The number of tablets and smartphones in use worldwide (in...

The number of tablets and smartphones in use worldwide (in millions) in year $t$ from 2010 through 2012 is approximately $$f(t)=128.1 t^{1.94} \quad(1 \leq t \leq 3)$$ where $t=1$ corresponds to 2010 a. How many tablets and smartphones were in use in $2011 ?$ b. How fast was the number of tablets and smartphones changing in $2011 ?$ Source: MIT Technology Review.

The number of tablets and smartphones in use worldwide (in millions) in year $t$ from 2010 through 2012 is approximately
$$f(t)=128.1 t^{1.94} \quad(1 \leq t \leq 3)$$
where $t=1$ corresponds to 2010
a. How many tablets and smartphones were in use in $2011 ?$
b. How fast was the number of tablets and smartphones changing in $2011 ?$
Source: MIT Technology Review.

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Key Concepts

Instantaneous Rate of Change

The instantaneous rate of change is given by the derivative of the function at a specific point and measures how quickly the quantity is changing at that exact moment. This concept is crucial for understanding the speed at which the number of devices is increasing during a particular year.

Differentiation

Differentiation is the process of calculating the derivative of a function, which represents the rate at which the function's output changes with respect to its input. The power rule, in particular, allows us to differentiate expressions of the form t^b by bringing down the exponent and reducing the power by one.

Function Evaluation

Evaluating a function involves substituting a specific value into the function to calculate the corresponding output. This is used to determine the number of devices in a given year by replacing the time variable with the appropriate value.

Power Functions

A power function, which has the form f(t) = a*t^b, is used to model phenomena where growth or decay is non-linear. In this case, the function models the number of devices over time, with the exponent indicating the degree of non-linearity in the growth trend.

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