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Chapter 4: Problem 119

$$\text { If } f(x)=x^{2}+5 x, \text { find and simplify } \frac{f(x)-f(c)}{x-c}, x \neq c$$

Short Answer

Expert verified
The simplified expression is \( x + c + 5 \).

Step by step solution

01

Understand the given function

The function given is \( f(x) = x^2 + 5x \).
02

Compute \( f(c) \)

Replace \( x \) with \( c \) in the function to find \( f(c) \): \( f(c) = c^2 + 5c \).
03

Set up the expression

Set up the expression for \( \frac{f(x) - f(c)}{x - c} \) using the values of \( f(x) \) and \( f(c) \): \[ \frac{f(x) - f(c)}{x - c} = \frac{x^2 + 5x - (c^2 + 5c)}{x - c} \].
04

Simplify the numerator

Simplify the numerator by distributing the negative sign and combining like terms: \[ x^2 + 5x - c^2 - 5c \]. This simplifies to \[ (x^2 - c^2) + 5(x - c) \].
05

Factor the numerator

Factor the numerator: \[ (x - c)(x + c) + 5(x - c) \]. Factor \( x - c \) from both terms: \[ (x - c)(x + c + 5) \].
06

Simplify the entire expression

Cancel \( x - c \) from the numerator and the denominator: \[ \frac{(x - c)(x + c + 5)}{x - c} = x + c + 5 \].

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Key Concepts

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

Difference Quotient
The difference quotient is a formula used in calculus to find the rate at which a function changes. It's essentially the foundation of the derivative. For a function given by \( f(x) \), the difference quotient is expressed as:
\[ \frac{f(x+h) - f(x)}{h} \]
In this case, we're working with two distinct points, so we use:
\[ \frac{f(x) - f(c)}{x - c} \]
This helps us find the slope of the secant line between these two points on the graph of the function. Understanding and mastering the difference quotient is crucial in understanding derivatives and the overall concept of differentiation in calculus.
Polynomial Functions
Polynomial functions are algebraic expressions that include terms in the form \( ax^n \), where \( n \) is a non-negative integer. The general form of a polynomial function is:
\[ a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \]
In simpler terms, it's a sum of multiple terms, each consisting of a variable raised to an integer power and multiplied by a coefficient. For example, the function in our problem is:
\( f(x) = x^2 + 5x \)
This is a polynomial of degree 2, since the highest power of \( x \) is 2. Polynomial functions are continuous and smooth, making them easier to work with compared to other types of functions.
Simplifying Expressions
Simplifying algebraic expressions involves reducing them to their most basic form. This includes combining like terms, factoring, and canceling common factors. Let's review the steps we took to simplify our given expression:
  • First, we substituted \( c \) into the polynomial to get \( f(c) = c^2 + 5c \).
  • Next, we set up the difference quotient \( \frac{f(x) - f(c)}{x - c} = \frac{x^2 + 5x - (c^2 + 5c)}{x - c} \).
  • We then simplified the numerator: \( x^2 + 5x - c^2 - 5c \) becomes \( (x^2 - c^2) + 5(x - c) \).
  • We factored this as \( (x - c)(x + c + 5) \) and canceled \( x - c \) in the numerator and denominator.
  • Finally, we are left with the simplified form: \( x + c + 5 \).

Simplifying such expressions makes it much easier to work with them in further calculations and helps in understanding the overall behavior of the function.

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Most popular questions from this chapter

Physics A ball is thrown vertically upward with an initial velocity of 80 feet per second. The distance \(s\) (in feet) of the ball from the ground after \(t\) seconds is \(s(t)=80 t-16 t^{2}\) (a) At what time \(t\) will the ball strike the ground? (b) For what time \(t\) is the ball more than 96 feet above the ground?

(a) find the vertex and the axis of symmetry of each quadratic function, and determine whether the graph is concave up or concave down. (b) Find the y-intercept and the \(x\) -intercepts, if any. (c) Use parts (a) and (b) to graph the function. (d) Find the domain and the range of the quadratic function. (e) Determine where the quadratic function is increasing and where it is decreasing. (f) Determine where \(f(x)>0\) and where \(f(x)<0\) \(f(x)=-2 x^{2}+2 x-3\)

Challenge Problem Runaway Car Using Hooke's Law, we can show that the work \(W\) done in compressing a spring a distance of \(x\) feet from its at-rest position is \(W=\frac{1}{2} k x^{2}\) where \(k\) is a stiffness constant depending on the spring. It can also be shown that the work done by a body in motion before it comes to rest is given by \(\tilde{W}=\frac{w}{2 g} v^{2},\) where \(w=\) weight of the object (in Ib), \(g=\) acceleration due to gravity \(\left(32.2 \mathrm{ft} / \mathrm{s}^{2}\right),\) and \(v=\) object's velocity \((\mathrm{in} \mathrm{ft} / \mathrm{s})\). A parking garage has a spring shock absorber at the end of a ramp to stop runaway cars. The spring has a stiffness constant \(k=9450 \mathrm{lb} / \mathrm{ft}\) and must be able to stop a \(4000-\mathrm{lb}\) car traveling at \(25 \mathrm{mph}\). What is the least compression required of the spring? Express your answer using feet to the nearest tenth. Source: www.sciforums com

Determine algebraically whether \(f(x)=\frac{-x}{x^{2}+9}\) is even, odd, or neither.

A projectile is fired at an inclination of \(45^{\circ}\) to the horizontal, with a muzzle velocity of 100 feet per second. The height \(h\) of the projectile is modeled by $$h(x)=\frac{-32 x^{2}}{100^{2}}+x$$ where \(x\) is the horizontal distance of the projectile from the firing point. (a) At what horizontal distance from the firing point is the height of the projectile a maximum? (b) Find the maximum height of the projectile. (c) At what horizontal distance from the firing point will the projectile strike the ground? (d) Graph the function \(h, 0 \leq x \leq 350\). (e) Use a graphing utility to verify the results obtained in parts (b) and (c). (f) When the height of the projectile is 50 feet above the ground, how far has it traveled horizontally?
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