Limits and estimating slope

In summary, the question is asking for an estimate of the slope of the tangent line to the graph of the exponential function y=2^x at the point (0,1). This can be found by plugging a small x value into the formula (2^x-1)/x. The limit itself is more difficult to find.
  • #1
fstam2
10
0
Hey there, I need a little push in the right direction.
Here is the question:
The slope of the tangent line to the graph of the exponential function
y=2^x at the point (0,1) is lim x approaches 0 (2^x-1)/x.
Estimate the slope to three decimal places.
Where I am getting confused is which formula do I plug (x) into to find a secant slope?
I hope I asked the right question.
Thanks
 
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  • #2
fstam2 said:
Hey there, I need a little push in the right direction.
Here is the question:
The slope of the tangent line to the graph of the exponential function
y=2^x at the point (0,1) is lim x approaches 0 (2^x-1)/x.
Estimate the slope to three decimal places.
Where I am getting confused is which formula do I plug (x) into to find a secant slope?
I hope I asked the right question.
Thanks

Yes, since you are only asked to "estimate the slope to three decimal places" you just need to plug a small enough x into that secant formula which is exactly what you wrote: (2x-1)/x. That should take about 5 seconds using a calculator!

(Finding the limit itself in order to find the actual formula is much harder!)
 
  • #3
Thank you for your help
 

1. What is the definition of a limit in calculus?

A limit is defined as the value that a function or sequence approaches as the input or index approaches a certain value. It represents the behavior of a function near a specific point or as the input approaches a certain value.

2. How do you find the limit of a function?

To find the limit of a function, you can use several techniques such as direct substitution, factoring, rationalization, and the squeeze theorem. These techniques help to simplify the function and determine the value it approaches as the input approaches a specific value.

3. What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the behavior of a function as the input approaches a specific value from one side, either the left or the right. A two-sided limit considers the behavior of a function as the input approaches the value from both sides, the left and the right.

4. How do you estimate the slope of a curve using limits?

The slope of a curve can be estimated by finding the derivative of the function at a specific point. The derivative represents the slope of the tangent line at that point, which can be calculated using the limit definition of the derivative. This involves finding the limit of the function as the change in the input approaches zero.

5. Why is it important to understand limits and estimating slope in calculus?

Limits and estimating slope are important concepts in calculus because they allow us to analyze the behavior of functions and make predictions about their values. They also help us to find the maximum and minimum values of a function, which is useful in many real-world applications such as optimization and modeling. Additionally, understanding limits and slope is crucial for further studying advanced calculus topics, such as derivatives and integrals.

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