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Chapter 3: Problem 14

Graph the function on your grapher using a screen with smaller and smaller dimensions about the point \((c, f(c))\) until the graph looks like a straight line. Find the approximate slope of this line. What is \(f^{\prime}(c) ?\) $$ y=x^{3}, c=2 $$

Short Answer

Expert verified
The approximate slope at \((2, 8)\) is 12, so \(f'(2) = 12\).

Step by step solution

01

Graph the Original Function

Start by graphing the function \(y = x^3\) on a standard viewing screen that includes the point \((2, f(2))\). The graph of \(y = x^3\) is a smooth curve where the slope changes throughout its length.
02

Determine the Point of Interest

Evaluate the function at \(x = 2\): \(f(2) = 2^3 = 8\). The point of interest here is \((2, 8)\). This is the point around which we will zoom in to determine the slope.
03

Zoom in on the Graph

Start reducing the window size around the point \((2, 8)\). Each time you zoom, make adjustments to the viewing range on your grapher to focus in closer on \((2, 8)\). Continue this process until the curve appears to be a straight line.
04

Estimate the Slope

With the graph now looking like a straight line around \((2, 8)\), use your graphing tool to estimate the slope by taking two points very close to \((2, 8)\). Calculate the rise over run (\(\Delta y / \Delta x\)) between these points to find the slope.
05

Analytical Derivation (Optional Verification)

To verify the slope obtained, alternatively, calculate the derivative analytically: The derivative of \(y = x^3\) is \(f'(x) = 3x^2\). Substitute \(x = 2\) to find \(f'(2) = 3(2)^2 = 12\).

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Key Concepts

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

Graphing Functions
Graphing functions helps you visualize how they behave. It's like drawing the shape of the function's equation. For the function \( y = x^3 \), the graph is a smooth continuous curve with points that rise and fall as the values of \( x \) change. When you initially graph this function, it includes the point \((2, 8)\). This point is important because it tells us where on the graph we are focusing as we explore its slope. Graphing the entire function can help us see:
  • The overall shape, which in this case is cubic with a twisting curve.
  • How the slope of the tangent line changes at each point along the graph.
  • The point \((2, 8)\) where our analysis will be centered.
Slope Estimation
Slope estimation involves finding how steep a line is at a certain point. When zooming into the graph of \( y = x^3 \) around \((2, 8)\), it begins to resemble a straight line. This is because, as you zoom in infinitely, most curves look like a straight line. To estimate the slope of the graph at this point:
  • Choose two points very close to \((2, 8)\) on the graph.
  • Use these points to calculate the 'rise over run' or the change in \( y \) over the change in \( x \).
Estimation provides a numerical value that represents the slope at that point, which tells us how fast the function's value is changing. This process provides an approximate rate of change that can be confirmed by other methods like analytical derivation.
Zooming Technique
The zooming technique allows you to investigate the behavior of a function more thoroughly at a specific point. By focusing the graph around \((2, 8)\), we reduce the visible area on the graph. This makes it easier to see the function's detail at that part of the graph. Steps involved in zooming include:
  • Gradually make the graph's visible area smaller, centering around the point of interest.
  • Observe how the curvature begins to look linear as you increase the zoom.
  • Finely adjust the graph's window so the curve is straight enough to estimate a slope.
The zooming technique amplifies the section of interest, allowing for more precise observations. It's particularly useful in considering how local changes occur in a function's behavior.
Analytical Derivation
Analytical derivation is a process where you use calculus to find the exact slope of a function at a given point. This is typically more accurate than purely visual or estimation techniques. For the function \( y = x^3 \), we derive its slope using calculus rules:
  • The rule for finding the derivative of \( x^3 \) is \( 3x^2 \).
  • Substitute \( x = 2 \) into the derivative \( f'(x) = 3x^2 \).
  • Compute \( f'(2) = 3(2)^2 = 12 \).
The outcome is the exact slope of the function at \( x = 2 \). By confirming our estimation with this calculation, we gain confidence that our visual slope estimate closely approximated the true slope. Analytical derivation thus acts as a verification tool, enhancing our understanding of the function's behavior at specific points.

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Most popular questions from this chapter

Cost The cost equation for a certain product is given by \(C(x)=\sqrt{x^{3}}(x+100)+1000 .\) Use the linear approximation formula to estimate the change in costs as \(x\) changes from 900 to \(904 .\) Find the slope of the tangent line at \(x=900\) by using your computer or graphing calculator to plot the tangent line at the appropriate point or use a screen with smaller and smaller dimensions until the graph looks like a straight line. (See the Technology Resource Manual.)

Find the limits graphically. Then confirm algebraically. $$ \lim _{h \rightarrow 0} \frac{(2+h)^{2}-4}{h} $$

Cost Curve Dean \(^{39}\) made a statistical estimation of the average cost- output relationship for a shoe chain for 1938 . He found that if \(x\) is the output in thousands of pairs of shoes and \(y\) is the average cost in dollars, \(y\) was approximately given by \(y=0.06204 x^{2}-4.063 x+112.76 .\) Graph on your computer or graphing calculator using a screen with dimensions [0,75.2] by \([0,120] .\) Have your computer or graphing calculator obtain the tangent line when \(x\) is \(12,\) \(20,32,40,\) and \(52 .\) In each case, relate the slope of the tangent to the rate of change. Interpret what each of these numbers means. On the basis of this model, what happens to average costs as output increases?

Corn Yield per Acre Atwood and Helmers \(^{33}\) studied the effect of nitrogen fertilizer on corn yields in Nebraska. Nitrate contamination of groundwater in Nebraska has become such a serious problem that actions such as nitrogen rationing and taxation are under consideration. Nitrogen fertilizer is needed to increase yields, but an excess of nitrogen has been found not to help increase yields. Atwood and Helmers created a mathematical model given approximately by the equation \(Y(N)=59+\) \(0.8 N-0.003 N^{2},\) where \(N\) is the amount of fertilizer in pounds per acre and \(Y\) is the yield in bushels per acre. Graph this equation on the interval \([50,200] .\) Find the instantaneous rate of change of yield with respect to the amount of fertilizer when (a) \(N=100\) (common rate), (b) \(N=200 .\) Give units and interpret your answer.

Growing Season Start Jobbagy and colleagues \(^{35}\) determined a mathematical model given by the equation \(G(T)=\) \(278-7.1 T-1.1 T^{2},\) where \(G\) is the start of the growing season in the Patagonian steppe in Julian days (day \(1=\) January 1 ) and \(T\) is the mean July temperature in degrees Celsius. Graph this equation on the interval \([-3,5] .\) Find the instantaneous rate of change of the day of the start of the growing season with respect to the mean July temperature when (a) \(T=-3,\) (b) \(T=0\). Give units and interpret your answer.
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