On May 7, 1992, the space shuttle Endeavour was launched on...

On May 7, 1992, the space shuttle Endeavour was launched on mission STS-49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters. (TABLE CAN'T COPY) (a) Use a graphing calculator or computer to find the cubic polynomial that best models the velocity of the shuttle for the time interval $t \in[0,125]$. Then graph this polynomial. (b) Find a model for the acceleration of the shuttle and use it to estimate the maximum and minimum values of the acceleration during the first 125 seconds.

On May 7, 1992, the space shuttle Endeavour was launched on mission STS-49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters.
(TABLE CAN'T COPY)
(a) Use a graphing calculator or computer to find the cubic polynomial that best models the velocity of the shuttle for the time interval $t \in[0,125]$. Then graph this polynomial.
(b) Find a model for the acceleration of the shuttle and use it to estimate the maximum and minimum values of the acceleration during the first 125 seconds.

Key Concepts

Polynomial Regression

Polynomial regression is a statistical technique for modeling the relationship between a dependent variable and one or more independent variables using a polynomial equation. It is commonly used to fit curves to data points, enabling smoother approximations of complex trends in the data even when the relationship is nonlinear.

Cubic Polynomial Model

A cubic polynomial model is a specific type of polynomial regression where the fitted function is a third degree polynomial. This type of model can capture changes in curvature and inflection points in the data, making it useful for modeling phenomena that exhibit non-linear growth or deceleration, such as velocity over time in dynamic systems.

Differentiation

Differentiation is a fundamental concept in calculus that measures how a function changes as its input changes. When applied to kinematics, the derivative of the velocity function with respect to time gives the acceleration function, which describes the rate of change of velocity, providing insights into the underlying forces and dynamics.

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function, often by setting its derivative equal to zero and analyzing critical points along with endpoints. This technique is essential for determining extreme behaviors in models, such as identifying the highest and lowest acceleration values over a specified interval.

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