sorry for the probably simple question, but i haven't done this stuff in a long, long time.
thought i had this right, but again not finding the answer
which of the following gives the slope of the tangent line to the graph of y=2^(1-x) at x=2?
a)-1/2
b)1/2
c)-2
d)2
e)-(ln2)/2
so... it thought y=2^1-x)
then y'=(1-x)(2^(1-x-1))
at x=2
then y'=(1-2)(2^(1-2-1))
y'= -1(2^(-2))
y'= -1/4
any help figuring out where i went wrong would be greatly appreciated.
thank you
y = 2^(1-x) is an exponential function, which is differentiated using natural logarithms
dy/dx = 2^(1-x) (-1) ( ln2)
so when x = 2,
dy/dx = -ln2(2^(1-2)
= -ln2 (1/2) = (-1/2)ln2 , looks like e
as soon as I had my derivative which contained ln2, I knew that the first 4 choices were not correct.
You found the derivative as if it was a polynomial function where the terms are such that the base is the variable, such as 3x^5
e.g.
compare 4^x with x^4
if y = x^4
dy/dx = 4x^3
but
if y = 4^x
dy/dx = ln4 (4^x)
Thanks Reiny,really appreciate the extra explanation. get it now. thank you
No problem at all! Let's go through the steps to find the slope of the tangent line to the graph of y = 2^(1-x) at x = 2.
To find the slope of the tangent line, we need to find the derivative of the function y = 2^(1-x) and evaluate it at x = 2.
Step 1: Finding the derivative of y = 2^(1-x)
To differentiate the function y = 2^(1-x), we will use the chain rule. The chain rule states that if we have a composition of functions, f(g(x)), the derivative is given by f'(g(x)) * g'(x).
Let g(x) = 1-x and f(x) = 2^x. Applying the chain rule, we get:
y' = f'(g(x)) * g'(x)
= (2^g(x)) * (-1)
Step 2: Evaluating the derivative at x = 2
Now we plug in x = 2 into the derivative formula we found in Step 1.
y'(2) = (2^(g(2))) * (-1)
= (2^(1-2)) * (-1)
= (2^(-1)) * (-1)
= (-1/2).
Therefore, the slope of the tangent line to the graph of y = 2^(1-x) at x = 2 is -1/2.
Looking at the options given:
a) -1/2 (correct)
b) 1/2 (incorrect)
c) -2 (incorrect)
d) 2 (incorrect)
e) -(ln2)/2 (incorrect)
So, the correct answer is option a) -1/2.
You did a great job calculating the derivative correctly, it seems you made a simple mistake when evaluating the derivative at x = 2. I hope this helps clarify the solution for you!