/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 $$ \text { Find } \lim _{h \ri... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 21

$$ \text { Find } \lim _{h \rightarrow 0} \frac{(3+h)^{2}-9}{h} $$

Short Answer

Expert verified
The limit as \(h\) approaches 0 of the given function is 6.

Step by step solution

01

Simplification

Expand the numerator \( (3+h)^{2}-9 \) which simplifies to \( h^{2} + 6h + 9 - 9 \). This further simplifies to \( h^{2} + 6h \)
02

Calculation of the limit

So the initial function \(\frac{(3+h)^{2}-9}{h}\) simplifies to \(\frac{h^{2} + 6h}{h}\), which further simplifies to \(h + 6\). As \(h\) tends to 0, \(h + 6\) will tend to 6

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Key Concepts

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

Simplifying Expressions
In calculus, simplifying expressions is about reducing complex mathematical expressions into more manageable forms. This simplification aids in calculations and helps uncover hidden patterns or properties. For the given exercise, we start with the expression \(\frac{(3+h)^2 - 9}{h}\). Our goal is to simplify the numerator, \((3+h)^2 - 9\), by expanding and then reducing the expression.

Why is simplification important? Well, it makes analyzing limits and solving equations much easier. Here are the steps we used:
  • First, we expanded the polynomial \( (3+h)^2 \) to get \( 9 + 6h + h^2 \).
  • Next, we subtracted 9 from the expanded form, as indicated in the expression, leading us to \( h^2 + 6h \).
Now that the expression is simplified, it becomes clearer and easier to evaluate, especially when involving limits.
Polynomial Expansion
Polynomial expansion is a mathematical technique used to express a power of a binomial as a sum of terms. This technique is valuable for simplifying expressions like \((3+h)^2\). Understanding polynomial expansion helps in recognizing and simplifying complex expressions.

For \((3+h)^2\), use the formula for expanding \((a+b)^2\), which is \(a^2 + 2ab + b^2\). Here's how we applied it:
  • Identify \(a\) as 3 and \(b\) as \(h\).
  • Apply the formula: \(3^2 + 2(3)(h) + h^2\).
  • This results in \(9 + 6h + h^2\).
With the expression expanded, you can simplify by combining like terms and preparing it for further calculations, such as evaluating limits.
Evaluating Limits
Evaluating limits is a fundamental concept in calculus. It involves determining the value that a function approaches as the input approaches a certain point. In our example, we want to find the limit of \( \frac{(3+h)^2 - 9}{h} \) as \( h \rightarrow 0 \).

Simplification and polynomial expansion have prepared our expression for this step. Initially, the function might seem undefined at \(h = 0\) due to division by zero. But after simplifying:
  • The expression becomes \(\frac{h^2 + 6h}{h}\), which simplifies to \(h + 6\).
  • By eliminating \(h\) from the denominator, we avoid division by zero.
  • This simplified form \(h + 6\) makes it straightforward to evaluate the limit.
As \(h\) approaches zero, \(h + 6\) approaches 6. Hence, \( \lim \limits_{h \to 0} (h + 6) = 6 \). This shows that understanding limits is critical for discovering the behavior of functions near specific points.

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Most popular questions from this chapter

Boats A and B leave the same place at the same time. Boat A heads due north at 12 km/hr. Boat B heads due east at 18 km/hr. After 2.5 hours, how fast is the distance between the boats increasing (in km/hr)? (A) 21.63 (B) 31.20 (C) 75.00 (D) 9.84

The acceleration of a particle moving along the \(x\) -axis at time \(t\) is given by \(a(t)=4 t-12 .\) If the velocity is 10 when \(t=0\) and the position is 4 when \(t=0\) , then the particle is changing direction at (A) \(t=1\) (B) \(t=3\) (C) \(t=5\) (D) \(t=1\) and \(t=5\)

Now evaluate the following integrals. \(\int \frac{\sin 2 x}{\cos x} d x\)

Find \(\frac{d}{d x} \int_{1}^{x} \sin ^{2} t d t\)

Find the area of the region between the two curves in each problem, and be sure to sketch each one. (We gave you only endpoints in one of them) The answers are in Chapter 19 . The curve \(y=x^{2}-2\) and the line \(y=2\).
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