Question

A function \(f\) and \(a\) value \(a\) are given. Approximate the limit of the difference quotient, \(\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h},\) using \(h=\pm 0.1, \pm 0.01 .\) $$ f(x)=-7 x+2, \quad a=3 $$

Short answer

The limit of the difference quotient is -7.

Step-by-step solution

Identify the Formula

The difference quotient is given by \( \frac{f(a+h)-f(a)}{h} \). We need to evaluate this expression for different values of \( h \).

Evaluate \( f(a) \)

Substitute \( a = 3 \) into the function \( f(x) = -7x + 2 \). This gives us \( f(3) = -7(3) + 2 = -21 + 2 = -19 \).

Calculate \( f(a+h) \) for \( h = 0.1 \)

Substitute \( a + h = 3.1 \) into the function. \( f(3.1) = -7(3.1) + 2 = -21.7 + 2 = -19.7 \).

Calculate \( \frac{f(3.1) - f(3)}{0.1} \)

The difference quotient is \( \frac{-19.7 - (-19)}{0.1} = \frac{-0.7}{0.1} = -7 \).

Calculate \( f(a+h) \) for \( h = -0.1 \)

Substitute \( a + h = 2.9 \) into the function. \( f(2.9) = -7(2.9) + 2 = -20.3 + 2 = -18.3 \).

Calculate \( \frac{f(2.9) - f(3)}{-0.1} \)

The difference quotient is \( \frac{-18.3 - (-19)}{-0.1} = \frac{0.7}{-0.1} = -7 \).

Calculate \( f(a+h) \) for \( h = 0.01 \)

Substitute \( a + h = 3.01 \) into the function. \( f(3.01) = -7(3.01) + 2 = -21.07 + 2 = -19.07 \).

Calculate \( \frac{f(3.01) - f(3)}{0.01} \)

The difference quotient is \( \frac{-19.07 - (-19)}{0.01} = \frac{-0.07}{0.01} = -7 \).

Calculate \( f(a+h) \) for \( h = -0.01 \)

Substitute \( a + h = 2.99 \) into the function. \( f(2.99) = -7(2.99) + 2 = -20.93 + 2 = -18.93 \).

Calculate \( \frac{f(2.99) - f(3)}{-0.01} \)

The difference quotient is \( \frac{-18.93 - (-19)}{-0.01} = \frac{0.07}{-0.01} = -7 \).

Determine Limit of Difference Quotient

Since all computed difference quotients for different values of \( h \) are \( -7 \), the limit of the difference quotient as \( h \rightarrow 0 \) is \( -7 \).

Key concepts

Limit Calculation

Understanding limit calculations is crucial when working with difference quotients. In this exercise, the goal is to approximate the limit of \( \lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h} \). When \( h \) approaches zero, you're effectively finding the slope of the tangent line (or the derivative) at a specific point. This kind of limit helps us better understand the behavior of functions at specific points, especially how they grow or shrink.

To calculate this limit, substituting small values for \( h \) (like 0.1 and 0.01) is key. By observing the results of these substitutions, you can estimate the limit as \( h \) gets infinitely close to zero. This approach forms the foundation for understanding continuity and smoothness of function graphs. Each substitution gives a clue about what happens at the exact point \( a \).

Remember:

• Limits describe how a function behaves as it approaches a certain point from both sides.
• Calculating the limit of a difference quotient enables finding a curve's instantaneous rate of change at a given point.

Derivative Approximation

The derivative of a function at a point \( a \) can be approximated using the limit of the difference quotient. In this exercise, we used different values like \( h = 0.1 \) and \( h = 0.01 \) to get closer to the actual derivative.

The derivative gives insight into how a function's output value changes concerning changes in input values. Essentially, it's the slope of the function at a particular point. By approximating the derivative using the difference quotient, we visually approximate the tangent's slope at that point.

As smaller values of \( h \) are used, the approximation aligns closer to the actual derivative, which, through precise math, was determined to be -7 in this case. Thus, at point \( a = 3 \), the function \( f(x)=-7x+2 \) has a constant rate of change (slope) of -7, which is an indication of how steep the line is at that point.

Step-by-Step Differentiation

Differentiation can initially seem complex, but breaking it down into steps makes it more manageable. In this context, step-by-step differentiation involves using small increments \( h \) to methodically approach the derivative of a function.

The process begins with identifying the function's value at the point of interest. Critical steps include plugging in values either slightly above or below the point. You then compute the difference quotient for each substitution. This method approximates the derivative through straightforward arithmetic while observing how closely each result hones in on a fixed value.

In our example, the derivative of \( f(x) = -7x + 2 \) at \( a = 3 \) was calculated using sequential values of \( h \). Every computation returned -7, affirming that with decreasing \( h \), our result honed onto this true derivative. By dealing with these smaller calculations and adjustments, differentiation becomes less daunting and more approachable for students and beginners. It's a methodical journey to uncover the precise rate of change at any chosen point on a curve.