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Chapter 2: Problem 80

Sketch the graph of a differentiable function \(f\) such that \(f>0\) and \(f^{\prime}<0\) for all real numbers \(x\).

Short Answer

Expert verified
The graph of a function that is always positive and decreasing would start in the first quadrant and approach the X-axis as it moves to the right, never touching or crossing it.

Step by step solution

01

Graph Initialization

The graph initialization is done by marking the X and Y axes. Consider the intersection of X and Y axes as the origin, where \(x = 0\) and \(y = 0\). You can also consider adding tick marks along the axes in order to better visualize the scale of the graph.
02

Determine Function Behavior

As given, \(f > 0\) and \(f' < 0\) for all real numbers \(x\). This implies that on the graph, all points of the function will be above the X-axis (since \(f > 0\)), and the function shows decreasing behavior (since \(f' < 0\)).
03

Sketch the Graph

Given that the function has no real roots and is always decreasing, start the sketch from any point in the 1st quadrant (where both X and Y values are positive) and draw a decreasing curve toward the X-axis. However, it's crucial to make sure the curve doesn't touch or cross the X-axis as the function is positive for all real numbers, meaning it never becomes zero or negative.

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Most popular questions from this chapter

Consider the equation \(x^{4}=4\left(4 x^{2}-y^{2}\right)\). (a) Use a graphing utility to graph the equation. (b) Find and graph the four tangent lines to the curve for \(y=3\). (c) Find the exact coordinates of the point of intersection of the two tangent lines in the first quadrant.

The displacement from equilibrium of an object in harmonic motion on the end of a spring is \(y=\frac{1}{3} \cos 12 t-\frac{1}{4} \sin 12 t\) where \(y\) is measured in feet and \(t\) is the time in seconds. Determine the position and velocity of the object when \(t=\pi / 8\).

Find the second derivative of the function. \(g(x)=\sqrt{x}+e^{x} \ln x\)

Find the derivative of the function. \(h(x)=\log _{3} \frac{x \sqrt{x-1}}{2}\)

Linear and Quadratic Approximations The linear and quadratic approximations of a function \(f\) at \(x=a\) are \(P_{1}(x)=f^{\prime}(a)(x-a)+f(a)\) and \(P_{2}(x)=\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}+f^{\prime}(a)(x-a)+f(a)\) \(\begin{array}{llll}\text { In Exercises } & 133-136, & \text { (a) find the specified linear and }\end{array}\) quadratic approximations of \(f,\) (b) use a graphing utility to graph \(f\) and the approximations, (c) determine whether \(P_{1}\) or \(P_{2}\) is the better approximation, and (d) state how the accuracy changes as you move farther from \(x=a\). $$ \begin{array}{l} f(x)=x \ln x \\ a=1 \end{array} $$
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