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Chapter 2: Problem 14

Sketch the graph of \(f(x),\) and use this graph to sketch the graph of \(f^{\prime}(x).\) $$f(x)=x^{2}$$

Short Answer

Expert verified
The graph of \(f(x)=x^2\) is a parabola, and \(f^{\prime}(x)=2x\) is a straight line.

Step by step solution

01

Sketch the Graph of \(f(x)\)

Begin by sketching the graph of the function \( f(x) = x^2 \). This is a standard parabola opening upwards with the vertex at the origin \((0,0)\). The graph is symmetric about the y-axis, and as \(x\) moves away from the origin in either direction, \(f(x)\) increases.
02

Determine \(f^{ ext{'}(x)}\)

To sketch the derivative, we first find \(f^{ ext{'}(x)}\). The derivative of \( f(x) = x^2 \) is found using the power rule: \[ f^{ ext{'}(x)} = 2x. \] This tells us the slope of the tangent line to the curve at each point \(x\).
03

Sketch the Graph of \(f^{ ext{'}(x)}\)

Now sketch \(f^{ ext{'}(x)} = 2x\). This graph is a straight line passing through the origin with a slope of 2. The slope of the tangent line \(f^{ ext{'}(x)}\) is positive for \(x > 0\) and negative for \(x < 0\). The line itself is symmetric about the origin.

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Key Concepts

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

Parabola
A parabola is a U-shaped curve that is commonly seen in quadratic functions. In this case, the function is given by \( f(x) = x^2 \), which represents a standard parabola that opens upwards. This shape is symmetric, meaning that if you fold the parabola along its vertical axis of symmetry, both sides would match perfectly.

The vertex, which is the lowest point when the parabola opens upwards, is at the origin \((0,0)\) for the function \( f(x) = x^2 \).
As \( x \) moves away from zero, the value of \( f(x) \) increases.
  • At \( x = 0 \), the value of \( f(x) \) is 0.
  • For positive \( x \), \( f(x) \) increases at a squared rate.
  • For negative \( x \), the function value also increases but remains positive as it is squared.
Understanding the basic structure of a parabola helps in predicting how it behaves as input values change.
Derivative
Derivatives are vital in calculus as they represent the concept of a function's rate of change or the slope of the tangent line at any given point. For a function like \( f(x) = x^2 \), finding its derivative, denoted \( f'(x) \), involves understanding how the function behaves as \( x \) varies.

The derivative \( f'(x) = 2x \) shows that the slope changes at a rate proportional to \( x \).
  • For \( x > 0 \), \( f'(x) \) is positive, indicating an upward slope.
  • For \( x < 0 \), \( f'(x) \) is negative, indicating a downward slope.
  • At \( x = 0 \), the slope is zero.
This derivative tells us how steep the curve of \( f(x) \) is at any point, and it's essential for sketching the tangent line graph.
Power Rule
The power rule is a basic and powerful tool in calculus that simplifies finding derivatives of functions involving powers of \( x \). It states: for any function \( f(x) = x^n \), the derivative \( f'(x) \) is \( n \times x^{n-1} \).

Applying the power rule to \( f(x) = x^2 \), we get:
  • Take the power, which is 2, and multiply it by the coefficient (in this case 1, if not stated). This results in \( 2 \times x^{2-1} = 2x \).
  • This process simplifies the task of differentiation, especially when handling polynomials.
Using the power rule effectively allows quick and accurate computation of derivatives, a fundamental step in calculus for analyzing the behavior of functions.

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Most popular questions from this chapter

Are the statements true or false? If a statement is true, give an example illustrating it. If a statement is false, give a counterexample. If a function is continuous, then it is differentiable.

Give an example of:A continuous function which is always increasing and positive.

In May 2007 in the US, there was one birth every 8 seconds, one death every 13 seconds, and one new international migrant every 27 seconds. \(^{7}\) (a) Let \(f(t)\) be the population of the US, where \(t\) is time in seconds measured from the start of May 2007 Find \(f^{\prime}(0) .\) Give units. (b) To the nearest second, how long did it take for the US population to add one person in May \(2007 ?\)

Let \(f(t)\) be the number of centimeters of rainfall that has fallen since midnight, where \(t\) is the time in hours. Interpret the following in practical terms, giving units. (a) \(\quad f(10)=3.1\) (b) \(\quad f^{-1}(5)=16\) (c) \(\quad f^{\prime}(10)=0.4\) (d) \(\quad\left(f^{-1}\right)^{\prime}(5)=2\)

Chlorofluorocarbons (CFCs) were used as propellants in spray cans until their build up in the atmosphere started destroying the ozone, which protects us from ultraviolet rays. since the 1987 Montreal Protocol (an agreement to curb CFCs), the CFCs in the atmosphere above the US have been reduced from a high of 3200 parts per trillion (ppt) in 1994 to 2750 ppt in \(2010 .^{15}\) The reduction has been approximately linear. Let \(C(t)\) be the concentration of CFCs in ppt in year \(t\) (a) Find \(C(1994)\) and \(C(2010)\) (b) Estimate \(C^{\prime}(1994)\) and \(C^{\prime}(2010)\) (c) Assuming \(C(t)\) is linear, find a formula for \(C(t)\) (d) When is \(C(t)\) expected to reach 1850 ppt, the level before CFCs were introduced? (e) If you were told that in the future, \(C(t)\) would not be exactly linear, and that \(C^{\prime \prime}(t)>0,\) would your answer to part (d) be too early or too late?
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