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Chapter 3: Problem 2

Complete the following statement. If \(\frac{d y}{d x}\) is large, then small changes in \(x\) result in relatively __________ changes in the value of \(y\).

Short Answer

Expert verified
Question: If the derivative \(\frac{dy}{dx}\) is large, then small changes in \(x\) result in relatively ______ changes in the value of \(y\). Answer: large

Step by step solution

01

Understanding the concept of the derivative

The derivative \(\frac{dy}{dx}\) represents the rate of change of a function. In other words, the derivative tells us how \(y\) changes as we change \(x\).
02

Considering the magnitude of the derivative

If the derivative \(\frac{dy}{dx}\) is large, it means that \(y\) changes very rapidly with respect to \(x\). In this case, we need to determine whether small changes in \(x\) result in large or small changes in \(y\).
03

Relating changes in \(x\) to changes in \(y\)

Since the derivative is the rate of change of \(y\) with respect to \(x\), when the derivative is large, small changes in \(x\) will result in relatively large changes in \(y\). This is because the function is changing more rapidly, and even a small change in \(x\) will cause a comparatively larger change in \(y\).
04

Filling in the blank

So the correct word to fill in the blank is "large." The statement should read, "If \(\frac{dy}{dx}\) is large, then small changes in \(x\) result in relatively large changes in the value of \(y\)."

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Key Concepts

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

Derivative
The derivative of a function represents the engine behind understanding motion, growth, and change across a myriad of disciplines including physics, economics, biology, and more. It holds a profound application in describing the exact rate at which one quantity changes with respect to another. Imagine you are driving and your speedometer is broken, but you have a way to see how quickly the landscape is passing by; the derivative is like a mathematical speedometer telling you the rate of change of scenery as you drive. Simply put, in calculus, if we denote a function as 'y' which depends on 'x', then the derivative, symbolized by \(\frac{dy}{dx}\), quantifies precisely how y varies as x changes. It's a fundamental concept that converts the abstract idea of change into a precise mathematical quantity, allowing for predictions and deep understanding of natural phenomena.
Rate of Change
Rate of change is a concept central to understanding the dynamic world around us. It describes how a quantity, such as distance or temperature, varies over time or in relation to another variable. In a practical context, the rate of change can be observed when you fill a balloon with air and notice how rapidly it expands. The balloon’s expansion per second is its rate of change in size with respect to time. In mathematical terms, if we observe how a function 'y' depends on a variable 'x', the rate of change is how much 'y' will increase or decrease when 'x' is altered by a certain amount. This rate is not always constant; it can accelerate or decelerate, and the study of this variable rate is precisely what differential calculus elaborates upon.
Differential Calculus
Differential calculus is the branch of mathematics that deals with the calculation of derivatives and their application in solving problems involving rates of change. It is like the grammar rules for the language of change, providing a structured framework for understanding how different functions behave as their inputs vary. One common real-life application is determining the acceleration of a car at any moment during its journey; differential calculus enables us to predict the car's future position based on its current speed and acceleration patterns. The tools from differential calculus allow us to calculate slopes of curves, find maximum and minimum values of functions, and understand how changes in one variable can lead to changes in another, which are all key in tackling complex real-world problems.
Rate of Change of a Function
The rate of change of a function is like the heartbeat of calculus, it measures how quickly the function's output value 'y' changes as the input value 'x' is varied. This concept is akin to analyzing how fast the water level in a tank drops as water is poured out at variable speeds. If the function changes rapidly as the input changes, it indicates a steep curve on a graph, where a tiny shift in 'x' can lead to a substantial climb or descent in 'y'. The power of calculus lies in its ability to capture the rate of change at any instant – not just on average over an interval – which has profound implications in physics, economics, biology, and more. Thus, by mastering the concept of the rate of change of a function, one holds the key to unlocking the dynamic behaviors observed in countless applications across multiple fields.

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Most popular questions from this chapter

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas \(y=c x^{2}\) form orthogonal trajectories with the family of ellipses \(x^{2}+2 y^{2}=k,\) where \(c\) and \(k\) are constants (see figure). Find \(d y / d x\) for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. \(y=m x ; x^{2}+y^{2}=a^{2},\) where \(m\) and \(a\) are constants

General logarithmic and exponential derivatives Compute the following derivatives. Use logarithmic differentiation where appropriate. $$\frac{d}{d x}\left(1+x^{2}\right)^{\sin x}$$

Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure they actually lie on the curve. Confirm your results with a graph. $$x^{2}(y-2)-e^{y}=0$$

Suppose \(f\) is differentiable on an interval containing \(a\) and \(b\), and let \(P(a, f(a))\) and \(Q(b, f(b))\) be distinct points on the graph of \(f\). Let \(c\) be the \(x\) -coordinate of the point at which the lines tangent to the curve at \(P\) and \(Q\) intersect, assuming that the tangent lines are not parallel (see figure). a. If \(f(x)=x^{2},\) show that \(c=(a+b) / 2,\) the arithmetic mean of \(a\) and \(b\), for real numbers \(a\) and \(b\) b. If \(f(x)=\sqrt{x}\), show that \(c=\sqrt{a b}\), the geometric mean of \(a\) and \(b\), for \(a > 0\) and \(b > 0\) c. If \(f(x)=1 / x,\) show that \(c=2 a b /(a+b),\) the harmonic mean of \(a\) and \(b,\) for \(a > 0\) and \(b > 0\) d. Find an expression for \(c\) in terms of \(a\) and \(b\) for any (differentiable) function \(f\) whenever \(c\) exists.

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas \(y=c x^{2}\) form orthogonal trajectories with the family of ellipses \(x^{2}+2 y^{2}=k,\) where \(c\) and \(k\) are constants (see figure). Find \(d y / d x\) for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. \(y=c x^{2} ; x^{2}+2 y^{2}=k,\) where \(c\) and \(k\) are constants
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