Question

The twice–differentiable function f is defined for all real numbers and satisfies the following conditions:

f(0)=3

f′(0)=5

f″(0)=7

a)The function g is given by g(x)=e^ax+f(x) for all real numbers, where a is a constant.
Find g ′(0) and g ″(0) in terms of a.

b)The function h is given by h(x)=cos(kx)[f(x)]+sin(x) for all real numbers, where k is a constant. Find h'(x) and write an equation for the line tangent to the graph of h at x=0.

Answer 1

note e^0 = 1

g' = a e^ax + f'
g'(0) = a + 5

g" = a^2 e^ax + f"
g"(0) = a^2 + 7
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note cos(0) = 1 and sin(0) = 0

h = cos kx [f] + sin x
h' = cos kx f'(x) - f(x) sin kx + cos kx
h'(0) = 5 - 0 + 1 = 6

so the slope at x = 0 is 6 = m
in y = 6 x + b

b is h at x = 0
which is 3 + 0 = 3
so
y = 6 x + 3