Let f(x) = x + 8 and g(x) = x2 - 6x - 7. Find f(g(2)).
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(a)
The average rate of change of f on the interval 0 ≤ x ≤ π is
____________
(b)
The slope of the tangent line is
.
____________
(c)
The absolute minimum value of f occurs at a critical point where f'(x) = 0 or at endpoints.
Solving f'(x) = 0
Use zero factor property to solve.
so that factor will not generate solutions.
Set cos(x) - sin(x) = 0
cos(x) = 0 when x = π/2, 3π/2, but x = π/2. 3π/2 are not solutions to the equation. Therefore, we are justified in dividing both sides by cos(x) to make tan(x):
![\displaystyle\cos(x) = \sin(x) \implies 0 = \frac{\sin (x)}{\cos(x)} \implies 0 = \tan(x) \implies \\ \\ x = \pi/4,\ 5\pi/4\ \forall\ x \in [0, 2\pi]](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Ccos%28x%29%20%3D%20%5Csin%28x%29%20%5Cimplies%200%20%3D%20%5Cfrac%7B%5Csin%20%28x%29%7D%7B%5Ccos%28x%29%7D%20%5Cimplies%200%20%3D%20%5Ctan%28x%29%20%5Cimplies%20%5C%5C%20%5C%5C%0Ax%20%3D%20%5Cpi%2F4%2C%5C%205%5Cpi%2F4%5C%20%5Cforall%5C%20x%20%5Cin%20%5B0%2C%202%5Cpi%5D)
We check the values of f at the end points and these two critical numbers.
There is only one negative number.
The absolute minimum value of f <span>on the interval 0 ≤ x ≤ 2π is
____________
(d)
The function f is a continuous function as it is a product of two continuous functions. Therefore,
g is a differentiable function; therefore, it is a continuous function, which tells us
.
When we observe the limit
, the numerator and denominator both approach zero. Thus we use L'Hospital's rule to evaluate the limit.
thus
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The average rate of change of f on the interval 0 ≤ x ≤ π is
____________
(b)
The slope of the tangent line is
____________
(c)
The absolute minimum value of f occurs at a critical point where f'(x) = 0 or at endpoints.
Solving f'(x) = 0
Use zero factor property to solve.
Set cos(x) - sin(x) = 0
cos(x) = 0 when x = π/2, 3π/2, but x = π/2. 3π/2 are not solutions to the equation. Therefore, we are justified in dividing both sides by cos(x) to make tan(x):
We check the values of f at the end points and these two critical numbers.
There is only one negative number.
The absolute minimum value of f <span>on the interval 0 ≤ x ≤ 2π is
____________
(d)
The function f is a continuous function as it is a product of two continuous functions. Therefore,
g is a differentiable function; therefore, it is a continuous function, which tells us
When we observe the limit
thus