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Chapter 3: Problem 109

For each of the functions in Exercises \(109-112,\) graph \(f, f^{\prime}\) and \(f^{\prime \prime}\). $$f(x)=e^{x}$$

Short Answer

Expert verified
All three graphs of \( f(x) = e^x, f'(x) = e^x \), and \( f''(x) = e^x \) are identical and show exponential growth.

Step by step solution

01

- Graph the function f(x)

The function given is \( f(x) = e^{x} \). This is an exponential function with base e. To graph this function, plot a few key points such as (0,1), (1, e), and (-1, 1/e), and then draw a smooth curve passing through these points. The graph will show exponential growth.
02

- Determine the first derivative f'(x)

Find the first derivative of \( f(x) \). The first derivative of \( f(x) = e^x \) is \( f'(x) = e^x \). This means the function has the same rate of change as the original function.
03

- Graph the first derivative f'(x)

Since \( f'(x) = e^x \), the graph of the derivative will be identical to the graph of \( f(x) \). Use the same points (0,1), (1,e), and (-1,1/e) to plot this graph. The graph will also demonstrate exponential growth.
04

- Determine the second derivative f''(x)

Find the second derivative of \( f(x) \). The second derivative of \( f(x) = e^x \) is \( f''(x) = e^x \). Just like before, the rate of change of the first derivative is the same exponential function.
05

- Graph the second derivative f''(x)

Since \( f''(x) = e^x \), the graph of the second derivative will also be identical to \( f(x) \) and \( f'(x) \). Use the same key points to plot this graph. It demonstrates exponential growth just like the previous graphs.

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Key Concepts

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

graphing exponential functions
Exponential functions are unique in their rapid growth. The function given in the exercise is \( f(x) = e^x \). Here, \( e \) is a special mathematical constant approximately equal to 2.718. The graph of \( e^x \) is always increasing for all values of \( x \) and never touches the x-axis, but gets infinitely close to it as \( x \to -\infty \). To graph this function, we can choose a few key points:
  • \( f(0) = e^0 = 1 \)
  • \( f(1) = e^1 = e \)
  • \( f(-1) = e^{-1} = \frac{1}{e} \)
Plotting these points will show rapid exponential growth. Drawing a smooth curve through these points gives the graph of the function.
first derivative
The first derivative of a function tells us the rate at which the function's value changes. In this case, the first derivative of \( f(x) = e^x \) is \( f'(x) = e^x \). This indicates that the rate of change of the function is equal to the function itself. To graph \( f'(x) \), use the same key points as when graphing \( f(x) \):
  • (0,1)
  • (1,e)
  • (-1,1/e)
Drawing a smooth curve through these points will show an identical exponential growth as the original function.
second derivative
The second derivative of a function provides information about the curvature or concavity of the graph. For \( f(x) = e^x \), the second derivative \( f''(x) = e^x \) shows that the curvature of the function is also an exponential function. This means the graph of \( f''(x) \) will be identical to the graphs of both \( f(x) \) and \( f'(x) \). Using points like (0,1), (1,e), and (-1,1/e) and drawing a smooth curve will illustrate exponential growth, just as in the previous graphs.
calculus
Calculus is the mathematical study of change. When dealing with exponential functions like \( f(x) = e^x \), calculus helps us understand rates of change and curvature through derivatives.

Key concepts include:
  • First Derivative: The rate of change of the function, \( f'(x) \). For \( f(x) = e^x \), we find \( f'(x) = e^x \), meaning the growth rate matches the original function.
  • Second Derivative: The rate of change of the first derivative, \( f''(x) \). For \( f(x) = e^x \), we get \( f''(x) = e^x \), showing the curvature follows the same exponential growth.
Understanding these derivatives and their graphs helps us master the behavior and characteristics of exponential functions in calculus.

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Most popular questions from this chapter

Sunshine Gardens determines the following demand function during early summer for tomato plants: $$q=D(x)=\frac{2 x+300}{10 x+11}$$ where \(q\) is the number of plants sold per day when the price is \(x\) dollars per plant. (GRAPH CANNOT COPY) a) Find the elasticity. b) Find the elasticity when \(x=3\) c) At $$ 3$ per plant, will a small increase in price cause the total revenue to increase or decrease?

Solve \(P=P_{0} e^{k t}\) for \(t\)

Solve for \(t\). $$e^{-t}=0.01$$

The Hullian learning model asserts that the probability p of mastering a task after \(t\) learning trials is approximated by \(p(t)=1-e^{-k t}\) where \(k\) is a constant that depends on the task to be learned. Suppose that a new dance is taught to an aerobics class. For this particular dance, the constant \(k=0.28\) a) What is the probability of mastering the dance's steps in 1 trial? 2 trials? 5 trials? 11 trials? 16 trials? 20 trials? b) Find the rate of change, \(p^{\prime}(t)\) c) Sketch a graph of the function.

Iodine-131 has a decay rate of \(9.6 \%\) per day. The rate of change of an amount \(N\) of iodine- 131 is given by \(\frac{d N}{d t}=-0.096 N\) where \(t\) is the number of days since the decay began. a) Let \(N_{0}\) represent the amount of iodine-131 present at \(t=0 .\) Find the exponential function that models the situation. b) Suppose that \(500 \mathrm{g}\) of iodine- 131 is present at \(t=0\) How much will remain after 4 days? c) After how many days will half of the 500 g of iodine-131 remain?
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