Problem 17
Question
Sketch the graph of \(f(x),\) and use this graph to sketch the graph of \(f^{\prime}(x)\). $$f(x)=e^{x}$$
Step-by-Step Solution
Verified Answer
Both the graph of \(f(x)\) and \(f'(x)\) are the same exponential curve \(e^x\).
1Step 1: Understand the Function
We have the function \( f(x) = e^x \), which is the exponential function. It is a smooth and continuous curve that starts from near 0 when \( x \) is negative and increases rapidly as \( x \) becomes positive.
2Step 2: Sketch the Graph of \(f(x)\)
Draw the graph of the function \( f(x) = e^x \). This function has the y-intercept at \( (0, 1) \) because \( e^0 = 1 \) and is always increasing. It is approaches the x-axis on the negative side, never touching it, and rises exponentially as \( x \) increases.
3Step 3: Determine the Derivative Function
The derivative of \( f(x) = e^x \) is \( f'(x) = e^x \). This means that the rate of change (slope) of \( f(x) \) at any point \( x \) is the same as its value \( f(x) \) at that point.
4Step 4: Analyze the Derivative Function
Because \( f'(x) = e^x \) is the same as \( f(x) \), the derivative is also an exponentially increasing function. At every point on the graph of \( f(x) \), the slope is equal to the value of the function itself.
5Step 5: Sketch the Graph of \(f^{\prime}(x)\)
Since \( f'(x) = e^x \) is identical to \( f(x) = e^x \), the graph of the derivative is the same as the original graph of the exponential function. Draw \( f'(x) = e^x \) on the same axes, showing it as another upward-sloping curve starting at \( (0,1) \).
Key Concepts
Exponential functionDerivativeGraph sketching
Exponential function
The exponential function is a crucial concept in calculus. It is beneficial to understand its key characteristics. The function is written as \( f(x) = e^x \), where \( e \) is an irrational constant approximately equal to 2.71828. This function produces a graph that is smooth and continuous. One of its standout features is that
Recognizing the graph's shape is vital. It helps to predict how the function behaves without computing exact values, especially when starting your graph sketching. The exponential pattern of steep growth is what gives functions of this type their unique calculus properties.
- it never touches the x-axis but gets arbitrarily close as \( x \) becomes more negative,
- begins with a y-intercept at \( (0, 1) \) since \( e^0 = 1 \),
- rapidly increases as \( x \) moves into positive values.
Recognizing the graph's shape is vital. It helps to predict how the function behaves without computing exact values, especially when starting your graph sketching. The exponential pattern of steep growth is what gives functions of this type their unique calculus properties.
Derivative
In calculus, finding the derivative is like finding a function's equivalent of an instant snapshot of its change at a particular point. For \( f(x) = e^x \), the derivative is particularly straightforward: \( f'(x) = e^x \). This is because the rate of change of an exponential function with base \( e \) is identical to its current value.
This property underscores why \( e^x \) is such an essential function in mathematics. At any given point on the curve \( f(x) = e^x \), the slope of the tangent line (which presents the movement direction) is equal to the original function value itself. This unique feature means that as \( x \) changes, the derivative doesn't just indicate whether the function increases or decreases—it's a direct indicator of how rapid that increase is.
For students tackling derivatives, recognizing these properties in \( e^x \) simplifies understanding differential calculus, as it provides a clear, consistent model of how a function's rate of increase can be directly tied to its current state.
This property underscores why \( e^x \) is such an essential function in mathematics. At any given point on the curve \( f(x) = e^x \), the slope of the tangent line (which presents the movement direction) is equal to the original function value itself. This unique feature means that as \( x \) changes, the derivative doesn't just indicate whether the function increases or decreases—it's a direct indicator of how rapid that increase is.
For students tackling derivatives, recognizing these properties in \( e^x \) simplifies understanding differential calculus, as it provides a clear, consistent model of how a function's rate of increase can be directly tied to its current state.
Graph sketching
Graph sketching in calculus involves both visualizing the function and determining its key points, like intercepts and asymptotic behavior. Let's use \( f(x) = e^x \) to understand this.
To begin sketching, start by plotting the y-intercept at \( (0, 1) \). This provides an anchoring point. From this point, as you move to the right (positive \( x \)), let the curve rise sharply, indicative of the exponential increase. As the curve progresses leftward into negative \( x \), it approaches the x-axis without ever touching it, reflecting the negative asymptote nature of \( e^x \).
When sketching the derivative graph \( f'(x) = e^x \), it mirrors the original curve since the derivative was the same as the function itself. Hence, the graph shares all features with \( f(x) = e^x \) upon sketching, showcasing identical increase patterns and intercepts. This exercise demonstrates the intrinsic connection between a function and its derivative in visual terms and allows a clearer understanding of how changes in functions are expressed graphically.
By practicing graph sketching, students become more adept at predicting how functions behave and in assessing the results visually, thus solidifying their grasp of calculus concepts.
To begin sketching, start by plotting the y-intercept at \( (0, 1) \). This provides an anchoring point. From this point, as you move to the right (positive \( x \)), let the curve rise sharply, indicative of the exponential increase. As the curve progresses leftward into negative \( x \), it approaches the x-axis without ever touching it, reflecting the negative asymptote nature of \( e^x \).
When sketching the derivative graph \( f'(x) = e^x \), it mirrors the original curve since the derivative was the same as the function itself. Hence, the graph shares all features with \( f(x) = e^x \) upon sketching, showcasing identical increase patterns and intercepts. This exercise demonstrates the intrinsic connection between a function and its derivative in visual terms and allows a clearer understanding of how changes in functions are expressed graphically.
By practicing graph sketching, students become more adept at predicting how functions behave and in assessing the results visually, thus solidifying their grasp of calculus concepts.
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