- C The derivative of tan(u) = sec^2 . Here we need to use the Chain Rule.
tan^2 (4(x) = 2[tan (4(x)][sec^2 (4x)](4) = 8 [tan (4x)][sec^2 (4x)]
- A If we want to find the equation of the tangent line, first we need to find the y-coordinate that
corresponds to x = . It is y = sin^2.
Next, we need to find the derivative of the curve at x = , using the Chain Rule.
We get = 2 sin x cos x. At x = = 2sin = 1.
Now we have the slope of the tangent line and a point that it goes through. We can use the
point-slope formula for the equation of a line, (y − y 1 ) = m(x − x 1 ), and plug in what we have
just found. We get .
- B In order to solve this for b, we need f(x) to be differentiable at x = 1, which means that it must
be continuous at x = 1. If we plug x = 1 into both pieces of this piecewise function, we get f(x)
= , so we need 3a + 2b + 1 = a − 4b − 3, which can be simplified to 2a
+ 6b = −4.
Now we take the derivative of both pieces of this function.
Plug in x = 1 to get f′(x) = .
From there, we can simplify 6a + 2b = 4a − 8b − 3 to get 2a + 10b = − 3.
Solving the simultaneous equations, we get a = − and b = .
