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Chapter 3: Problem 58

Find the values of \(a\) such that \(f(x)=\cos x-a x+b\) is decreasing everywhere.

Short Answer

Expert verified
For the function \(f(x) = \cos x - ax + b\) to be decreasing everywhere, the value of \(a\) must be greater than or equal to \(1\).

Step by step solution

01

Find the derivative of the function

First, we need to take the derivative of the function \(f(x) = \cos x - ax + b\) with respect to \(x\). We treat \(a\) and \(b\) as constants in this case. The derivative of the cosine function is \(-\sin x\), the derivative of a constant times \(x\) is the constant itself, and the derivative of a constant is zero. Therefore, the derivative of \(f(x)\) is given by: \(f'(x) = -\sin x - a\)
02

Determine when the derivative is less than or equal to zero

We want the function to be decreasing everywhere, so the derivative should be less than or equal to zero for all values of \(x\). In other words, we want to find when the following inequality holds: \(-\sin x - a \leq 0\) Now, let's solve for \(a\).
03

Manipulate inequality to isolate \(a\)

To isolate \(a\), we add \(\sin x\) to both sides of the inequality: \(-a \leq \sin x\) Now, multiply both sides by \(-1\) to eliminate the negative sign on \(a\). Keep in mind that when multiplying an inequality by a negative value, the inequality sign must be flipped: \(a \geq -\sin x\)
04

Find the maximum value of \(-\sin x\)

We know that the sine function has a range of \([-1, 1]\), meaning that its minimum value is \(-1\) and its maximum value is \(1\). Therefore, the maximum value of \(-\sin x\) is \(1\).
05

Determine the condition for \(a\)

Since we want \(a \geq -\sin x\) and the maximum value of \(-\sin x\) is \(1\), we can conclude that \(a \geq 1\). So for the function \(f(x) = \cos x - ax + b\) to be decreasing everywhere, the value of \(a\) must be greater than or equal to \(1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivative of a Function
Understanding the concept of the derivative is crucial in calculus. It measures how a function's value changes as its input changes鈥攅ssentially, it gives the rate of change or the slope of the function at any point. For the given function, \( f(x) = \cos x - ax + b \), we identify each component of the function鈥擻(\cos x\), \(ax\), and \(b\)鈥攁nd find their derivatives separately.

The derivative of \(\cos x\) is \(-\sin x\) due to the trigonometric differentiation rule. For \(ax\), which is a linear term, the derivative is just the coefficient \(a\) because the derivative of \(x\) is 1. And since the derivative of a constant like \(b\) is 0, it disappears in the derivative. As a result, we combine these to find the overall derivative: \(f'(x) = -\sin x - a\), which we use for further analysis on the function's behavior.
First Derivative Test
The First Derivative Test is a powerful tool in calculus to determine whether a function is increasing or decreasing on certain intervals. A function is deduced to be decreasing where its first derivative is less than zero. For our function \(f(x)\), the derivative \(f'(x) = -\sin x - a\) tells us how the function behaves as \(x\) changes.

By setting the derivative less than or equal to zero, \( -\sin x - a \leq 0 \) , we can find the conditions on \(a\) that make the function decrease everywhere. This provides a precise method to assess how the parameter \(a\) affects the slope of the tangent line at any point on the graph of \(f(x)\), thereby offering insight into the overall trend of the function.
Solving Inequalities in Calculus
In calculus, solving inequalities often entails finding the range of values for a variable that satisfies a certain condition. In our exercise, we want to ensure that \(f'(x) \), the rate of change of \(f(x)\), remains negative or zero to guarantee that \(f(x)\) is decreasing everywhere. The inequality \( -\sin x - a \leq 0 \) leads us to isolate \(a\) by performing valid algebraic manipulations that maintain the inequality's direction.

We implement a critical step when we multiply by \(-1\), flipping the inequality to get \(a \geq -\sin x\). Since the sine function oscillates between \(-1\) and \(1\), the largest value \(-\sin x\) can attain is \(1\), which sets the condition for \(a\): it must be at least \(1\). This process not only provides the value of \(a\) for a decreasing function but also exemplifies how to maneuver through inequalities鈥攁 fundamental skill in calculus.

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Most popular questions from this chapter

Terminal Velocity A skydiver leaps from the gondola of a hotair balloon. As she free-falls, air resistance, which is proportional to her velocity, builds up to a point at which it balances the force due to gravity. The resulting motion may be described in terms of her velocity as follows: Starting at rest (zero velocity), her velocity increases and approaches a constant velocity, called the terminal velocity. Sketch a graph of her velocity \(v\) versus time \(t\).

Traffic Flow Analysis The speed of traffic flow in miles per hour on a stretch of Route 123 between 6 A.M. and 10 A.M. on a typical workday is approximated by the function $$ f(t)=20 t-40 \sqrt{t}+52 \quad 0 \leq t \leq 4 $$ where \(t\) is measured in hours and \(t=0\) corresponds to 6 A.M. Sketch the graph of \(f\) and interpret your results.

Optimal Inventory Control The equation $$ A(q)=\frac{k m}{q}+c m+\frac{h q}{2} $$ gives the annual cost of ordering and storing (as yet unsold) merchandise. Here, \(q\) is the size of each order, \(k\) is the cost of placing each order, \(c\) is the unit cost of the product, \(m\) is the number of units of the product sold per year, and \(h\) is the annual cost for storing each unit. Determine the size of each order such that the annual cost \(A(q)\) is as small as possible.

An observer stands on a straight path that is parallel to a straight test track. At \(t=0\) a Formula 1 car is directly opposite her and \(200 \mathrm{ft}\) away. As she watches, the car moves with a constant acceleration of \(20 \mathrm{ft} / \mathrm{sec}^{2}\), so it is at a distance of \(10 t^{2} \mathrm{ft}\) from the starting position after \(t \mathrm{sec}\), where \(0 \leq t \leq 15 .\) As the car moves, \(d \theta / d t\) increases, slowly at first, then faster, and finally it slows down again. The observer perceives the car to be moving at the fastest speed when \(d \theta / d t\) is maximal. Determine the position of the car at the moment she perceives it to be moving at the fastest speed.

Crime Rate The number of major crimes per 100,000 people committed in a city from the beginning of 2002 to the beginning of 2009 is approximated by the function $$ N(t)=-0.1 t^{3}+1.5 t^{2}+80 \quad 0 \leq t \leq 7 $$ where \(N(t)\) denotes the number of crimes per 100,000 people committed in year \(t\) and \(t=0\) corresponds to the beginning of 2002. Enraged by the dramatic increase in the crime rate, the citizens, with the help of the local police, organized Neighborhood Crime Watch groups in early 2007 to combat this menace. Sketch the graph of the function \(N\), and interpret your results. Is the Neighborhood Crime Watch program working?
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