how do i find if a transition is smooth between two functions using derivatives?

Question

i have two functions

one is an e^x function f(x) = -e^x+46.46 {-4 < x < 0} . the derivative is -e^x

the second is a straight line f(x)= -1.32x+45.46 {0 < x < 16.84}. the derivative is -1.32

i have been told i can check if the transition between the two function can be checked by using the derivatives.

how do i find this transition?

Answer 1

If the functions are continuous at the point in question AND if the two functions have the same slope at that point in question then the transition is smooth.

First find the intersection point of the two functions. In this case, that is given to us as x = 0.

Check to ensure the function values are the same. For f(x) = -e^x + 46.46, we get f(0) = 45,46. For f(x) = -1.32x + 45.46, we get f(0) = 45.46 so the combination is continuous at x = 0.

Now check the derivatives at x = 0 to ensure they both have the same values (i.e., the slopes are the same)

For f(x) = -e^x + 46.46, we get f '(x) = -e^x so f '(0) = -1. For f(x) = -1.32x + 45.46, f '(x) = -1.32 so f '(0) = -1.32. The slopes are NOT equal therefore the function is NOT smooth