Solving calculus problems is a great way to master the various rules, theorems, and calculations you encounter in a typical Calculus class. This Cheat Sheet provides some basic formulas you can refer to regularly to make solving calculus problems a breeze (well, maybe not a breeze, but definitely easier).
Useful Calculus Theorems, Formulas, and Definitions
Following are some of the most frequently used theorems, formulas, and definitions that you encounter in a calculus class for a single variable. The list isn’t comprehensive, but it should cover the items you’ll use most often.
Limit Definition of a Derivative
Definition: Continuous at a number a
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Calculus Problems And Solutions Pdf
The Intermediate Value Theorem
Definition of a Critical Number
A critical number of a function f is a number c in the domain of f such that either f‘(c)= 0 or f‘(c) does not exist.
Rolle’s Theorem
Let f be a function that satisfies the following three hypotheses:
f is continuous on the closed interval [a, b].
f is differentiable on the open interval (a, b).
f(a)= f(b).
Then there is a number c in (a, b) such that f‘(c)= 0.
The Mean Value Theorem
Let f be a function that satisfies the following hypotheses:
f is continuous on the closed interval [a, b].
f is differentiable on the open interval (a, b).
Newton’s Method Approximation Formula
Newton’s method is a technique that tries to find a root of an equation. To begin, you try to pick a number that’s “close” to the value of a root and call this value x1. Picking x1 may involve some trial and error; if you’re dealing with a continuous function on some interval (or possibly the entire real line), the intermediate value theorem may narrow down the interval under consideration. After picking x1, you use the recursive formula given here to find successive approximations:
A word of caution: Always verify that your final approximation is correct (or close to the value of the root). Newton’s method can fail in some instances, based on the value picked for x1. Any calculus text that covers Newton’s method should point out these shortcomings.
The Fundamental Theorem of Calculus
Suppose f is continuous on [a, b]. Then the following statements are true:
The Trapezoid Rule
where
Simpson’s Rule
where n is even and
Special Limit Formulas in Calculus
Many people first encounter the following limits in a Calculus textbook when trying to prove the derivative formulas for the sine function and the cosine function. These results aren’t immediately obvious and actually take a bit of work to justify. Any calculus text should provide more explanation if you’re interested in seeing it!
Derivative and Integration Formulas for Hyperbolic Functions
The hyperbolic functions are certain combinations of the exponential functions ex and e–x. These functions occur often enough in differential equations and engineering that they’re typically introduced in a Calculus course. Some of the real-life applications of these functions relate to the study of electric transmission and suspension cables.
Calculus
The calculus section of QuickMath allows you to differentiate and integrate almost any mathematical expression.
What is calculus?
Calculus is a vast topic, and it forms the basis for much of modern mathematics. The two branches of calculus are differential calculus and integral calculus.
Differential calculus is the study of rates of change of functions. At school, you are introduced to differential calculus by learning how to find the derivative of a function in order to determine the slope of the graph of that function at any point.
Integral calculus is often introduced in school in terms of finding primitive functions (indefinite integrals) and finding the area under a curve (definite integrals).
Differentiate
The differentiate command allows you to find the derivative of an expression with respect to any variable. In the advanced section, you also have the option of specifying arbitrary functional dependencies within your expression and finding higher order derivatives. The differentiate command knows all the rules of differential calculus, including the product rule, the quotient rule and the chain rule.
Integrate
The integrate command can be used to find either indefinite or definite integrals. If an indefinite integral (primitive function) is sought but cannot be found for a particular function, QuickMath will let you know. Definite integrals will always be given in their exact form when possible, but failing this QuickMath will use a numerical method to give you an approximate value.
Integrals were evaluated in the previous tutorial by identifying the integral with an appropriate area and then using methods from geometry to find the area. This procedure will succeed only for very simple integrals. The main result of this section, the fundamental theorem of calculus, includes a very important formula for evaluating integrals. This theorem shows us how to evaluate integrals by first evaluating antiderivatives. The theorem establishes an amazing relationship between the integral, which may be interpreted as an area, and the antiderivative, which is inversely related to the derivative; that is, it relates area and the derivative.
Theorem 1
Let f be a continuous function on [a, b ], and define a function F by
then
This result could also be written as
that is, the derivative of the integral of/ with respect to the upper limit of integration x is equal to f evaluated at x. We shall verify a special case of this theorem at the end of this section. Example I illustrates Theorem l.
Example 1
The following theorem is called the fundamental theorem and is a consequence of Theorem 1 .
The Fundamental Theorem of Calculus
Let f be continuous on [a. b ], and suppose G is any antiderivative of f on [a, b], that is. G'(x) = f(x) for x in [a. b]. Then,
To verify the fundamental theorem, let F be given by , as in Formula (1). Then by Theorem 1, F is an antiderivative of f. Since G is also an antiderivative of f, we know that there is a constant c for which F(x} = G(x) + c.
Since F(a) = J;' f(x) dx = 0, it follows that
Calculus Problems And Answers Pdf
0 = F (a) = G (a) + c
Math Problem Solving Calculus Ab&bc
and hence c = -G(a). Thus, F(x) = G(x) + c = G(x) - G(a), from which we see that
For convenience, we introduce the following notation:
With this notation, Formula (4) can be written as
The following equivalent formula demonstrates the convenience of using a symbol for the integral that resembles the one for the antiderivative.
Calculus Problem
This formula is a consequence of the fact that integral of f(x) dx is the most general anti derivative of f.