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Chapter 1: Problem 70

$$ \begin{array}{l} \text { For each function, find and simplify }\\\ \frac{f(x+h)-f(x)}{h} . \quad(\text { Assume } h \neq 0 .) \end{array} $$ $$ \begin{array}{l} f(x)=x^{4} \\ \text { [Hint: Use }(x+h)^{4}=x^{4}+4 x^{3} h+6 x^{2} h^{2}+ \\ \left.4 x h^{3}+h^{4} .\right] \end{array} $$

Short Answer

Expert verified
The simplified expression is \( 4x^3 + 6x^2h + 4xh^2 + h^3 \).

Step by step solution

01

Substitute and Expand

Substitute \( x + h \) for \( x \) in \( f(x) \). Therefore, \[ f(x+h) = (x+h)^4. \]Using the hint, expand this to get: \[ f(x+h) = x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4. \]
02

Compute Difference

Calculate the difference \( f(x+h) - f(x) \): \[ f(x+h) - f(x) = (x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4) - x^4. \]Simplifying, the \( x^4 \) terms cancel out, leaving: \[ 4x^3h + 6x^2h^2 + 4xh^3 + h^4. \]
03

Simplify the Expression

Divide the expression from Step 2 by \( h \): \[ \frac{f(x+h) - f(x)}{h} = \frac{4x^3h + 6x^2h^2 + 4xh^3 + h^4}{h}. \]Cancel out \( h \) from the numerator and denominator: \[ = 4x^3 + 6x^2h + 4xh^2 + h^3. \]
04

Final Simplified Expression

The simplified expression for \( \frac{f(x+h)-f(x)}{h} \) is: \[ 4x^3 + 6x^2h + 4xh^2 + h^3. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Evaluation
Evaluating a function involves substituting a specific value or expression in place of the variable in the function. Let's take a closer look at our exercise: the function is given by \( f(x) = x^4 \). To evaluate \( f(x+h) \), substitute \( x+h \) for \( x \) across the function.

Why substitute? Substitution helps us see how the function behaves for inputs slightly different from \( x \). This is crucial when working with operations like the difference quotient, which helps to understand the concept of a derivative in calculus.
  • Start with the original function: \( f(x) = x^4 \).
  • Substitute \( x+h \) into the function: \( f(x+h) = (x+h)^4 \).
  • Expand the substituted function: \( x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 \).
This process, while simple, lays the foundation for understanding changes in functions.
Binomial Expansion
Binomial expansion is a powerful algebraic tool that allows us to expand expressions that are raised to a power, such as \((x+h)^4\). Using the binomial theorem, an expression of the form \((a+b)^n\) can be expanded.

In our exercise, we deal with the expansion of \((x+h)^4\). Using binomial expansion:
  • The formula becomes: \( x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 \).
  • The expansion uses coefficients from Pascal's triangle: \( 1, 4, 6, 4, 1 \).
These coefficients multiply each term incrementally involving powers of \( x \) and \( h \). This results in terms that reflect combined effects of these variables on the function.

Understanding binomial expansion simplifies the process of handling powers of binomial expressions, making calculations for the difference quotient manageable and precise.
Simplification of Expressions
Simplifying expressions is a fundamental skill in algebra that involves reducing an expression to its simplest form. In our problem, simplification plays a key role.

Start with the difference we obtained: \( f(x+h) - f(x) = 4x^3h + 6x^2h^2 + 4xh^3 + h^4 \). Division by \( h \) (provided \( h eq 0 \)) cancels out the common \( h \) in all terms.
  • The expression \( \frac{f(x+h) - f(x)}{h} \) becomes \( 4x^3 + 6x^2h + 4xh^2 + h^3 \).
  • Every term has been divided by \( h \), simplifying the overall expression.
This step highlights the process of extracting the core structure of the changing relationship in the original function.

Through simplification, understanding the behavior of functions near a point becomes more intuitive, paving the way to grasp concepts like derivatives more effectively.

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Most popular questions from this chapter

An insurance company keeps reserves (money to pay claims) of \(R(v)=2 v^{0.3}\), where \(v\) is the value of all of its policies, and the value of its policies is predicted to be \(v(t)=60+3 t\), where \(t\) is the number of years from now. (Both \(R\) and \(v\) are in millions of dollars.) Express the reserves \(R\) as a function of \(t\). and evaluate the function at \(t=10\).

Explain why, if a quadratic function has two \(x\) intercepts, the \(x\) -coordinate of the vertex will be halfway between them.

GENERAL: Boiling Point At higher altitudes, water boils at lower temperatures. This is why at high altitudes foods must be boiled for longer times - the lower boiling point imparts less heat to the food. At an altitude of \(h\) thousand feet above sea level, water boils at a temperature of \(B(h)=-1.8 h+212\) degrees Fahrenheit. Find the altitude at which water boils at \(98.6\) degrees Fahrenheit. (Your answer will show that at a high enough altitude, water boils at normal body temperature. This is why airplane cabins must be pressurized - at high enough altitudes one's blood would boil.)

GENERAL: Tsunamis The speed of a tsunami (popularly known as a tidal wave, although it has nothing whatever to do with tides) depends on the depth of the water through which it is traveling. At a depth of \(d\) feet, the speed of a tsunami will be \(s(d)=3.86 \sqrt{d}\) miles per hour. Find the speed of a tsunami in the Pacific basin where the average depth is 15,000 feet.

A 5 -foot-long ramp is to have a slope of \(0.75\). How high should the upper end be elevated above the lower end? [Hint: Draw a picture.]
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