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Chapter 12: Problem 79

Sketch the graph of a function whose derivative is always positive.

Short Answer

Expert verified
A graph of a function where the derivative is always positive is any upward slanting line or curve. Note that the function must always be increasing.

Step by step solution

01

Identify properties of function

A function will always be increasing if its derivative is always positive. This is a fundamental principle of calculus.
02

Draw the function

Begin at any point on the y-axis, since this exercise does not specify where the function should start. Draw a line that keeps moving upwards, and not down or staying flat. Make the slope of the line positive, meaning it should rise to the right. A line drawing doesn't need to be perfect but it should clearly show that the function is always increasing.

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Key Concepts

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

Derivatives
When we talk about derivatives in calculus, we're essentially discussing the rate at which a function changes at any given point. A derivative is a fundamental concept that helps us understand how a function behaves concerning different variables.
  • The derivative of a function at a particular point is the slope of the tangent line to the graph of the function at that point.
  • For a function \( f(x) \), its derivative is often denoted as \( f'(x) \) or \( \frac{df}{dx} \).
  • The value of the derivative tells us how much the function's value changes for a small change in \( x \).
For example, if \( f'(x) = 3 \) at some point \( x = a \), it implies that for a tiny increase in \( x \), the function's value will increase by 3 times that tiny increase.
Increasing Functions
An increasing function is one where as you move from left to right on its graph, the function continually rises. In other words, for any two points \( x_1 \) and \( x_2 \) on the graph, if \( x_1 < x_2 \), then \( f(x_1) < f(x_2) \). This property is heavily tied to the concept of derivatives.
  • If the derivative \( f'(x) > 0 \) for all \( x \) in an interval, the function is increasing on that interval.
  • This is because a positive derivative indicates that the slope of the tangent line is positive, which means the function's graph is rising.
  • Conversely, if \( f'(x) < 0 \) in an interval, the function is decreasing in that interval.
Using the derivative to determine intervals of increase and decrease helps us piece together the overall shape and behavior of a function.
Graphing Functions
Graphing functions is a visual way to understand how functions behave. When graphing, your goal is to plot points calculated from the function and connect these points to form a curve or line that accurately represents the function.
  • To sketch a graph of a function known to be always increasing (as indicated by a positive derivative), start by selecting a point, any point, as the initial value.
  • From this starting point, ensure the slope of the curve is consistently moving upwards, reflecting that it's continuously increasing.
  • Remember, the function should not have any flat spots (where the derivative is zero) or downward slopes (where the derivative is negative), ensuring it keeps rising.
While specific plotting may not be perfect, the endeavor is to ensure the overall trajectory of the function aligns with its properties, providing a meaningful interpretation of its behavior.

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Most popular questions from this chapter

Evaluating a Limit at Infinity In Exercises \(9 - 28\) , find the limit (if it exists). If the limit does not exist, then explain why. Use a graphing utility to verify your result graphically. $$ \lim _ { t \rightarrow \infty } \frac { t ^ { 2 } } { t + 3 } $$

Evaluating a Limit at Infinity In Exercises \(9 - 28\) , find the limit (if it exists). If the limit does not exist, then explain why. Use a graphing utility to verify your result graphically. $$ \lim _ { x \rightarrow - \infty } \frac { 3 x ^ { 2 } + 1 } { 4 x ^ { 2 } - 5 } $$

The path of a ball thrown by a child is modeled by \(y=-x^{2}+5 x+2\) where \(y\) is the height of the ball (in feet) and \(x\) is the horizontal distance (in feet) from the point from which the child threw the ball. Using your knowledge of the slopes of tangent lines, show that the height of the ball is increasing on the interval \([0,2]\) and decreasing on the interval \([3,5] .\) Explain your reasoning.

Finding the Area of a Region, use the limit process to find the area of the region bounded by the graph of the function and the \(x\) -axis over the specified interval. $$\begin{array}{ll}{\text { Function }} & {\text { Interval }} \\\ {f(x)={\frac{1}{4}} \left(x^{2}+4 x\right){}} & {[1,4]}\end{array}$$

Finding the Limit of a Sequence In Exercises \(45 - 54\) , write the first five terms of the sequence and find the limit of the sequence (if it exists). If the limit does not exist, then explain why. Assume \(n\) begins with \(1 . \) $$ a _ { n } = \frac { n + 1 } { n ^ { 2 } + 1 } $$
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