Find the function f(x) described by the given initial value problem
f'(x)=6^x, f(1)=0
f(x)6^x ln(6)
To find f(x), first you need to integrate f'(x).
good start
f(x) = ln6*6^x + c
given : f(1) = 0 ----> implies that (1,0) satisfies the equation
0 = ln6(6^0) + c , but 6^0 = 1
c = -ln6
then f(x) = ....
Hmmm. seems to me that
∫ a^x dx = 1/lna * a^x + C
good catch
I multiplied instead of divided.
∫ 6^x dx = 1/ln(6) * 6^x + C ,f(1)=0
0=6^x dx= 1ln(6) * 6^x + C
is this right so far
To find the function f(x) described by the given initial value problem, we need to solve the differential equation f'(x) = 6^x with the initial condition f(1) = 0.
To solve the differential equation, we can integrate both sides of the equation with respect to x. This gives us:
∫ f'(x) dx = ∫ 6^x dx
Integration of f'(x) with respect to x gives us the original function f(x). The integral of 6^x with respect to x can be found using the formula for the integral of a power function. The integral of 6^x is (1/ln(6)) * 6^x.
So, we have:
f(x) = (1/ln(6)) * 6^x + C
Where C is the constant of integration. To find the value of C, we use the initial condition f(1) = 0. Substituting x = 1 and f(1) = 0 into the equation, we get:
0 = (1/ln(6)) * 6^1 + C
0 = 6/ln(6) + C
Solving for C, we have:
C = -6/ln(6)
The final function f(x) is:
f(x) = (1/ln(6)) * 6^x - 6/ln(6)