Calculus Assignment Help




homework assignment help is most useful online portal for students providing all type of Online Calculus assignment help Services. Calculus Assignment Help is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change,in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient.

AP calculus:

The AP calculus is advanced version of calculus. In high school, AP calculus is widely used. We need learn about general formulas for solving AP calculus problems. AP Calculus is classified as differential calculus and integral calculus. Different problems are used for solving the AP calculus problems. In this article, we deal with some differential and integral calculus problems.

Example Problems for Ap Calculus

AP calculus problem 1:

Find the integral value of the function g(x) = (9x4 + 5x2 - 3) dx

Solution:

The given function is g(x) = (9x4 + 5x2 - 3) dx

Integrate the given function with respect to x, we get

                                     G(x) = ∫ ( 9x4 + 5x2 - 3) dx

                                              = (9 / 5) x5 + (5 / 3) x3 - 3x + c

Answer:

The final answer is (9 / 5) x5 + (5 / 3) x3 - 3x + c

AP calculus problem 2:

Find the integral value of the function y = cos 2x sin 3x dx

Solution:

The given function is y = cos 2x sin 3x dx

We know that,

cosA sinB = (1 / 2) [ sin(A + B) - sin(A - B)]

Given cos 2x sin 3x

A = 2x, B = 3x

Therefore,

cos 2x sin 3x dx = (1 / 2)` (sin 5x - sin (- x)) dx

Integrate the given function with respect to x, we get

                                     G(x) = ∫ (1 / 2) (sin 5x - sin (- x)) dx

                                            = (1 / 2) [( - cos x) - (cos 5x) / 5) ]

                                            = `(1 / 10)` (- 5 cos x - cos (5x)) + c

Answer:

The final answer is (1 / 10) (- 5 cos x - cos (5x)) + c

AP calculus problem 3:

Differentiate the given function u = 4. 3x4 + 3.8x2 - 2.12x

Solution:

Given function u = 4. 3x4 + 3.8x2 - 2.12x

Differentiate the given function with respect to x, we get

              ((du) / dx) = (4.3 * 4)x3 + (3.8 * 2)x - 2.12

                           = 17.2x3 + 7.6x - 2.12

Answer:

The final answer is 17.2x3 + 7.6x - 2.12

AP calculus problem 4:

Differentiate the given function z = sin (5x)

Solution:

Differentiate the given function with respect to x, we get

                      (dz / dx) = 5 cos (5x)

Answer:

The final answer is 5 cos (5x)

Practice Problems for Ap Calculus

AP calculus problem 1:

Find the differentiation value of the given function u(x) = cos (3x) dx

Answer:

The final answer is - 3 sin (3x)

AP calculus problem 2:

Find the integral value of the given function f(x) = sin (3x) cos (4x) dx

Answer:

 The final answer is(1 / 14) (7 cosx - cos(7x))

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Functions Limits and Continuity

Left Hand Limit: Let f(x) tend to a limit l1 as x tends to a through values less than a, then l1 is called the left hand limit.

Right Hand Limit: Let f(x) tend to a limit l2 as x tends to 'a' through values greater than 'a', then l2 is called the right hand limit.

Differentiation

The derivative, measures the rate at which the dependent variable changes with respect to the independent variable. It is one of the most important ideas in Calculus. The differentiation of functions are widely used in science, economics, medicine and computer science.

Differential Equations

Differential Equation: A differential equation is a relation between the independent, dependent variables and their differential coefficients.

Indefinite Integrals

The expression ∫ f(x) dx is read "the indefinite integral of f(x) with respect to x," and stands for the set of all antiderivatives of f.

Thus, ∫ f(x) dx is a collection of functions; it is not a single function, nor a number.

Definite Integrals

Let f (x) be a single valued continuous function defined in the interval [a,b] where b > 0 and let the interval [a,b] be divided into n equal parts each of length h, so that nh = b - a; then we define

Application of Derivatives

Differential calculus can be considered as mathematics of motion, growth and change where there is a motion, growth, change. Whenever there is variable forces producing acceleration, differential calculus is the right mathematics to apply. Application of derivatives are used to represent and interpret the rate at which quantities change with respect to another variable.

Exponential and Logarithmic Series

The sum of the infinite series 1 + 1/1! + 1/2! + 1/3! + 1/4! + ...¥ is called the exponential number.

If x is any complex number then the series img is called the exponential series. It can be proved mathematically that this exponential series has a sum and we denote it by ex.

Functions: 

In mathematics, a function is the idea that one quantity (called the input) completely determines another quantity, often called the output. Modern mathematics defines functions using sets. A function can then be seen as a rule. With this rule, each element of the set of inputs will be assigned one element of the set of outputs. 

Types of Limit:

Left Hand Limit:

Let f(x) tend to a limit l1 as x tends to a through values less than a, then l1 is called the left hand limit.

Right Hand Limit:

Let f(x) tend to a limit l2 as x tends to 'a' through values greater than 'a', then l2 is called the right hand limit.

We say that limit of f(x) exists at x = a, if l1 and l2 are both finite and equal.

Limits of Trigonometric Functions and Sandwich Theorem:
img

for all x in some open interval containing c and suppose

img

img

Since f is sandwiched between two functions g and h, the above theorem is known as sandwich theorem.

Limits at infinity:

If x is a variable such that it can take any real value how much ever img

If x is a variable such that it can take any real value how much ever img

The two important properties of these one-sided limits that:

i) If the left hand limit and right hand limit of a function at a point exists, but are not equal, then we conclude that the limit at that point does not exist.

ii) If LHL and RHL of a function at a point (say a) exist and they are equal, we conclude that limit at that point exists and we write

Real Functions and their Graphs

     Functions: f is a function from set A to a set B if each element x in A can be associated with a unique element in B.
img

     Domain: In the above definition of the function, set A is called domain.

     Co-Domain: In the above definition of the function, set B is called co-domain.

     Real Function: A real valued function f : A to B or simply a real function f is a rule which associates to each possible real number xA, a unique real number f(x)imgB, when A and B are subsets of R, the set of real numbers.

     Value of a Function: If f is a function and x is an element in the domain of f, then image f(x) of x under f is called the value of f at x.

     Types of Function and their Graphs: Constant function, Identity function, Polynomial function, Modulus function, Square root function, Greatest integer function or Step function (Floor function), Smallest integer function (Ceiling function), Exponential function, Logarithmic function, Trigonometric functions, Inverse functions, Signum functions, Odd function, Even function, Reciprocal function.

Operation on Real Functions

     The following are the Operation on Real Functions: Sum Function, Difference Function, Product Function, Quotient Function, Scalar Multiplication Function, Composite Functions, Inverse Functions.

Continuity at a Point

     1. We say that f(x) is continuous if f(x) is continuous at every point in its domain.

     2. If f and g are two continuous functions then f + g, f - g, fg are continuous functions.

     3. Every polynomial function is continuous.

     4. Every rational function is continuous at each point of its domain.

     5. Composition of two continuous functions is continuous.

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