Given F(2)=1,F'(2)=5,F(4)=3,F'(4)=7 and G(4)=2,G'(4)=6,G(3)=4,G'(3)=8, find:
a)H(4)if H(x)= F(G(x))
b)H'(4)if H(x)= F(G(x))
c)H(4)if H(x)= G(F(x))
d)H'(4)if H(x)= G(F(x))
e)H'4 if H(x)= F(x)/G(x)
Please Help!
a) H(4) = F(G(4)) = F(2) = 1
b) H'(4) = F'(G(4)) * G'(4) = F'(2) * 6 = 5*6 = 30
c) H(4) = G(F(4)) = G(3) = 4
d) H'(4) = G'(F(4)) * F(4) = G'(3) * 3 = 8*3 = 24
e) H'(4) = (F'(4) G(4) - F(4) G'(4))/[G(4)]^2 = (7*2-3*6)/4 = -1
Thank you
Sure, I can help you with that!
To find the values of H(4) and H'(4) in each case, we need to apply the appropriate rules of calculus and use the given information about the functions and their derivatives at specific points.
a) H(x) = F(G(x))
To find H(4), we substitute x = 4 into the function H(x) = F(G(x)).
First, find G(4) and substitute it into F(x):
G(4) = 2.
Now, substitute G(4) = 2 into F(x):
F(2) = 1.
Therefore, H(4) = F(G(4)) = F(2) = 1.
b) H(x) = F(G(x))
To find H'(4), we apply the chain rule. The chain rule states that if H(x) = F(G(x)), then H'(x) = F'(G(x)) * G'(x).
First, find G'(4) and substitute it into F(x):
G'(4) = 6.
Now, substitute G'(4) = 6 into F'(x):
F'(4) = 7.
Therefore, H'(4) = F'(G(4)) * G'(4) = F'(2) * 6 = 5 * 6 = 30.
c) H(x) = G(F(x))
To find H(4), we substitute x = 4 into the function H(x) = G(F(x)).
First, find F(4) and substitute it into G(x):
F(4) = 3.
Now, substitute F(4) = 3 into G(x):
G(3) = 4.
Therefore, H(4) = G(F(4)) = G(3) = 4.
d) H(x) = G(F(x))
To find H'(4), we apply the chain rule. The chain rule states that if H(x) = G(F(x)), then H'(x) = G'(F(x)) * F'(x).
First, find F'(4) and substitute it into G(x):
F'(4) = 7.
Now, substitute F'(4) = 7 into G'(x):
G'(4) = 6.
Therefore, H'(4) = G'(F(4)) * F'(4) = G'(3) * 7 = 8 * 7 = 56.
e) H(x) = F(x) / G(x)
To find H'(4), we apply the quotient rule. The quotient rule states that if H(x) = F(x) / G(x), then H'(x) = (F'(x) * G(x) - F(x) * G'(x)) / (G(x))^2.
First, find F(4), G(4), F'(4), and G'(4):
F(4) = 3,
G(4) = 2,
F'(4) = 7,
G'(4) = 6.
Now, substitute these values into the quotient rule:
H'(x) = (F'(x) * G(x) - F(x) * G'(x)) / (G(x))^2
= (7 * 2 - 3 * 6) / (2)^2
= (14 - 18) / 4
= -4 / 4
= -1.
Therefore, H'(4) = -1.
So, the values of H(4) and H'(4) are as follows:
a) H(4) = 1
b) H'(4) = 30
c) H(4) = 4
d) H'(4) = 56
e) H'(4) = -1.