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Chapter 3: Problem 2

Using the definition, calculate the derivatives of the functions. $$F(x)=(x-1)^{2}+1 ; \quad F^{\prime}(-1), F^{\prime}(0), F^{\prime}(2)$$

Short Answer

Expert verified
The derivatives are \( F'(-1) = -4 \), \( F'(0) = -2 \), and \( F'(2) = 2 \).

Step by step solution

01

Identify the Function

The given function is \( F(x) = (x-1)^2 + 1 \). We need to find its derivative using the definition of the derivative.
02

Recall the Definition of Derivative

The derivative of a function \( F(x) \) at a point \( a \) is given by \( F'(a) = \lim_{h \to 0} \frac{F(a + h) - F(a)}{h} \). We'll use this formula to find \( F'(-1) \), \( F'(0) \), and \( F'(2) \).
03

Differentiate Using the Definition 鈥� General Derivative

We first compute the general derivative. Using the definition, consider \( F(x + h) = ((x+h)-1)^2 + 1 \) which simplifies to \( ((x-1) + h)^2 + 1 = ((x-1)^2 + 2h(x-1) + h^2 + 1) \). The difference \( F(x+h) - F(x) = 2h(x-1) + h^2 \). Hence, the difference quotient is \( \frac{F(x+h) - F(x)}{h} = 2(x-1) + h \). Taking the limit as \( h \to 0 \), we get \( F'(x) = 2(x-1) \).
04

Calculate F'(-1)

Substitute \( x = -1 \) into \( F'(x) = 2(x-1) \). So, \( F'(-1) = 2(-1-1) = 2(-2) = -4 \).
05

Calculate F'(0)

Substitute \( x = 0 \) into \( F'(x) = 2(x-1) \). So, \( F'(0) = 2(0-1) = 2(-1) = -2 \).
06

Calculate F'(2)

Substitute \( x = 2 \) into \( F'(x) = 2(x-1) \). So, \( F'(2) = 2(2-1) = 2(1) = 2 \).

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Key Concepts

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

Definition of Derivative
The derivative of a function at a certain point provides us with the rate at which the function's value changes as its input changes. In essence, it offers a snapshot of the function's behavior at a precise moment, much like a speedometer tells us how fast a car is going at an instant in time.

The definition of the derivative is formally expressed as a limit:
  • For a function \( F(x) \), the derivative at point \( a \) is given by \( F'(a) = \lim_{h \to 0} \frac{F(a + h) - F(a)}{h} \).
  • This formula looks at the ratio of the change in the function value to the change in input, as the change in input (\( h \)) approaches zero.
This foundational concept allows us to "zoom in" on a graph to see whether the curve is going up, going down, or lying flat at the point \( a \).
Limit Process
The limit process is a fundamental part of finding a derivative. It mathematically expresses the idea of getting infinitely close to a point. When evaluating the limit it approaches an incredibly small distance, which helps in precisely calculating the derivative without actually ever touching the undefined points.

To understand the limit process in the definition of the derivative, it helps to follow these steps:
  • First, substitute \( a + h \) into the function to evaluate \( F(a + h) \).
  • Next, find the difference \( F(a + h) - F(a) \).
  • Divide this difference by \( h \) to get the difference quotient.
  • Finally, let \( h \to 0 \) and find the limit of the difference quotient. This will yield \( F'(a) \).
This process is at the heart of differentiating any function, as it focuses on precise changes at exact points. It's like squinting really hard to see the exact slope of a curvy road.
Differentiation
Differentiation is the process of finding a derivative. It turns a complex function into a much simpler expression that reveals the rate at which something changes. Differentiating functions is a critical tool in many fields, such as physics, engineering, and economics, where it鈥檚 necessary to understand change.

Here's a simplified walkthrough of the differentiation process using the definition:
  • For \( F(x) = (x-1)^2 + 1 \), we found its derivative \( F'(x) = 2(x-1) \).
  • By replacing \( x \) in the general derivative \( F'(x) = 2(x-1) \) with specific points, such as \(-1, 0,\) and \(2\), we calculated \( F'(-1), \) \( F'(0), \) and \( F'(2) \).
  • For instance:
    • \( F'(-1) = 2(-1-1) = -4\)
    • \( F'(0) = 2(0-1) = -2\)
    • \( F'(2) = 2(2-1) = 2\)
Through these steps, you see differentiation in action as it transforms functions for further analysis and application.

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Most popular questions from this chapter

Suppose that \(f(x)=x^{2}\) and \(g(x)=|x| .\) Then the Suppose that \(f(x)=x^{2}\) and \(g(x)=|x| .\) Then the composites $$(f \circ g)(x)=|x|^{2}=x^{2} \quad \text { and } \quad(g \circ f)(x)=\left|x^{2}\right|=x^{2}$$ are both differentiable at \(x=0\) even though \(g\) itself is not differentiable at \(x=0 .\) Does this contradict the Chain Rule? Explain.

The amount of work done by the heart's main pumping chamber, the left ventricle, is given by the equation $$W=P V+\frac{V \delta v^{2}}{2 g}$$, where \(W\) is the work per unit time, \(P\) is the average blood pressure, \(V\) is the volume of blood pumped out during the unit of time, \(\delta(\text { "delta") is the weight density of the blood, } v\) is the average velocity of the exiting blood, and \(g\) is the acceleration of gravity. When \(P, V, \delta,\) and \(v\) remain constant, \(W\) becomes a function of \(g,\) and the equation takes the simplified form $$ W=a+\frac{b}{g}(a, b \text { constant }) $$ As a member of NASA's medical team, you want to know how sensitive \(W\) is to apparent changes in \(g\) caused by flight maneuvers, and this depends on the initial value of \(g\). As part of your investigation, you decide to compare the effect on \(W\) of a given change \(d g\) on the moon, where \(g=1.6 \mathrm{m} / \mathrm{s}^{2}\), with the effect the same change \(d g\) would have on Earth, where \(g=9.8 \mathrm{m} / \mathrm{s}^{2} .\) Use the simplified equation above to find the ratio of \(d W_{\text {moon }}\) to \(d W_{\text {Earth }}\)

A highway patrol plane flies 3 km above a level, straight road at a steady \(120 \mathrm{km} / \mathrm{h}\). The pilot sees an oncoming car and with radar determines that at the instant the line-of-sight distance from plane to car is \(5 \mathrm{km},\) the line-of-sight distance is decreasing at the rate of \(160 \mathrm{km} / \mathrm{h}\). Find the car's speed along the highway.

Find the tangent to \(y=\sqrt{x^{2}-x+7}\) at \(x=2\)

Find both \(d y / d x\) (treating \(y\) as a differentiable function of \(x\) ) and \(d x / d y\) (treating \(x\) as a differentiable function of \(y\) ). How do \(d y / d x\) and \(d x / d y\) seem to be related? Explain the relationship geometrically in terms of the graphs. $$x^{3}+y^{2}=\sin ^{2} y$$
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