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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.6.86b
Chapter 3, Problem 3.6.86b

Temperatures in Fairbanks, Alaska The graph in the accompanying figure shows the average Fahrenheit temperature in Fairbanks, Alaska, during a typical 365-day year. The equation that approximates the temperature on day x is


y = 37 sin[(2π/365)(x − 101)] + 25


and is graphed in the accompanying figure.


b. About how many degrees per day is the temperature increasing when it is increasing at its fastest?


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Verified step by step guidance
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Step 1: To determine the rate at which the temperature is increasing at its fastest, we need to find the derivative of the given temperature function y = 37 sin[(2π/365)(x − 101)] + 25. The derivative represents the rate of change of temperature with respect to time (x).
Step 2: Differentiate the function y with respect to x. Using the chain rule, the derivative of y = 37 sin[(2π/365)(x − 101)] + 25 is given by dy/dx = 37 * cos[(2π/365)(x − 101)] * (2π/365). The constant 25 disappears because its derivative is zero.
Step 3: Simplify the derivative expression. The derivative becomes dy/dx = (74π/365) * cos[(2π/365)(x − 101)]. This represents the rate of change of temperature at any given day x.
Step 4: To find the maximum rate of increase, observe that the cosine function achieves its maximum value of 1. Substitute cos[(2π/365)(x − 101)] = 1 into the derivative expression. This gives the maximum rate of change as dy/dx = (74π/365).
Step 5: Interpret the result. The maximum rate of temperature increase occurs when the cosine term equals 1, which corresponds to the steepest part of the graph. The value of (74π/365) represents the maximum degrees per day the temperature is increasing.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative of a Function

The derivative of a function represents the rate at which the function's value changes with respect to its input. In the context of the temperature function, the derivative will indicate how quickly the temperature is increasing or decreasing on a given day. Calculating the derivative of the given sinusoidal function will help determine the rate of temperature change.
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Derivatives of Other Trig Functions

Sinusoidal Functions

Sinusoidal functions, such as sine and cosine, are periodic functions that oscillate between a maximum and minimum value. The given temperature function is a sine function, which models the cyclical nature of temperature changes over a year. Understanding the properties of sinusoidal functions, including amplitude, period, and phase shift, is crucial for analyzing the temperature pattern in Fairbanks.
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Properties of Functions

Critical Points and Maximum Rate of Change

Critical points occur where the derivative of a function is zero or undefined, indicating potential maxima, minima, or points of inflection. To find when the temperature increases at its fastest, we need to identify the point where the derivative reaches its maximum positive value. This involves analyzing the derivative to find where the rate of change is greatest, which corresponds to the steepest slope on the graph.
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Critical Points
Related Practice
Textbook Question

Consider the function


f(x) = { x² cos(2/x), x ≠ 0

0, x = 0


b. Determine f' for x ≠ 0.

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Textbook Question

The radius r of a circle is measured with an error of at most 2%. What is the maximum corresponding percentage error in computing the circle’s


b. area?

Textbook Question

Right circular cylinder The total surface area S of a right circular cylinder is related to the base radius r and height h by the equation S = 2πr² + 2πrh.


b. How is dS/dt related to dh/dt if r is constant?

Textbook Question

Computer Explorations


Use a CAS to perform the following steps in Exercises 55–62.


b. Using implicit differentiation, find a formula for the derivative dy/dx and evaluate it at the given point P.


x√(1 + 2y) + y = x², P(1,0)

Textbook Question

Generalizing the Product Rule The Derivative Product Rule gives the formula


d/dx (uv) = u (dv/dx) + (du/dx) v


for the derivative of the product uv of two differentiable functions of x.


b. What is the formula for the derivative of the product u₁u₂u₃u₄ of four differentiable functions of x?

Textbook Question

Area The area A of a triangle with sides of lengths a and b enclosing an angle of measure θ is

A = (1/2) ab sinθ.


b. How is dA/dt related to dθ/dt and da/dt if only b is constant?