Textbook Question
Find an equation of the line tangent to the following curves at the given value of x.
y = 4 sin x cos x; x = π/3
2. Intro to Derivatives
Tangent Lines and Derivatives
4:22 minutesProblem 99c
Textbook Question
The population of the United States (in millions) by decade is given in the table, where t is the number of years after 1910. These data are plotted and fitted with a smooth curve y = p(t) in the figure. <IMAGE><IMAGE>
Estimate the instantaneous rate of growth in 1985.
Verified step by step guidance1
Identify the function y = p(t) that models the population growth. This function is given as a smooth curve fitting the data points from the table.
Determine the year 1985 in terms of t, where t is the number of years after 1910. Calculate t by subtracting 1910 from 1985, which gives t = 75.
To estimate the instantaneous rate of growth at t = 75, you need to find the derivative of the function y = p(t), denoted as p'(t). The derivative represents the rate of change of the population with respect to time.
Evaluate the derivative p'(t) at t = 75. This will give you the instantaneous rate of growth of the population in millions per year at the year 1985.
Interpret the result: The value of p'(75) will indicate how fast the population was growing at that specific year, providing insight into the demographic trends during that period.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Instantaneous Rate of Change
The instantaneous rate of change of a function at a given point is defined as the derivative of the function at that point. It represents how the function's output value changes as the input value changes infinitesimally. In the context of population growth, this means calculating the derivative of the population function p(t) at t corresponding to the year 1985.
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Derivatives
A derivative is a fundamental concept in calculus that measures how a function changes as its input changes. It is often denoted as f'(x) or dp/dt for a function p(t). To estimate the instantaneous rate of growth of the population in 1985, one would need to compute the derivative of the population function p(t) at the specific value of t that represents that year.
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Population Growth Models
Population growth models are mathematical representations that describe how a population changes over time. These models can be linear, exponential, or logistic, depending on the factors influencing growth. Understanding the model used to fit the population data is crucial for accurately estimating the instantaneous rate of growth, as it determines the form of the function p(t) and its derivative.
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