Let f be the function defined by f(x) = e^(x) cos x.

(a) find the average rate of change of f on the interval 0 ≤ x ≤ π.

(b) what is the slope of the line tangent to the graph of f at x = 3π/2 ?

(c) find the absolute minimum value of f on the interval 0 ≤ x ≤ 2π. justify your answer.

(d) let g be a differentiable function such that g(π/2) = 0. the graph of g', the derivative of g, is shown below. find the value of or state that it does not exist. justify your answer.

(this was on the no-calculator section of the recently-released ap calculus ab 2018 exam so i appreciate it if you tried to limit calculator usage)

(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):



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 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