Find functions ƒand g such that ƒ(g(x)) = (x² +1)⁵ . Find a different pair of functions ƒ and g that also satisfy ƒ(g(x)) = (x² +1)⁵
Table of contents
- 0. Functions7h 52m
- Introduction to Functions16m
- Piecewise Functions10m
- Properties of Functions9m
- Common Functions1h 8m
- Transformations5m
- Combining Functions27m
- Exponent rules32m
- Exponential Functions28m
- Logarithmic Functions24m
- Properties of Logarithms34m
- Exponential & Logarithmic Equations35m
- Introduction to Trigonometric Functions38m
- Graphs of Trigonometric Functions44m
- Trigonometric Identities47m
- Inverse Trigonometric Functions48m
- 1. Limits and Continuity2h 2m
- 2. Intro to Derivatives1h 33m
- 3. Techniques of Differentiation3h 18m
- 4. Applications of Derivatives2h 38m
- 5. Graphical Applications of Derivatives6h 2m
- 6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
- 7. Antiderivatives & Indefinite Integrals1h 26m
- 8. Definite Integrals3h 25m
0. Functions
Combining Functions
Problem 3.1.63cBriggs - 3rd Edition
Textbook Question
Another way to approximate derivatives is to use the centered difference quotient: f' (a) ≈ f(a+h) - f(a- h) / 2h. Again, consider f(x) = √x.
c. Explain why it is not necessary to use negative values of h in the table of part (b).
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Step 1: Understand the centered difference quotient formula: The centered difference quotient is given by . This formula is used to approximate the derivative of a function at a point .
Step 2: Recognize the role of : In the formula, is a small positive number that represents a small change in . The choice of affects the accuracy of the approximation.
Step 3: Consider symmetry in the formula: The centered difference quotient uses both and , which means it inherently considers both positive and negative changes around .
Step 4: Analyze the effect of negative : Using a negative would simply reverse the roles of and , but since the formula already accounts for both directions, negative values of are redundant.
Step 5: Conclude on the necessity of negative : Since the centered difference quotient already incorporates both forward and backward differences through its symmetric structure, using negative values of does not provide additional information or improve the approximation.
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