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These questions involve evaluating velocities, rates of change, and derivatives of functions representing real-world phenomena, such as an object's movement along an axis and the cost and rate of change in a production process. The tasks include finding average velocities over specific intervals, calculating instantaneous velocities at particular points, determining the average and instantaneous rates of change of a cost function, and computing difference quotients for polynomial functions. These exercises probe understanding of foundational calculus concepts such as average velocity, instantaneous velocity, difference quotient, and derivatives, which are key to analyzing dynamic systems and functions in mathematics and applied sciences.
The problems are structured to test the ability to interpret function-based models, perform algebraic manipulations, and apply calculus principles to compute rates of change both over intervals and at specific points. Additionally, the questions help to develop intuition about the properties of derivatives by examining the limiting process inherent in the difference quotient and understanding how the derivative relates to a tangent line’s slope at a point.
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The first problem introduces a function y = 4x^2 + 8x, describing an object moving along the y-axis where x represents time in seconds, and y represents position in meters. To find the average velocity over an interval, we employ the difference quotient formula for average rate of change: (change in y) / (change in x). Specifically, for x changing from 2 to 9 seconds, the average velocity is computed as:
$$
\text{Average velocity} = \frac{y(9) - y(2)}{9 - 2}
$$ which simplifies to the calculation of y at specific points and division by the interval length, providing an average rate of change in meters per second.
For the interval from 5 to 5+h seconds, the same formula applies: the difference in y-values at points x = 5 and x = 5+h, divided by h. This gives the average velocity over that small interval, which approaches the instantaneous velocity as h approaches zero.
To find the instantaneous velocity at x=5 seconds, the limit of the average velocity as h approaches zero is calculated via the derivative of y with respect to x. Since y = 4x^2 + 8x, the derivative y' = 8x + 8 is used at x=5, yielding the instantaneous velocity.
The second problem involves a cost function C(x) = 6500 + 19x + 0.02x^2, modeling cost in dollars. Computing the average rate of change over specific intervals involves evaluating the difference quotient for C(x) at given x-values. For instance, from x=100 to x=105, the average rate of change is:
$$ \frac{C(105) - C(100)}{105 - 100}
$$
similarly with other bounds. The instantaneous rate of change at x=100, known as the marginal cost, is obtained by differentiating C(x) with respect to x and evaluating at x=100.
The third problem concerns the function f(x) = x^3 - 2x and asks for the difference quotient f(3+h) - f(3) for various small values of h. Computing these involves algebraic expansion and substitution, illustrating the process of approaching the derivative as h tends to zero. Examining the limiting behavior of this difference quotient as h approaches zero enables inference about the derivative at the point x=3.
The subsequent problems follow similar patterns with different functions and points, reinforcing the understanding of the derivative as the limit of the difference quotient and as the slope of the tangent line at a specific point. For example, for the function f(x) = 1, the difference quotient involves constants and their simplifications, with the derivative expected to be zero since the function is constant. For f(x) = x + 5, the difference quotient simplifies to 1 regardless of small h, indicating that the derivative of a linear function with slope 1 is constant everywhere.
Overall, these exercises highlight essential calculus concepts: the computation of average and instantaneous rates of change, the use of limits to define derivatives, and the interpretation of derivatives as slopes of tangent lines. Mastery of these topics allows for modeling and analyzing various physical and economical systems where rates of change are crucial for understanding behavior over time or across different conditions.
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