The exponential function y(x)=Ce^ax satisfies the conditions y(0)=6 and y(1)=7.
(a) Find the constants C and a. Enter the exact value of a.
(b) Find y(2). Enter the exact answer. Simplify the answer so it does not contain any logarithms.
To find the constants C and a, we can use the given conditions:
(a)
We are given that y(0) = 6, which means when x = 0, y = 6.
Substituting these values into the exponential function:
y(0) = Ce^a(0) = C(1) = C = 6
So, we have found C = 6.
Now, we are given that y(1) = 7, which means when x = 1, y = 7.
Substituting these values into the exponential function:
y(1) = 6e^a(1) = 6e^a = 7
To find the value of a, we divide both sides of the equation by 6:
e^a = 7/6
To solve for a, we take the natural logarithm (ln) of both sides:
ln(e^a) = ln(7/6)
Using the property of logarithms (ln(e^x) = x):
a = ln(7/6)
Hence, the exact value of a is ln(7/6).
(b)
To find y(2), we substitute x = 2 into the exponential function and use the values we found for C and a:
y(2) = 6e^(ln(7/6)*2)
Using the property of exponents (e^(ln(x)) = x):
y(2) = 6(7/6)^2
Simplifying the expression:
y(2) = 6(49/36)
Simplifying further:
y(2) = 49/6
Therefore, the exact value of y(2) is 49/6.
To find the constants C and a in the exponential function y(x)=Ce^ax, we can use the given conditions y(0)=6 and y(1)=7.
(a) Finding C and a:
Step 1: Using the condition y(0)=6:
Substituting x=0 into the equation, we get:
y(0)=Ce^a(0)=C=6
Step 2: Using the condition y(1)=7:
Substituting x=1 into the equation, we get:
y(1)=Ce^a(1)=6e^a=7
Step 3: Solving for a:
To find the exact value of a, we divide both sides of the equation by 6:
e^a=7/6
To isolate a, we can take the natural logarithm (ln) of both sides:
ln(e^a)=ln(7/6)
a=ln(7/6)
So, we have found the constants C=6 and a=ln(7/6).
(b) Finding y(2):
Using the equation y(x)=Ce^ax and the values of C and a that we found previously, we can find y(2) by substituting x=2:
y(2)=6e^(ln(7/6)*2)=6(7/6)^2
To simplify the answer, we can compute the exponential and perform the calculations:
y(2)=6(49/36)
y(2)=49/6
Therefore, the exact value of y(2) is 49/6.
6 = Ce^0
C = 6
7 = 6e^a
a = ln(7/6)
y(x) = 6e^(ln(7/6)x) = 6(e^(ln(7/6)))^x = 6*(7/6)^x
y(2) = 6(7/6)^3 = 343/36