Question

Calculus Q:
Assume f and g are differentiable functions with h(x) = f(g(x)).
Suppose the equation of the line tangent to the graph of g at the point (4,7) is y = 3x-5 and the equation of the line tangent to the graph of f at (7,9)
is y = -2x + 23.
(a) Calculate h(4) and h'(4).
(b) Determine an equation of the line tangent to the graph of h at the point on the graph where x = 4.

Answer 1

The slope of the tangent line to \(g\) at \(x=4\) is \(3\), i.e. \(g'(4)=3\).
Similarly, you see that \(f'(7)=-2\).

So, if \(h(x)=f(g(x))\), then \(h(4)=f(g(4))=f(7)=9\).

Applying the chain rule, you have \(h'(x)=f'(g(x))\cdot g'(x)\), so \(h'(4)=f'(g(4))\cdot g'(4)=f'(7)g'(4)=-2(3)=-6\).

You can use this info to find the equation of the tangent line to \(h\) when \(x=4\).

Answer 2

so basicall this is what I did:
g(4) = 7
g'(4) = 12

f(7) = 9
f'(7) = -14

h(x)=f(g(x))
h'(x)=f'(g(x))*g'(x)
h(4) = f(g(4)) = f(7) = 9
h'(4) = f(g(4))*g'(4) = f(7) * 12 = 84

Answer 3

but the slope is way too big!!

Answer 4

How did you get that value for g'(4)? and f'(7)?

Answer 5

ahhh. ok so the function is y=3x-5 and to get g'4 I took the 3 and subbed in 4 for x to get 12

Answer 6

and to get g(4) ...i just subbed in 4 for x and got 7.

Answer 7

my derivative for g'4 is wrong...how did you get your value?

Answer 8

The derivative of g at x=4 is the same as the slope of the tangent line at x=4. You need only check the slope of the tangent line.

Answer 9

eeek thank-you! and would the equation be: y-9=-6(x-4)

Answer 10

You're welcome, and yes.