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Chapter 1: Problem 6

Find the first derivatives. $$x=16 t^{2}+45 t+10$$

Short Answer

Expert verified
\( \frac{dx}{dt} = 32t + 45 \)

Step by step solution

01

Understand the Question

The task is to find the first derivative of the given function: \(x = 16t^2 + 45t + 10\). The first derivative measures how the value of the function changes as the variable \(t\) changes.
02

Apply the Power Rule

To find the derivative of a polynomial, use the power rule, which states that the derivative of \(t^n\) is \(n t^{n-1}\).
03

Differentiate Each Term Separately

Differentiate each term in the function separately:\( \frac{d}{dt}(16t^2) = 32t \)\( \frac{d}{dt}(45t) = 45 \)\( \frac{d}{dt}(10) = 0 \)
04

Combine the Results

Add the derivatives of each term to get the first derivative of the entire function:\( \frac{dx}{dt} = 32t + 45 \)

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Key Concepts

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

Power Rule
Understanding the power rule is crucial for differentiating polynomials. The power rule states that the derivative of a term of the form \(t^n\) is given by \(n t^{n-1}\). In simpler terms, you multiply by the exponent and then decrease the exponent by one.

For example:
  • If you have \(t^3\), its derivative is \(3t^2\).
  • For \(t^5\), the derivative is \(5t^4\).
  • And \(t^1\) becomes \(1\), since any term to the power of zero is 1.
Each term in a polynomial can be differentiated individually using this rule.
Polynomial Differentiation
Polynomial differentiation involves differentiating functions that are sums of terms consisting of variables raised to powers, such as \(t^2\) or \(t^3\). Each term can be differentiated separately following the power rule. Then, these derivative terms are summed up.

Applying polynomial differentiation to the function from the exercise:
  • The first term, \(16t^2\), becomes \(32t\) (derivative of \(t^2\) is \(2t\), multiplied by 16).
  • The second term, \(45t\), becomes \(45\) (derivative of \(t\) is 1, multiplied by 45).
  • The constant term, 10, has a derivative of 0 (derivative of any constant is zero).
Combining these gives the first derivative: \(32t + 45\).

This process generalizes to polynomials of any length.
Calculus Basics
Understanding the basics of calculus is essential for tackling differentiation problems.

  • **Derivative**: It represents how a function changes as its input changes. In simple terms, it is the slope of the function at any given point.
  • **First Derivative**: The process of finding the first derivative answers the question: How does the function behave concerning its variable? For a function \(f(t)\), its first derivative, denoted as \(f'(t)\) or \(\frac{df}{dt}\), gives the rate of change of \(f(t)\) with respect to \(t\).
In our exercise example, the first derivative \(32t + 45\) shows how the function \(16t^2 + 45t + 10\) changes with respect to \(t\). Understanding these fundamental concepts will give you a strong foundation to solve more complex differentiation problems.

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Most popular questions from this chapter

(a) Draw two graphs of your choice that represent a function \(y=f(x)\) and its vertical shift \(y=f(x)+3.\) (b) Pick a value of \(x\) and consider the points \((x, f(x))\) and \((x, f(x)+3) .\) Draw the tangent lines to the curves at these points and describe what you observe about the tangent lines. (c) Based on your observation in part (b), explain why $$\frac{d}{d x} f(x)=\frac{d}{d x}(f(x)+3)$$

The third derivative of a function \(f(x)\) is the derivative of the second derivative \(f^{\prime \prime}(x)\) and is denoted by \(f^{\prime \prime \prime}(x) .\) Compute \(f^{\prime \prime \prime}(x)\) for the following functions: (a) \(f(x)=x^{5}-x^{4}+3 x\) (b) \(f(x)=4 x^{5 / 2}\)

A toy rocket fired straight up into the air has height \(s(t)=160 t-16 t^{2}\) feet after \(t\) seconds. (a) What is the rocket's initial velocity (when \(t=0\) )? (b) What is the velocity after 2 seconds? (c) What is the acceleration when \(t=3 ?\) (d) At what time will the rocket hit the ground? (e) At what velocity will the rocket be traveling just as it smashes into the ground?

Table 2 gives a car's trip odometer reading (in miles) at 1 hour into a trip and at several nearby times. What is the average speed during the time interval from 1 to 1.05 hours? Estimate the speed at time 1 hour into the trip. $$\text { Table 2 Trip Odometer Readings at Several Times }$$ $$\begin{array}{|c|c|}\hline \text { Time } & \text { Trip Meter } \\\\\hline 96 & 43.2 \\\97 & 43.7 \\\\.98 & 44.2 \\\\.99 & 44.6 \\\1.00 & 45.0 \\\1.01 & 45.4 \\\1.02 & 45.8 \\\1.03 & 46.3 \\\1.04 & 46.8 \\\1.05 & 47.4 \\\\\hline\end{array}$$

Let \(P(x)\) be the profit (in dollars) from manufacturing and selling \(x\) cars. Interpret \(P(100)=90,000\) and \(P^{\prime}(100)=1200 .\) Estimate the profit from manufacturing and selling 99 cars.
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