Questions: Given F(1)=4, F'(1)=3, F(4)=6, F'(4)=3 and G(1)=4, G'(1)=5, G(4)=7, G'(4)=7, find each of the following. (Enter dne for any derivative that cannot be computed from this information alone.) A. H(1) if H(x)=F(G(x)) B. H'(1) if H(x)=F(G(x)) C. H(1) if H(x)=G(F(x)) D. H'(1) if H(x)=G(F(x)) E. H'(1) if H(x)=F(x) / G(x)

Solution Steps

To solve the given problems, we need to apply the chain rule and the quotient rule for derivatives, as well as evaluate functions at given points.

A. To find \( H(1) \) where \( H(x) = F(G(x)) \), substitute \( x = 1 \) into \( G(x) \) to find \( G(1) \), then substitute this result into \( F(x) \).

B. To find \( H^{\prime}(1) \) where \( H(x) = F(G(x)) \), use the chain rule: \( H^{\prime}(x) = F^{\prime}(G(x)) \cdot G^{\prime}(x) \). Evaluate this at \( x = 1 \).

C. To find \( H(1) \) where \( H(x) = G(F(x)) \), substitute \( x = 1 \) into \( F(x) \) to find \( F(1) \), then substitute this result into \( G(x) \).

To find \( H(1) \), we first evaluate \( G(1) \): \[ G(1) = 4 \] Next, we substitute this value into \( F \): \[ H(1) = F(G(1)) = F(4) = 6 \]

Using the chain rule, we find \( H^{\prime}(1) \): \[ H^{\prime}(1) = F^{\prime}(G(1)) \cdot G^{\prime}(1) \] Substituting the known values: \[ H^{\prime}(1) = F^{\prime}(4) \cdot G^{\prime}(1) = 3 \cdot 5 = 15 \]

To find \( H(1) \), we first evaluate \( F(1) \): \[ F(1) = 4 \] Next, we substitute this value into \( G \): \[ H(1) = G(F(1)) = G(4) = 7 \]

Final Answer