Calculus Of One Real Variable – By Pheng Kim Ving Chapter 7: The Exponential And Logarithmic Functions – Section 7.7: The Inverse Hyperbolic Functions 7.7 The Inverse Hyperbolic Functions Return To Contents Go To Problems & Solutions The Inverse Hyperbolic Sine Function The graph of the hyperbolic sine function y = sinh x is sketched […]

Calculus Of One Real Variable – By Pheng Kim Ving Chapter 3: Rules Of Differentiation – Section 3.3: Differentiation Of Compositions Of Functions – The Chain Rule 3.3 Differentiation Of Compositions Of Functions – The Chain Rule Return To Contents Go To Problems & Solutions 1. Compositions Of Functions Consider an example. Let f(x) = […]

12.5 Distance And Displacement Return To Contents Go To Problems & Solutions 1. Distance Travelled And Displacement We’re now going to find the distance travelled by and the displacement of an object moving on a straight line given that its velocity v = v(t) at any time t is known. Fig. 1.1 Distance Travelled And […]

5.4 Concavity And Inflection Return To Contents Go To Problems & Solutions Let f be differentiable on (a, b). See Fig. 1.1. The graph of f is bending upward. We say that the graph of f is concave up. We note that as x increases from the near right of a to the near left […]

6.1.2 Trigonometric Identities Return To Contents Go To Problems & Solutions 1. Sine And Cosine Values Of Special Angles 360o), and the negatives of these angles. We’ll find the sine and cosine values for the positive angles. Those for the negative ones can be derived from those for the positive ones by the identities sin […]

8.6 Return To Contents Go To Problems & Solutions 1. The Cauchy Mean-Value Theorem For a review of the mean-value theorem see Section 5.1 Theorem 5.1. where D2, C2, G2, and F2 are the counterparts for point c of D, C, G, and F respectively. The property expressed by Eq. [1.1] is called the Cauchy […]

Calculus Of One Real Variable – By Pheng Kim Ving Chapter 9: The Integral – Section 9.1: Summation Notation And Formulas 9.1 Summation Notation And Formulas Return To Contents Go To Problems & Solutions Example 1.1 Write out these sums: EOS The lower limit of the sum is often 1. It may also be any […]

8.7 More Indeterminate Forms Return To Contents Go To Problems & Solutions 1. The Indeterminate Quotient Forms In Section 1.1.3 we discussed the indeterminate quotient form 0/0, and in Section 1.1.7 we handled the indeterminate Go To Problems & Solutions     Return To Top Of Page 2. The Indeterminate Sum And Difference Forms which is just […]

Calculus Of One Real Variable – By Pheng Kim Ving Chapter 11: Techniques Of Integration – Section 11.2: Improper Integrals 11.2 Improper Integrals Return To Contents Go To Problems & Solutions 1. Proper And Improper Integrals Let the function f be continuous on the closed bounded interval [a, b] ([a, b] is bounded if both […]

16.1.2 Variables-Separable Equations Return To Contents Go To Problems & Solutions 1. Variables-Separable Differential Equations All the sections in Chapter 12 concentrated on applications of the definite integral. In this chapter we’ll focus on an application of the indefinite integral, the one to differential equations. In the remainder of this section the abbreviation “DE” stands […]

13.2.3 Area By Polar Curves Return To Contents Go To Problems & Solutions Fig. 1.1 Area Bounded By A Polar Curve And 2 Rays. Fig. 1.2 Regular Partition Of Order n = 5 Of An Angle Interval. Fig. 1.3 Area Of Wedge In Light-Blue Color. Differential Area Fig. 1.4 Differential Area In Light-Blue Color Using […]

Calculus Of One Real Variable – By Pheng Kim Ving Chapter 1: Limits And Continuity – Section 1.1.5: Limits At Infinity And Infinite Limits 1.1.5 Limits At Infinity And Infinite Limits Return To Contents Go To Problems & Solutions The formal definitions of limits at infinity are stated as follows: Example 1.1 Find this limit: […]

Calculus Of One Real Variable – By Pheng Kim Ving Chapter 8: Applications Of The Derivative Part 2 – Section 8.4: Approximations Of Errors In Measurement 8.4 Approximations Of Errors In Measurement Return To Contents Go To Problems & Solutions If a quantity x (eg, side of a square) is obtained by measurement and a […]

Calculus Of One Real Variable – By Pheng Kim Ving Chapter 12: Applications Of The Integral – Section 12.4: Finding Volumes By Using Cylindrical Shells 12.4 Finding Volumes By Using Cylindrical Shells Return To Contents Go To Problems & Solutions   region R bounded by f, y = 0, x = a, and x = […]