How do you find the equation of the line tangent to #y=2^x# that passes through the point (1,0)?

Answer 1
For #y=2^x#, we have #y'=2^x ln2#
The tangent line at #x=a#, has slope #m = 2^a ln2# and passes through the point #(a, 2^a)#.
So the equation of the tangent at the point #(a, 2^a)# is:
#y-2^a=(2^aln2)(x-a) #
We want #(1,0)# to be on the tangent line, so we want #(1,0)# to be a solution to the equation of the line.

That is:

We require: #0-2^a=(2^aln2)(1-a) #
Solve for #a = 1/ln2 +1 = lne/ln2 + 1 = log_2e +1#
So, #2^a = 2e#
#y-2^a=(2^aln2)(x-a) # becomes: #y-2e=(2eln2)(x-1/ln2 -1 ) #
Which we can solve for #y# to get:
#y=(2eln2)x- 2eln2 #
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Answer 2

To find the equation of the line tangent to y=2^x that passes through the point (1,0), we need to find the slope of the tangent line at the point (1,0) and then use the point-slope form of a linear equation.

The slope of the tangent line can be found by taking the derivative of the function y=2^x with respect to x. The derivative of 2^x is ln(2) * 2^x.

Substituting x=1 into the derivative, we get ln(2) * 2^1 = 2ln(2).

So, the slope of the tangent line at the point (1,0) is 2ln(2).

Using the point-slope form of a linear equation, y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope, we can substitute the values to find the equation of the tangent line.

Substituting (1,0) and 2ln(2) into the equation, we get y - 0 = 2ln(2)(x - 1).

Simplifying, the equation of the line tangent to y=2^x that passes through the point (1,0) is y = 2ln(2)(x - 1).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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