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Chapter 3: Problem 61

Show that the curve \(y=6 x^{3}+5 x-3\) has no tangent line with slope \(4 .\)

Short Answer

Expert verified
The curve has no tangent line with slope 4, as the required equation is impossible to solve with real numbers.

Step by step solution

01

Find the derivative of the curve

To determine the slope of the tangent line at any point on the curve, we first need the derivative of the function. The given function is \( y = 6x^3 + 5x - 3 \). Using the power rule \( \frac{d}{dx}[x^n] = nx^{n-1} \), calculate the derivative \( y' \). Thus, \( y' = 18x^2 + 5 \).
02

Set the derivative equal to the desired slope

The slope of the tangent line is given by the derivative \( y' \). We are looking for a tangent line with a slope of 4. Therefore, set the derivative equal to 4: \( 18x^2 + 5 = 4 \).
03

Solve the equation for x

Now solve the equation obtained in Step 2: \( 18x^2 + 5 = 4 \). Subtract 4 from both sides to get \( 18x^2 = -1 \).
04

Analyze the equation \( 18x^2 = -1 \)

Observe that the equation \( 18x^2 = -1 \) is not possible because \( 18x^2 \) gives a non-negative result for any real \( x \), meaning it can't equal \( -1 \). This indicates there are no real solutions for \( x \) such that the tangent slope is 4.

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Key Concepts

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

Tangent Line
Understanding tangent lines is crucial in calculus. A tangent line is a straight line that touches a curve at a single point without crossing over it. This single point of contact is where the slope of the tangent line is the same as the slope of the curve.
A useful way to visualize a tangent line is to imagine a line drawn just touching a curve, like pressing a ruler against a curved piece of paper so that it only touches at one point. The direction of this line represents the instantaneous rate of change or slope of the curve at that precise point.
In practical applications, tangent lines help define the slope at a particular moment, which can be used to predict behaviors and trends in a variety of scientific settings.
Derivative
The derivative is a fundamental concept tied to tangent lines. When calculating the slope of a curve at a specific point, the derivative becomes essential.
In simple terms, the derivative represents the rate of change of a function concerning its variable. This tells us how steep the curve is at any given point. For the curve in the given exercise, the derivative is found using differentiation rules.
More specifically, it relates to the original function as the function's slope formula. Differentiation involves applying guidelines like the power rule to determine this slope function, which we'll explore next.
Power Rule
The power rule is one of the most straightforward differentiation techniques. It's used to find derivatives efficiently when dealing with polynomial functions.
To apply the power rule, take any term in the form of \( x^n \) and differentiate it. The rule states that the derivative is \( nx^{n-1} \). This means you multiply the exponent by the coefficient and reduce the exponent by one.
For example, in the exercise, the power rule is applied to each term of the function \( y = 6x^3 + 5x - 3 \). Thus, \( 6x^3 \) becomes \( 18x^2 \) and \( 5x \) simplifies to just \( 5 \), leading to the full derivative \( y' = 18x^2 + 5 \). Mastering this rule means taking quick and accurate derivatives of many common functions.

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Most popular questions from this chapter

The biomass \(B(t)\) of a fish population is the total mass of the members of the population at time \(t .\) It is the product of the number of individuals \(N(t)\) in the population and the average mass \(M(t)\) of a fish at time \(t .\) In the case of guppies, breeding occurs continually. Suppose that at time \(t=4\) weeks the population is 820 guppies and is growing at a rate of 50 guppies per week, while the average mass is 1.2 \(\mathrm{g}\) and is increasing at a rate of 0.14 \(\mathrm{g} / \mathrm{week}\) . At what rate is the biomass increasing when \(t=4 ?\)

A tangent line is drawn to the hyperbola \(x y=c\) at a point \(P\). (a) Show that the midpoint of the line segment cut from this tangent line by the coordinate axes is \(P .\) (b) Show that the triangle formed by the tangent line and the coordinate axes always has the same area, no matter where \(P\) is located on the hyperbola.

Find \(d y / d x\) by implicit differentiation. \(\sin x+\cos y=\sin x \cos y\)

Let $$r(x)=f(q(h(x))),\( where \)h(1)=2, q(2)=3$$ $$h^{\prime}(1)=4, q^{\prime}(2)=5,\( and \)f^{\prime}(3)=6 .\( Find \)r^{\prime}(1)$$

At what point on the curve \(y=1+2 e^{x}-3 x\) is the tangent line parallel to the line \(3 x-y=5 ?\) Illustrate by graphing the curve and both lines.
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