Suppose F ( T ) = 8 T − 9 F(t)=\frac{8}{\sqrt{t-9}} F ( T ) = T − 9 ​ 8 ​ .a. Find The Derivative Of F ( T F(t F ( T ].b. Find An Equation For The Tangent Line To The Graph Of F ( T F(t F ( T ] At The Point ( T , Y ) = ( 34 , 8 5 (t, Y)=(34, \frac{8}{5} ( T , Y ) = ( 34 , 5 8 ​ ].

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Introduction

In this article, we will delve into the world of derivatives and tangent lines, exploring the mathematical concepts that underlie these fundamental ideas. We will examine the derivative of a given function, f(t)=8t−9f(t)=\frac{8}{\sqrt{t-9}}, and use it to find an equation for the tangent line to the graph of f(t)f(t) at a specific point.

The Derivative of a Function

The derivative of a function represents the rate of change of the function with respect to its input. In other words, it measures how fast the output of the function changes when the input changes. The derivative is denoted by the symbol dydx\frac{dy}{dx} or f′(x)f'(x).

To find the derivative of a function, we can use various rules and techniques, including the power rule, product rule, and quotient rule. In this case, we will use the quotient rule to find the derivative of f(t)=8t−9f(t)=\frac{8}{\sqrt{t-9}}.

The Quotient Rule

The quotient rule states that if we have a function of the form f(x)=g(x)h(x)f(x)=\frac{g(x)}{h(x)}, then the derivative of f(x)f(x) is given by:

f′(x)=h(x)g′(x)−g(x)h′(x)[h(x)]2f'(x)=\frac{h(x)g'(x)-g(x)h'(x)}{[h(x)]^2}

In our case, we have g(t)=8g(t)=8 and h(t)=t−9h(t)=\sqrt{t-9}. We can find the derivatives of g(t)g(t) and h(t)h(t) using the power rule:

g′(t)=0g'(t)=0

h′(t)=12t−9h'(t)=\frac{1}{2\sqrt{t-9}}

Now, we can plug these values into the quotient rule formula:

f′(t)=(t−9)(0)−(8)(12t−9)(t−9)2f'(t)=\frac{(\sqrt{t-9})(0)-(8)(\frac{1}{2\sqrt{t-9}})}{(\sqrt{t-9})^2}

Simplifying the expression, we get:

f′(t)=−4(t−9)3/2f'(t)=-\frac{4}{(t-9)^{3/2}}

The Tangent Line

The tangent line to the graph of a function at a point (a,f(a))(a, f(a)) is the line that passes through the point and has the same slope as the function at that point. In other words, it is the line that best approximates the function at the point.

To find the equation of the tangent line, we need to know the slope of the line, which is given by the derivative of the function at the point. In this case, we have:

f′(34)=−4(34−9)3/2=−4(25)3/2=−4125f'(34)=-\frac{4}{(34-9)^{3/2}}=-\frac{4}{(25)^{3/2}}=-\frac{4}{125}

The equation of the tangent line is given by:

y−f(34)=−4125(t−34)y-f(34)=-\frac{4}{125}(t-34)

Simplifying the expression, we get:

y−85=−4125(t−34)y-\frac{8}{5}=-\frac{4}{125}(t-34)

Rearranging the terms, we get:

y=−4125t+136125+85y=-\frac{4}{125}t+\frac{136}{125}+\frac{8}{5}

Simplifying further, we get:

y=−4125t+136+200125y=-\frac{4}{125}t+\frac{136+200}{125}

y=−4125t+336125y=-\frac{4}{125}t+\frac{336}{125}

Conclusion

In this article, we have explored the concept of derivatives and tangent lines, using the function f(t)=8t−9f(t)=\frac{8}{\sqrt{t-9}} as a case study. We have found the derivative of the function using the quotient rule and used it to find an equation for the tangent line to the graph of f(t)f(t) at the point (34,85)(34, \frac{8}{5}). The equation of the tangent line is given by:

y=−4125t+336125y=-\frac{4}{125}t+\frac{336}{125}

This equation represents the line that best approximates the function at the point (34,85)(34, \frac{8}{5}). We hope that this article has provided a clear and concise introduction to the concepts of derivatives and tangent lines, and has inspired readers to explore these fascinating topics further.

References

  • [1] Calculus, 3rd edition, Michael Spivak
  • [2] Calculus, 2nd edition, James Stewart
  • [3] Derivatives and Tangent Lines, Khan Academy

Glossary

  • Derivative: The rate of change of a function with respect to its input.
  • Tangent line: The line that passes through a point on the graph of a function and has the same slope as the function at that point.
  • Quotient rule: A rule for finding the derivative of a function of the form f(x)=g(x)h(x)f(x)=\frac{g(x)}{h(x)}.
  • Power rule: A rule for finding the derivative of a function of the form f(x)=xnf(x)=x^n.
  • Product rule: A rule for finding the derivative of a function of the form f(x)=g(x)h(x)f(x)=g(x)h(x).

Further Reading

  • For more information on derivatives and tangent lines, see the references listed above.
  • For a more in-depth exploration of the quotient rule, see the Khan Academy video on the quotient rule.
  • For a more in-depth exploration of the power rule, see the Khan Academy video on the power rule.
  • For a more in-depth exploration of the product rule, see the Khan Academy video on the product rule.
    Derivatives and Tangent Lines: A Q&A Guide =====================================================

Introduction

In our previous article, we explored the concept of derivatives and tangent lines, using the function f(t)=8t−9f(t)=\frac{8}{\sqrt{t-9}} as a case study. We found the derivative of the function using the quotient rule and used it to find an equation for the tangent line to the graph of f(t)f(t) at the point (34,85)(34, \frac{8}{5}). In this article, we will answer some frequently asked questions about derivatives and tangent lines.

Q&A

Q: What is the derivative of a function?

A: The derivative of a function represents the rate of change of the function with respect to its input. In other words, it measures how fast the output of the function changes when the input changes.

Q: How do I find the derivative of a function?

A: There are several rules and techniques for finding the derivative of a function, including the power rule, product rule, and quotient rule. The power rule states that if we have a function of the form f(x)=xnf(x)=x^n, then the derivative of f(x)f(x) is given by f′(x)=nxn−1f'(x)=nx^{n-1}. The product rule states that if we have a function of the form f(x)=g(x)h(x)f(x)=g(x)h(x), then the derivative of f(x)f(x) is given by f′(x)=g′(x)h(x)+g(x)h′(x)f'(x)=g'(x)h(x)+g(x)h'(x). The quotient rule states that if we have a function of the form f(x)=g(x)h(x)f(x)=\frac{g(x)}{h(x)}, then the derivative of f(x)f(x) is given by f′(x)=h(x)g′(x)−g(x)h′(x)[h(x)]2f'(x)=\frac{h(x)g'(x)-g(x)h'(x)}{[h(x)]^2}.

Q: What is the tangent line to a function?

A: The tangent line to a function at a point (a,f(a))(a, f(a)) is the line that passes through the point and has the same slope as the function at that point. In other words, it is the line that best approximates the function at the point.

Q: How do I find the equation of the tangent line to a function?

A: To find the equation of the tangent line to a function, we need to know the slope of the line, which is given by the derivative of the function at the point. We can then use the point-slope form of a line to find the equation of the tangent line.

Q: What is the point-slope form of a line?

A: The point-slope form of a line is given by y−y1=m(x−x1)y-y_1=m(x-x_1), where (x1,y1)(x_1, y_1) is a point on the line and mm is the slope of the line.

Q: How do I use the point-slope form to find the equation of the tangent line?

A: To use the point-slope form to find the equation of the tangent line, we need to know the point (a,f(a))(a, f(a)) and the slope m=f′(a)m=f'(a). We can then plug these values into the point-slope form to get the equation of the tangent line.

Q: What is the significance of the derivative in calculus?

A: The derivative is a fundamental concept in calculus, and it has many important applications in physics, engineering, and economics. It is used to model real-world phenomena, such as the motion of objects, the growth of populations, and the behavior of financial markets.

Q: How do I apply the derivative in real-world problems?

A: To apply the derivative in real-world problems, we need to identify the function that models the problem and find its derivative. We can then use the derivative to solve the problem and make predictions about the behavior of the system.

Conclusion

In this article, we have answered some frequently asked questions about derivatives and tangent lines. We have discussed the concept of the derivative, the rules for finding the derivative, and the significance of the derivative in calculus. We have also provided examples of how to apply the derivative in real-world problems. We hope that this article has provided a clear and concise introduction to the concepts of derivatives and tangent lines, and has inspired readers to explore these fascinating topics further.

References

  • [1] Calculus, 3rd edition, Michael Spivak
  • [2] Calculus, 2nd edition, James Stewart
  • [3] Derivatives and Tangent Lines, Khan Academy

Glossary

  • Derivative: The rate of change of a function with respect to its input.
  • Tangent line: The line that passes through a point on the graph of a function and has the same slope as the function at that point.
  • Quotient rule: A rule for finding the derivative of a function of the form f(x)=g(x)h(x)f(x)=\frac{g(x)}{h(x)}.
  • Power rule: A rule for finding the derivative of a function of the form f(x)=xnf(x)=x^n.
  • Product rule: A rule for finding the derivative of a function of the form f(x)=g(x)h(x)f(x)=g(x)h(x).

Further Reading

  • For more information on derivatives and tangent lines, see the references listed above.
  • For a more in-depth exploration of the quotient rule, see the Khan Academy video on the quotient rule.
  • For a more in-depth exploration of the power rule, see the Khan Academy video on the power rule.
  • For a more in-depth exploration of the product rule, see the Khan Academy video on the product rule.