term obtained from its derivatives about a point. The effectiveness of Taylors function in approximating the function is based on the fact that, it sums up all the finite derivatives of the function and thus accommodating all the functions properties (Dawkins 2014).For instance, given a function f(x) = 2x3, about x=0.

The function can be estimated by Taylor series as follows;Suppose we start by f(x) =f (1) = 2which gives us a horizontal line as shown in the graph. (line1)By equating the first derivative while still holding the condition above, we obtain, F(x) = f (1) + f’ (1) .x Thus And still passes through point (2, 16)(line 2)The second derivative; With a slope of 6 at point(1, 2) (line 30The process continues until the last derivative is obtained and included. The importance of using Taylor series is that it converts all functions into polynomial series which can easily be estimated.

Moreover, one just needs to obtain the first few derivatives and use them to generate the general formula.Question 4Integration refers to the aspect of obtaining integrals. It is the reverse process of differentiation and a very important tool in calculus. Integration can be used to determine the area under a given curve, determine average function values as well as obtaining volumes of several shapes. While indefinite integral gives a generalized function that can be used to solve problems listed above, definite integrals give exact figures.

ReferencesIkenaga, B. (2005). LHopitals Rule. Retrieved May 25, 2014, from http://www.millersville.edu/~bikenaga/calculus/lhopit/lhopit.htmlPaul Dawkins (2014). Online Math Notes. Retrieved May 25, 2014, from http://tutorial.math.lamar.edu/