6.3.2 Transcendency Of The Trigonometric Functions Return To Contents Go To Problems & Solutions 1. Transcendency Of The Trigonometric Functions In this section we’re going to show that the trigonometric functions and their inverses are transcendental. For the definition of transcendental functions, see Section 6.3.1 Definition 3.1. We see in Section 6.3.1 Part 3 that […]

Calculus Of One Real Variable – By Pheng Kim Ving Chapter 9: The Integral – Section 9.3: The Definite Integral 9.3 The Definite Integral Return To Contents Go To Problems & Solutions 1. Riemann Sums For General Continuous Functions In Section 9.2 we dealt only with continuous functions that are non-negative. For a function f […]

16.1.1 Introduction To Differential Equations Return To Contents Go To Problems & Solutions 1. Differential Equations In Section 5.8 Example 5.1, we were given x” = 4, and in Part 6 of the same section, we had y” = – g. We solved these equations to determine the functions x and y respectively. These are […]

15.2 Derivatives And Integrals Of Power Series Return To Contents Go To Problems & Solutions 1. Derivatives Of Power Series Centre, Radius, And Interval Of Convergence Of Derivative Power Series Recall that if two functions are equal on an interval, their derivatives must also be equal on that interval, except at one or both endpoints […]

Calculus Of One Real Variable – By Pheng Kim Ving Chapter 5: Applications Of The Derivative Part 1 – Section 5.6: Sketching Graphs Of Functions 5.6 Sketching Graphs Of Functions Return To Contents Go To Problems & Solutions Horizontal Asymptotes The horizontal line y = L is called a horizontal asymptote of the graph of […]

Return To Contents Go To Problems & Solutions Refer to Fig. 1.1. When x increases from a to b, y = f (x) changes from f (a) to f (b). The quantity y may go from f (a) straight to f (b), as in Fig. 1.1, where it increases from f (a) straight to f […]

Calculus Of One Real Variable By Pheng Kim Ving Chapter 7: The Exponential And Logarithmic Functions Section 7.3: General Exponential And Logarithmic Functions 7.3 General Exponential And Logarithmic Functions Return To Contents Go To Problems & Solutions 1. The General Exponential Functions Since the natural exponential function ex is differentiable, it’s continuous. Its range is […]

Calculus Of One Real Variable – By Pheng Kim Ving Chapter 4: More On The Derivative – Section 4.2: Implicit Differentiation 4.2 Implicit Differentiation Return To Contents Go To Problems & Solutions 1. Explicitly And Implicitly Defined Functions Consider the equation y = x2. See Fig. 1.1. Clearly each value of x is mapped to […]

Return To Contents Go To Problems & Solutions Consider the (infinite) sequence {an} = {a1, a2, a3, …} of real numbers. The addition of all the terms of {an}: We can have the index start from 1 by adjusting the expression for the general term. In example 4, the kth term corresponds to index k […]

Calculus Of One Real Variable – By Pheng Kim Ving Chapter 3: Rules Of Differentiation – Section 3.4: Differentiation Of Inverse Functions 3.4 Differentiation Of Inverse Functions Return To Contents Go To Problems & Solutions Fig. 1.1 f is one-to-one. g isn’t one-to-one. function, one x corresponds to exactly one y. Now we also have […]

Calculus Of One Real Variable – By Pheng Kim Ving Chapter 12: Applications Of The Integral – Section 12.3: Finding Volumes By Slicing 12.3 Finding Volumes By Slicing Return To Contents Go To Problems & Solutions 1. Volumes Of Solids Of Revolution Let f be a continuous function on [a, b]. See Fig. 1.1. Let […]

Calculus Of One Real Variable – By Pheng Kim Ving Chapter 3: Rules Of Differentiation – Section 3.2: Differentiation Of Products And Quotients 3.2 Differentiation Of Products And Quotients Return To Contents Go To Problems & Solutions 1. Differentiation Of Products Of Functions Recall from algebra that the product  of functions f and g is […]

14.6 The Alternating-Series And Absolute-Convergence Tests Return To Contents Go To Problems & Solutions Alternating Series A series may have finitely many negative terms, and thus it must have infinitely many positive-or-0-valued terms; they’re ultimately positive series. A series may have infinitely many negative terms, and thus it may have either no or finitely or […]

Calculus Of One Real Variable – By Pheng Kim Ving Chapter 5: Applications Of The Derivative Part 1 – Section 5.8: Motion 5.8 Motion Return To Contents Go To Problems & Solutions A car travelling on a straight road can be viewed as a point or an object moving along a line. An apple falling […]