Table of contents

0. Functions7h 52m
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 Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

0. Functions

Combining Functions

Calculus

0. Functions

Combining Functions

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3:55 minutes

Problem 1.1.66

Textbook Question

Simplify the difference quotient ƒ(x+h)-ƒ(x)/h
ƒ(x) = 2x² -3x +1

Verified step by step guidance

Step 1: Start by substituting the function \( f(x) = 2x^2 - 3x + 1 \) into the difference quotient \( \frac{f(x+h) - f(x)}{h} \).

Step 2: Calculate \( f(x+h) \) by replacing \( x \) with \( x+h \) in the function: \( f(x+h) = 2(x+h)^2 - 3(x+h) + 1 \).

Step 3: Expand \( f(x+h) \): \( 2(x+h)^2 = 2(x^2 + 2xh + h^2) = 2x^2 + 4xh + 2h^2 \) and \( -3(x+h) = -3x - 3h \). Combine these to get \( f(x+h) = 2x^2 + 4xh + 2h^2 - 3x - 3h + 1 \).

Step 4: Substitute \( f(x+h) \) and \( f(x) \) into the difference quotient: \( \frac{(2x^2 + 4xh + 2h^2 - 3x - 3h + 1) - (2x^2 - 3x + 1)}{h} \).

Step 5: Simplify the expression by canceling out like terms and dividing each term by \( h \). This will give you the simplified form of the difference quotient.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Difference Quotient

The difference quotient is a formula used to find the average rate of change of a function over an interval. It is expressed as (ƒ(x+h) - ƒ(x))/h, where h represents a small change in x. This concept is fundamental in calculus as it leads to the definition of the derivative, which measures the instantaneous rate of change.

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. In this case, we need to evaluate ƒ(x+h) and ƒ(x) for the given function ƒ(x) = 2x² - 3x + 1. Understanding how to correctly substitute and simplify expressions is crucial for manipulating the difference quotient.

Algebraic Simplification

Algebraic simplification is the process of reducing expressions to their simplest form. This involves combining like terms, factoring, and canceling common factors. In the context of the difference quotient, simplifying the expression after substituting ƒ(x+h) and ƒ(x) is essential to arrive at a clear and concise result, especially as h approaches zero.