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Chapter 16: Problem 40

The slope of the curve \(y=2 x^{3}-3 x\) at \(x=1\) is a) 3 b) 5 c) 6 d) 8

Short Answer

Expert verified
Answer: (a) 3

Step by step solution

01

Find the derivative of the given function.

To find the derivative of the function \(y = 2x^3 - 3x\), apply the power rule for differentiation, which states that if \(y = ax^n\), then the derivative of y with respect to x is \(y' = nax^{(n-1)}\). Therefore, we have: \(y' = \frac{d}{dx}(2x^3 - 3x) = \frac{d}{dx}(2x^3) - \frac{d}{dx}(3x)\) Applying the power rule: \(y' = 3(2x^{(3-1)}) - 1(3x^{(1-1)})\)
02

Simplify the derivative.

Now, simplify the derivative by performing the necessary operations: \(y' = 3(2x^2) - 3x^0 = 6x^2 - 3\)
03

Evaluate the derivative at x = 1.

To find the slope of the curve at x = 1, plug in x = 1 into the derivative and simplify: \(y'(1) = 6(1)^2 - 3 = 6 - 3 = 3\) So, the slope of the curve at x = 1 is 3. The correct answer is (a) 3.

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Most popular questions from this chapter

The included angle between two successive sides of a regular eight-sided polygon is a) \(150^{\circ}\) b) \(135^{\circ}\) c) \(120^{\circ}\) d) \(75^{\circ}\)

Solve for \(\left[x_{i}\right]\) $$ \begin{aligned} 3 x_{1}+2 x_{2} &=-2 \\ x_{1}-x_{2}+x_{3} &=0 \\ 4 x_{1}+2 x_{3} &=4 \end{aligned} $$ a) \(\left[\begin{array}{r}2 \\ 4 \\ -6\end{array}\right]\) \(\left[\begin{array}{l}2 \\ 4 \\ 8\end{array}\right]\) d) \(\left[\begin{array}{r}8 \\ 14\end{array}\right]\)

Determine \(\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right]\left[\begin{array}{rr}-1 & 0 \\ 1 & 2\end{array}\right]\). a) \(\left[\begin{array}{rr}1 & 4 \\ -1 & 2\end{array}\right]\) b) \(\left[\begin{array}{rr}1 & -1 \\ 4 & 2\end{array}\right]\) c) \(\left[\begin{array}{r}1 \\ -1\end{array}\right]\) d) \(\left[\begin{array}{l}4 \\ 2\end{array}\right]\)

Find the root of \((1+i)^{1 / 5}\) with the smallest argument. a) \(0.168+1.06 i\) c) \(1.06-0.168 i\) b) \(1.06+0.168 i\) d) \(0.168-1.06 i\)

Resolve \(\frac{2}{x\left(x^{2}-3 x+2\right)}\) into partial fractions. a) \(\frac{1}{x}+\frac{1}{x-2}-\frac{2}{x-1}\) c) \(\frac{1}{x}-\frac{2}{x-2}+\frac{1}{x-1}\) b) \(\frac{1}{x}-\frac{2}{x-2}+\frac{1}{x-1}\) d) \(\frac{1}{x}+\frac{1}{x-2}+\frac{1}{x-1}\)
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