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Chapter 3: Problem 10

In Exercises \(7-12,\) find the horizontal tangents of the curve. $$y=4 x^{3}-6 x^{2}-1$$

Short Answer

Expert verified
The horizontal tangents of the curve \(y = 4x^{3} - 6x^{2} - 1\) occur at points where \(x = 0\) and \(x = 1\).

Step by step solution

01

Compute the derivative

The derivative of the function \(y = 4x^{3} - 6x^{2} - 1\) needs to be calculated. The derivative \(y'\) can be calculated as follows using the power rule of differentiation: \(y' = (3*4)x^{3-1} - (2*6)x^{2-1} = 12x^{2} - 12x\).
02

Set the derivative equal to zero

The x-values where the derivative is zero give the points on the curve where the tangent is horizontal. Therefore, set the derivative equal to zero and solve for x, which gives: \(0 = 12x^{2} - 12x\). Dividing through by 12 simplifies this to \(0 = x^{2} - x\).
03

Solve the resulting equation

Solving the equation \(x^{2} - x = 0\) gives \(x(x - 1) = 0\). According to the zero factor property, if a product of factors equals zero, then at least one of the factors must be zero. So, the solutions are \(x = 0\) and \(x = 1\).

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Key Concepts

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

Horizontal Tangents
A horizontal tangent is a line that runs parallel to the x-axis. It indicates a point on the curve where the slope is zero. In simpler terms, this is where the curve "levels out" momentarily, creating a flat section. Finding these tangents involves determining where the curve stops increasing or decreasing and changes direction. To find a horizontal tangent, you must locate where the derivative equals zero.
  • The concept of a horizontal tangent is crucial in calculus because it allows us to identify maxima, minima, or points of inflection on a graph.
  • It signifies no vertical rise, meaning the function behaves differently around these points, often changing direction.
Understanding where these points occur helps in sketching curves and analyzing the behavior of functions.
Power Rule of Differentiation
The power rule of differentiation is an essential tool in calculus for finding the derivative of a function that is a power of x. It is a simple, yet powerful, method that enables you to differentiate polynomials with ease. The power rule states that if you have a term of the form \(ax^n\), the derivative is \(anx^{n-1}\).
Applying the power rule simplifies finding the rate of change of functions significantly, especially when you deal with polynomials.
  • For example, in our exercise, the derivative of \(4x^3\) is \((3*4)x^{3-1} = 12x^2\), and the derivative of \(-6x^2\) is \((2*6)x^{2-1} = 12x\).
  • Power functions often appear in mathematical models, and their differentiation using the power rule allows for swift and accurate analysis.
This technique is one of the first rules taught in differentiation, due to its straightforward application and consistent results.
Setting Derivative to Zero
When you set the derivative of a function to zero, you are essentially solving for the values of \(x\) where the slope of the tangent line to the curve is zero. This is the mathematical way to find points on the curve where the direction changes, indicating a local maximum, minimum, or saddle point.
The derivation step is crucial when finding horizontal tangents, as it helps locate key transition points that determine the function's graphical behavior.
  • In our case, setting the derivative \(12x^2 - 12x\) to zero gives the equation \(12x^2 - 12x = 0\).
  • Simplifying it, we obtain \(x(x-1) = 0\), and solving this gives \(x = 0\) and \(x = 1\) as the points where the horizontal tangents occur.
Setting the derivative to zero is a fundamental step in calculus, enabling us to thoroughly explore the geometry of functions.

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Most popular questions from this chapter

The Derivative of sin 2\(x\) Graph the function \(y=2 \cos 2 x\) for \(-2 \leq x \leq 3.5 .\) Then, on the same screen, graph $$\quad y=\frac{\sin 2(x+h)-\sin 2 x}{h}$$ for \(h=1.0,0.5,\) and \(0.2 .\) Experiment with other values of \(h,\) including negative values. What do you see happening as \(h \rightarrow 0 ?\) Explain this behavior.

Multiple Choice Which of the following gives \(d y / d x\) if \(y=\log _{10}(2 x-3) ? \quad \) (A) $$\frac{2}{(2 x-3) \ln 10} \quad\left(\( B ) \)\frac{2}{2 x-3} \quad\( (C) \)\frac{1}{(2 x-3) \ln 10}\right.\( (D) \)\frac{1}{2 x-3} \quad\( (E) \)\frac{1}{2 x}$$

Finding \(f\) from \(f^{\prime}\) Let $$f^{\prime}(x)=3 x^{2}$$ (a) Compute the derivatives of \(g(x)=x^{3}, h(x)=x^{3}-2,\) and \(t(x)=x^{3}+3 .\) (b) Graph the numerical derivatives of \(g, h,\) and \(t\) (c) Describe a family of functions, \(f(x),\) that have the property that \(f^{\prime}(x)=3 x^{2}\) . (d) Is there a function \(f\) such that \(f^{\prime}(x)=3 x^{2}\) and \(f(0)=0 ?\) If so, what is it? (e) Is there a function \(f\) such that \(f^{\prime}(x)=3 x^{2}\) and \(f(0)=3 ?\) If so, what is it?

Multiple Choice Find \(y^{\prime \prime}\) if \(y=x \sin x\) (A) \(-x \sin x\) (B) \(x \cos x+\sin x\) (C) \(-x \sin x+2 \cos x\) (D) \(x \sin x\) (E) \(-\sin x+\cos x\)

Inflating a Balloon The volume \(V=(4 / 3) \pi r^{3}\) of a spherical balloon changes with the radius (a) At what rate does the volume change with respect to the radius when \(r=2 \mathrm{ft} ?\) (b) By approximately how much does the volume increase when the radius changes from 2 to 2.2 \(\mathrm{ft}\) ?
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