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Chapter 23: Problem 27

Find the slope of a line tangent to the curve of each of the given functions for the given values of \(x\) $$y=35 x-2 x^{4} \quad(x=2)$$

Short Answer

Expert verified
The slope of the tangent line at \( x = 2 \) is \(-29\).

Step by step solution

01

Understand the problem

We need to find the slope of the tangent line to the curve at a specific point, which corresponds to the derivative of the function evaluated at that point. The function is given as \( y = 35x - 2x^4 \) and the point of interest is \( x = 2 \).
02

Differentiate the function

Find the derivative of the function \( y = 35x - 2x^4 \). Use the power rule, which states that the derivative of \( x^n \) is \( nx^{n-1} \). The derivative of \( y \) is \( y' = \frac{d}{dx}(35x) - \frac{d}{dx}(2x^4) \).
03

Apply differentiation rules

Differentiate the given function: \[ \frac{d}{dx}(35x) = 35 \] and \[ \frac{d}{dx}(2x^4) = 8x^3 \]. Therefore, \( y' = 35 - 8x^3 \).
04

Evaluate the derivative at the given point

Substitute \( x = 2 \) into the derivative \( y' = 35 - 8x^3 \). Compute \[ y'(2) = 35 - 8(2)^3 = 35 - 64 = -29 \].
05

Interpret the result

The slope of the tangent line to the curve at \( x = 2 \) is \(-29\). This means the line is decreasing steeply at that point.

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Key Concepts

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

Tangent Line
Let's explore what a tangent line is all about. A tangent line is like a straight path that just touches a curve at a specific point, without crossing through it. Imagine you have a roller coaster track. If you could draw a perfectly flat piece of track that only touches the roller coaster at one point, that's your tangent line.
  • It gives us the direction in which the curve is headed at that exact point.
  • The tangent line can slope upwards or downwards, getting its direction from the curve.
The slope of the tangent line at any point on the curve is what we use to understand how a function is behaving locally. It tells us if the function is increasing or decreasing at that point. When we say the slope is -29, as in your exercise, it shows the tangent is sloping downwards steeply.
Slope of a Function
The slope of a function at any given point is akin to how steep or flat the tangent line is at that specific spot. We can think of the slope as a way to measure how a function is changing. Is it going up, going down, or staying flat?
  • If the slope is positive, the function is rising as you move along the x-axis.
  • If the slope is negative, like -29 in your exercise, the function is falling.
  • A slope of zero means the function is flat at that point.
To find the slope, we calculate the derivative of a function at a certain point. This derivative is a formula that tells us the rate of change of the function's output as its input changes. By plugging in the desired value of x, you get the slope (or steepness) of the function at that exact point.
Power Rule
The power rule is one of the most straightforward rules in calculus and a powerful tool for finding derivatives quickly. It helps in finding out how a function changes by allowing us to take derivatives of power functions very efficiently.Here's how the power rule works:
  • If you have a function of the form \( x^n \), its derivative is \( nx^{n-1} \).
  • Also applies to functions multiplied by coefficients, as seen in the original exercise with terms like \( 2x^4 \).
By applying the power rule, we simplify complex differentiation problems. In your exercise, it was used first to find \( \frac{d}{dx}(2x^4) = 8x^3 \), allowing us to hurdle past intricate calculations efficiently when determining the slope. This makes it quicker to see how steep or flat a curve will be at any given point. Remember, practicing with the power rule will make it second nature over time!

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Most popular questions from this chapter

Find the second derivative of each of the given functions. $$f(x)=\frac{7.5}{\sqrt{3-4 x}}$$

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