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Chapter 23: Problem 33

Find the point(s) where the slope of a tangent line to the given curve has the given value. In Exercises \(35-38,\) solve the given problems. $$y=12 x-\frac{1}{3} x^{3} ; m_{\mathrm{tan}}=-4$$

Short Answer

Expert verified
Points: \((4, \frac{80}{3})\) and \((-4, -\frac{80}{3})\)."

Step by step solution

01

Find the Derivative

To find the slope of the tangent line to the curve, we need the derivative of the function. Given \( y = 12x - \frac{1}{3}x^3 \), the derivative \( y' \) is obtained using the power rule:\[\frac{dy}{dx} = 12 - x^2\]
02

Set Derivative Equal to Given Slope

The problem specifies that the slope of the tangent line should be \(-4\). Therefore, we set the derivative equal to \(-4\):\[12 - x^2 = -4\]
03

Solve for x

Solve the equation \( 12 - x^2 = -4 \) to find the values of \( x \):\[12 - x^2 = -4 \12 + 4 = x^2 \16 = x^2 \x = \pm 4\]
04

Find Corresponding y-values

Substitute \( x = 4 \) and \( x = -4 \) back into the original equation to find the corresponding \( y \)-values.For \( x = 4 \):\[y = 12(4) - \frac{1}{3}(4)^3 = 48 - \frac{1}{3}(64) = 48 - \frac{64}{3} = \frac{144}{3} - \frac{64}{3} = \frac{80}{3}\]For \( x = -4 \):\[y = 12(-4) - \frac{1}{3}(-4)^3 = -48 + \frac{1}{3}(64) = -48 + \frac{64}{3} = -\frac{144}{3} + \frac{64}{3} = -\frac{80}{3}\]
05

Conclusion

The coordinates of the point(s) on the curve where the slope of the tangent line is \(-4\) are \((4, \frac{80}{3})\) and \((-4, -\frac{80}{3})\).

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Key Concepts

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

Derivative
The derivative of a function is a critical concept in calculus. It represents how a function changes or, in other words, the rate of change of the function's value with respect to changes in its input value. Imagine driving a car; the derivative tells you your speed at any given moment.
  • In mathematical terms, the derivative of a function at a certain point is the slope of the tangent line to the curve at that point.
  • To find a derivative, standard rules and techniques, like the power rule and product rule, are often applied.
The goal with derivatives is to simplify the function to help you analyze and understand its behavior more easily. It's at the heart of finding maximum or minimum values of functions, among many other applications.
Tangent Line
A tangent line is a straight line that just "touches" a curve at a given point without crossing it. Think of the tangent line as a mirror of the curve's behavior at a specific spot. Sometimes, a curve can resemble a rolling wave, with the tangent line at the peak of each wave showing the immediate direction of travel.
  • The slope of the tangent line gives you a snapshot of the curve's steepness (or flatness) right there at the point of contact.
  • This concept is crucial for understanding how graphs and models behave instantaneously, especially in real-world scenarios such as physics or economics.
Tangent lines provide local linear approximations to curves, making complex shapes and relationships more intuitive and manageable.
Slope of a Curve
The slope of a curve at a particular point means the steepness of the line that is tangent to the curve at that point. Just like a mountain has varying inclines, a curve changes in steepness, and the slope gives a numerical value to that steepness.
  • This slope is determined by the derivative: the derivative tells you how steep the curve is at any point.
  • If the slope is positive, the curve is going upward; if it鈥檚 negative, the curve is going downward.
Understanding the slope allows for predictions about the curve's behavior over small intervals, akin to knowing whether you need to pedal harder going up or ease off going downhill on a bike ride.
Power Rule
The Power Rule is a shorthand way of determining derivatives of polynomials. It states that if you have a term in the form of \[x^n\],its derivative is:\[nx^{n-1}\].In simpler terms, the power gets multiplied by the coefficient in front of \(x\), and then the power of \(x\) is reduced by one.
  • This rule is especially useful in efficiently differentiating a wide variety of polynomial terms, saving time and effort.
  • For example, for the function \(f(x) = x^3\), its derivative using the power rule is \(3x^2\).
The Power Rule is fundamental to calculus, forming the basis of many more complex derivative calculations.

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Most popular questions from this chapter

\(\lim _{x \rightarrow a^{-}} f(x)\) means to find the limit as x approaches a from the left only, and \(\lim _{x \rightarrow a^{+}} f(x)\) means to find the limit as \(x\) approaches a from the right only. These are called one-sided limits. Solve the following problems. Is there a difference between \(\lim _{x \rightarrow 2^{-}} \frac{1}{x-2}\) and \(\lim _{x \rightarrow 2^{+}} \frac{1}{x-2} ?\)

Evaluate the second derivative of the given function for the given value of \(x\). $$f(x)=x-\frac{2}{x^{3}}, x=-1$$

Solve the given problems by finding the appropriate derivatives. Find the derivative of \(y=\frac{x^{2}-1}{x-1}\) by (a) the quotient rule, and (b) by first simplifying the function.

Find the second derivative of each of the given functions. $$y=\frac{8 x}{\sqrt{9-x^{2}}}$$

Evaluate the indicated limits by direct evaluation as in Examples \(10-14 .\) Change the form of the function where necessary. $$\lim _{x \rightarrow \infty} \frac{x-27}{7 x+4}$$
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