Painguy said:Homework Statement
For what values of a are y=a^x and y=1+x tangent at x=0? ExplainHomework Equations
y1=1+x
y2=a^x
y2'=a^xln(a)
y1'=1The Attempt at a Solution
Since both equations are tangent at x=0 i set their derivatives equal to each other in hopes of getting a a^xln(a)=1 I then substitute x in that equation with 0 and end up with ln(a)=1.
I end up with a = e. Is this correct?
No i did not, but in hindsight i should have done that. Thanks for the reminder :)Dick said:It sure is correct. You also checked that y1(0)=y2(0), yes?
To find the values of A for two functions to be tangent at a point, you will need to use the derivative of both functions. Set the derivatives equal to each other and solve for A. This will give you the value of A that makes the two functions tangent at the given point.
Finding the values of A for two functions to be tangent at a point is important because it allows us to determine the point of tangency between the two functions. This point represents the point of intersection between the two functions where their slopes are equal.
No, two functions can only have one value of A that makes them tangent at a point. This is because the derivative of a function is unique at a given point, and thus the value of A that makes the derivative of both functions equal will also be unique.
To verify if the values of A you have found make the two functions tangent at a point, you can plug in the value of A into both functions and calculate their derivatives. If the derivatives are equal at the given point, then the two functions are tangent at that point.
One limitation of using this method is that it only works for continuous functions. If one or both of the functions are not continuous at the given point, then this method cannot be used to find the values of A that make them tangent. Additionally, this method may not work for more complex functions that cannot be easily differentiated.