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Chapter 1: Problem 6

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

Short Answer

Expert verified
The first derivative is \(32t + 45\).

Step by step solution

01

Identify the Given Function

Given the function for which the first derivative needs to be found: ewline ewline \(x = 16t^2 + 45t + 10\)
02

Differentiate Each Term with Respect to t

Apply the power rule to each term in the function. The power rule states that \((d/dt)[t^n] = n t^{n-1}\):ewline \((d/dt)[16t^2] \rightarrow 16 \times 2 \times t^{2-1} = 32t\) ewline \((d/dt)[45t] \rightarrow 45\) ewline \((d/dt)[10] \rightarrow 0\)
03

Combine the Results

Add the derivatives of each term to get the first derivative of the given function: ewline \(dx/dt = 32t + 45\)

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Key Concepts

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

Power Rule in Differentiation
The power rule is a fundamental concept in calculus that makes finding derivatives easier. The power rule states that if you have a function of the form \( f(t) = t^n \), then its derivative is given by \( f'(t) = n t^{n-1} \).
For example, if you have something like \( 16t^2 \), applying the power rule:
  • Multiply the exponent by the coefficient (2 * 16 = 32).
  • Reduce the exponent by 1 (2 - 1 = 1).

So, the derivative of \( 16t^2 \) becomes \( 32t \).
For linear terms like \( 45t \), the derivative of \( t^1 \) is simply the coefficient 45. For a constant like 10, the derivative is 0.
Combining these, we get the result \( 32t + 45 \).
Understanding the power rule is crucial for simplifying differentiation.
What is Differentiation?
Differentiation is a process in calculus used to find the rate at which a function is changing at any given point. This rate of change is known as the derivative.
To differentiate a function means to compute its derivative. In the process outlined in the exercise:
  • We started with the function \( x = 16t^2 + 45t + 10 \).
  • We then used the power rule to find the derivative of each term.
  • Finally, we combined the derivatives to get \( dx/dt = 32t + 45 \).

The result tells us how the function \( x \) changes concerning \( t \). Differentiation is a key tool used in various fields like physics, engineering, and economics, helping us understand relationships between variables.
Introduction to Single-Variable Calculus
Single-variable calculus is a branch of calculus that deals with functions of a single variable. It includes concepts like limits, derivatives, and integrals.
In the context of the exercise:
  • We dealt with a function involving a single variable, \( t \).
  • We calculated the first derivative, providing information about the function鈥檚 rate of change with respect to \( t \).

Single-variable calculus lays the groundwork for more advanced studies in mathematics and science. It focuses primarily on understanding the behavior of functions, which are crucial for solving real-world problems.

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Most popular questions from this chapter

A helicopter is rising straight up in the air. Its distance from the ground \(t\) seconds after takeoff is \(s(t)\) feet, where \(s(t)=t^{2}+t\). (a) How long will it take for the helicopter to rise 20 feet? (b) Find the velocity and the acceleration of the helicopter when it is 20 feet above the ground.

Use a derivative routine to obtain the value of the derivative. Give the value to 5 decimal places. \(f^{\prime}(1)\), where \(f(x)=\frac{1}{1+x^{2}}\)

Estimate how much the function $$f(x)=\frac{1}{1+x^{2}}$$ will change if \(x\) decreases from 1 to \(.9 .\)

g(x) is the tangent line to the graph of \(f(x)\) at \(x=a\). Graph \(f(x)\) and \(g(x)\) and determine the value of \(a\). \(f(x)=x^{3}-12 x^{2}+46 x-50, g(x)=14-2 x\)

(a) Let \(A(x)\) denote the number (in hundreds) of computers sold when \(x\) thousand dollars is spent on advertising. Represent the following statement by equations involving \(A\) or \(A^{\prime}:\) When $$\$ 8000$$ was spent on advertising, the number of computers sold was 1200 and it was rising at the rate of 50 computers for each $$\$ 1000$$ spent on advertising. (b) Estimate the number of computers that will be sold if $$\$ 9000$$ is spent on advertising.
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