Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Given \(s(t)=-16 t^{2}+3 t,\) find the difference quotient \(\frac{s(t)-s(1)}{t-1}\)

Short answer

The difference quotient is \[ \frac{-16t^2 + 3t + 13}{t - 1} \].

Step-by-step solution

- Evaluate s(1)

First, calculate the value of the function at t = 1. Plug in 1 into the function: \[ s(1) = -16(1)^2 + 3(1) \] This simplifies to: \[ s(1) = -16 + 3 = -13 \]

- Substitute s(t) and s(1)

Now substitute the expressions for s(t) and s(1) into the difference quotient: \[ \frac{s(t) - s(1)}{t - 1} = \frac{(-16t^2 + 3t) - (-13)}{t - 1} \]

- Simplify the numerator

Simplify the expression in the numerator: \[ (-16t^2 + 3t + 13) \] So the difference quotient becomes: \[ \frac{-16t^2 + 3t + 13}{t - 1} \]

Key concepts

difference quotient

The difference quotient is a fundamental concept in calculus. It helps us find the average rate of change of a function over an interval. Essentially, it serves as a stepping stone towards understanding derivatives.
To compute the difference quotient, we use the formula: \ \ \[ \frac{f(x+h)-f(x)}{h} \ \ \] In our exercise, we use a similar form, given as: \ \ \[ \frac{s(t) - s(a)}{t - a} \ \ \] where in this case, a is replaced by 1.
Remember:

• The function value at two different points is calculated.
• The difference is taken for these points.
• It is then divided by the difference of the points themselves.

Understanding the difference quotient is crucial because it directly connects to the concept of a derivative, which represents the instantaneous rate of change.

algebraic function

An algebraic function is a function that can be defined using algebraic expressions. These expressions can involve variables, constants, and operations like addition, subtraction, multiplication, division, and raising to a power.
In the given exercise, we are dealing with an algebraic function represented by: \ \ \[ s(t) = -16t^2 + 3t \ \ \] An algebraic function usually expects:

• Substituting values directly for calculations.
• Simplifying complex expressions through operations.

For instance, calculating s(1) involves substituting t with 1 in the function and performing algebraic operations:
\