Problem 33 $t^{3}+4 t^{2}+4 t, t^{2}-4 t$... [FREE SOLUTION]

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Elementary Algebra Within Reach

Ron Larson, Kimberly Nolting

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 7: Problem 33

$t^{3}+4 t^{2}+4 t, t^{2}-4 t$

Short Answer

Expert verified

The derivative of $t^{3}+4 t^{2}+4 t$ is $3*t^{2}+8 t+4$ and the derivative of $t^{2}-4 t$ is $2t-4$.

Step by step solution

- Find the derivative of the first function

Differentiate $t^{3}+4 t^{2}+4 t$ term by term using the power rule for each term separately. The derivative of $t^n$ is $n*t^{n-1}$, so the derivative of $t^{3}$ is $3*t^{2}$. The derivative of $4t^{2}$ is $2*4*t=8t$, and the derivative of $4t$ is just $4$. Therefore, the derivative of $t^{3}+4 t^{2}+4 t$ is $3*t^{2}+8 t+4$.

- Find the derivative of the second function

Again, differentiate $t^{2}-4 t$ term by term using the power rule. The derivative of $t^{2}$ is $2*t$ and the derivative of $-4t$ is $-4$. Therefore, the derivative of $t^{2}-4 t$ is $2t-4$.

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Key Concepts

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

Power Rule

The power rule is a fundamental concept in calculus, used primarily for finding the derivative of functions of the form $ t^n $, where $ n $ is any real number. To apply the power rule, you multiply the exponent $ n $ by the term's coefficient and reduce the exponent by one. In formula form, the derivative of $ t^n $ is given by $ n \times t^{n-1} $.

For example, let's take $ t^3 $:

The exponent is 3.
Using the power rule, bring down the 3, multiply it by the coefficient (1 in this case), and reduce the exponent by 1.
This results in the derivative $ 3 \times t^{3-1} = 3t^2 $.

This approach makes polynomial differentiation straightforward and manageable.

Term by Term Differentiation

Term by term differentiation simplifies the process of finding the derivative for polynomial functions. It involves applying the power rule to each term independently, which allows for easy calculations.

Suppose we want to differentiate the polynomial $ t^3 + 4t^2 + 4t $:

For $ t^3 $, the derivative is $ 3t^2 $ as calculated using the power rule.
For $ 4t^2 $, the coefficient 4 is multiplied by the derivative of $ t^2 $, giving $ 8t $.
Finally, for $ 4t $, the derivative is simply the constant 4.

Combine these results to get the complete derivative: $ 3t^2 + 8t + 4 $. Each term is handled separately, illustrating the simplicity of this method.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. In a single-variable polynomial like $ t^3 + 4t^2 + 4t $, each term contributes to the function's overall degree, which is the highest exponent present.

Polynomials are generally easy to differentiate because they follow straightforward rules like the power rule. Here are some key characteristics:

Polynomials are continuous and differentiable everywhere on the real number line.
The degree of the polynomial dictates the function's behavior and the number of roots it may have.
Each term's variable exponent dictates how it contributes to the polynomial's overall rate of change.

Understanding polynomials helps simplify calculus operations, making topics like term by term differentiation more intuitive.

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91影视

\(t^{3}+4 t^{2}+4 t, t^{2}-4 t\)

Short Answer

Step by step solution

- Find the derivative of the first function

- Find the derivative of the second function

Key Concepts

Power Rule

Term by Term Differentiation

Polynomial Functions

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