Question

Find the function The following limits represent the slope of a curve \(y=f(x)\) at the point \((a, f(a)) .\) Determine a possible function \(f\) and number \(a ;\) then calculate the limit. $$\lim _{x \rightarrow 2} \frac{5 x^{2}-20}{x-2}$$

Short answer

Answer: The possible function f(x) is $$f(x) = \frac{5}{2}x^2 + 5x + C$$. The number a is 2, and the limit is 20.

Step-by-step solution

Identify the limit expression and type

We are given the limit expression: $$\lim_{x \rightarrow 2} \frac{5 x^{2}-20}{x-2}.$$ From this, we can identify that this is the limit as x approaches 2 and that we are dealing with a limit of the form: $$ \lim_{x \rightarrow a} \frac{f(x) - f(a)}{x - a} $$

Identify the possible function f(x) and number a

Since the limit expression represents the slope of the curve, it gives us the derivative of the function. We are given: $$ \lim_{x \rightarrow 2} \frac{5 x^{2}-20}{x-2} $$ We notice that the numerator is a difference of squares, so we can factor it as follows: $$ 5x^2 - 20 = 5(x^2 - 4) = 5(x - 2)(x + 2) $$ Now our limit expression becomes: $$ \lim_{x \rightarrow 2} \frac{5(x-2)(x+2)}{x-2} $$ We can cancel out the (x-2) factors: $$ \lim_{x \rightarrow 2} 5(x+2) $$ Now our expression is in the form of the limit as x approaches a, from which the derivative of the function has been derived. Given the above limit, the possible function f(x) can be the antiderivative of the simplified limit expression: $$ f'(x) = 5(x+2) $$ Taking the antiderivative, we find f(x): $$ f(x) = \frac{5}{2}x^2 + 5x + C $$ Where C is a constant. As for the number a, we can see that the limit expression is defined when x approaches 2. Hence, a = 2.

Calculate the given limit

Since we have already simplified the limit expression and found the function f(x), we can now evaluate the limit: $$ \lim_{x \rightarrow 2} 5(x+2) $$ By substituting x = 2, we find: $$= 5(2+2) = 5(4) = 20$$ Thus, the limit is equal to 20. In summary, the possible function f(x) is: $$ f(x) = \frac{5}{2}x^2 + 5x + C $$ The number a is 2, and the limit is 20.

Key concepts

Limit Definition of Derivative

The limit definition of a derivative provides us with a powerful way to define the slope of a tangent line to a curve at any given point. When you see the expression:

• \( \lim_{x \rightarrow a} \frac{f(x) - f(a)}{x - a} \)

it might seem a bit complex at first, but its goal is simple. It's about finding how much a function changes as you make tiny shifts around a particular point. Here, \(a\) is that special point we're interested in, and this expression is our foundation for determining the derivative, which is simply the slope at point \((a, f(a))\).
By calculating this limit, we develop our understanding of how a function behaves precisely at a moment rather than over an interval. This foundational concept is crucial for all of calculus.

Differentiation

Differentiation is essentially the process of finding the derivative of a function. In simpler terms, it's about finding the rate at which one quantity changes with respect to another. Suppose you've formed an equation from the limit process like we did:

• \( \frac{5(x-2)(x+2)}{x-2} \)

You realize that the \((x-2)\) terms cancel out, simplifying the expression. What's left? Just the direct, straightforward derivative:
5(x+2).
Differentiation has countless applications: predicting the growth of a population, measuring the speed of a moving car, or even determining your coffee's cooling rate. Each time, we seek how rapidly something is changing at any moment.

Continuity

Continuity is an essential concept when talking about limits and derivatives because for a function to have a derivative at a point, it must first be continuous at that point. A function is continuous at a specific value of x if there is no sudden jump or break in its graph at that point.
Continuity can be easily understood with a simple rule: if you can draw the graph of a function without lifting your pencil, it's continuous. Now, why do we care about this in calculus? Because if a function isn't continuous at a given point, the limit, and therefore the derivative, may not exist there.
In a nutshell, continuity assures us that our function behaves well and predictably, paving the way for us to explore its derivative.

Polynomial Functions

Polynomial functions are among the simplest yet most vital types of functions you'll encounter in calculus. They are expressions made up of variables and coefficients, where the only arithmetic operations are addition, subtraction, multiplication, and non-negative integer exponentiation of variables.

• Example: \(5x^2, 3x + 2,\) and \(x^3 - 4x + 7\)

Features of polynomial functions make them a favorite in calculus. They're easy to differentiate because we can apply basic rules like the power rule. For instance, the derivative of \(5x^2\) is \(10x\).
Understanding polynomial functions is crucial as they often serve as the building blocks for more complex equations and are frequently used to approximate other functions' behavior. These functions are crucial in modeling real-world phenomena, from physics to economics, making them indispensable in any calculus toolkit.