6.1.3 Limits Of Trigonometric Functions Return To Contents Go To Problems & Solutions The following infinite limits can be visualized easily in Fig. 1.1. Of course these limits can be proved by using the definitions of the functions in terms of the sine and cosine functions. For example: Go To Problems & Solutions Return To […]
Calculus Of One Real Variable – By Pheng Kim Ving Chapter 9: The Integral – Section 9.4: The Fundamental Theorem Of Calculus 9.4 The Fundamental Theorem Of Calculus Return To Contents Go To Problems & Solutions 1. Integrals And Antiderivatives In Section 9.3 Example 3.1 we calculated definite integrals directly from the definition by using […]
10.1 Integration By Inspection Return To Contents Go To Problems & Solutions 1. Integrals And Integration The fundamental theorem of calculus, as presented in Section 9.4 Theorem 2.1, provides us a powerful method to calculate definite integrals, by the formula: where F is any antiderivative of f. This method places the burden of calculation squarely […]
Return To Contents Go To Problems & Solutions Example 1.1 Calculate: Solution EOS Integrals whose integrands involve the quadratic expression ax2 + bx + c but aren’t polynomials can often be handled as follows. First, complete the square: Go To Problems & Solutions Return To Top Of Page 2. Elimination Of All Fractional Exponents Example […]
10.7 The Method Of Partial Fractions Return To Contents Go To Problems & Solutions Recall from algebra that a linear function is a polynomial of degree 1, ie a function of the form ax + b (its graph is a line). A quadratic function is a polynomial of degree 2, ie a function of the […]
15.5 Taylor Polynomials And Taylor Theorem Return To Contents Go To Problems & Solutions Consider the Taylor series T(x) of a function f(x) centered at c: Taylor series of f(x) centered at c: Taylor polynomial of f(x) centered at c: Clearly Tn(x) is a polynomial in x – c that involves derivatives of f(x) of […]
Return To Contents Go To Problems & Solutions In general, when two or more quantities are related to each other, their rates of change with respect to time (speeds) are also related to each other. In this section, we’ll solve problems of finding a rate of change with respect to time by searching for how […]
Calculus Of One Real Variable – By Pheng Kim Ving Chapter 7: The Exponential And Logarithnic Functions – Section 7.4: Logarithmic Differentiation 7.4 Logarithmic Differentiation Return To Contents Go To Problems & Solutions 1. Differentiation Of The Function y = ( f (x)) g (x) Let’s differentiate the function y = ( f (x))g(x), where […]
Calculus Of One Real Variable – By Pheng Kim Ving Chapter 11: Techniques Of Integration – Section 11.1: Approximate Numerical Integration 11.1 Approximate Numerical Integration Return To Contents Go To Problems & Solutions Suppose we wish to evaluate the definite integral: of f. That’s why we spent a considerable amount of time in the last […]
14.3 The Comparison Tests Return To Contents Go To Problems & Solutions 1. The Standard Comparison Test (SCT) In Section 14.2 we introduced some series that were shown to converge or diverge. To determine the behavior (convergence or divergence) of most of them, we algebraically manipulated the sequence of their partial sums, expressing the general […]
6.1.7 The Simple Harmonic Motion Return To Contents Go To Problems & Solutions 1. The Simple Harmonic Motion Consider a mass m suspended vertically by an elastic spring attached to a beam. See Fig. 1.1. It has stretched the spring for some distance. It hangs unmoving in its position of rest, which is its equilibrium […]
14.4 The Root And Ratio Tests Return To Contents Go To Problems & Solutions This section discusses two more convergence tests: the Root Test and the Ratio Test. First we prove a limit that’ll be used later on in this section. The Limit Of x1/x As x Approaches Infinity Proof EOP The Root Test If […]
6.2.1 The Inverse Trigonometric Functions Return To Contents Go To Problems & Solutions 1. The Principal-Value Sine Function And Its Inverse Arcsine Let f(x) = x2. Clearly dom ( f ) = R. Consider the “ part ” of f on [0, 1]. Certainly this part isn’t the same function as f, because its domain […]
