AP Calculus AB Unit 5 — Analytical Applications of Differentiation Practice Test

Let f be the function given by f(x) = x ³ . What are all values of c that satisfy the conclusion of the Mean Value Theorem on the closed interval [-1,2] ?

Let . Below is Rafael’s attempt to write a formal justification for the fact that the equation has a solution where ; Is Rafael’s justification complete? If not, why? Rafael’s justification:

Given a function g( x) , if g ‘ ‘ ( x )=0 for a certain value of  x , then g(x) has ____________________ at x.

Let h ( x )=e ^{2 x−6} −e . Where does that apply. h have critical points? Choose all answers

Given a function, f (x) , if f ‘ ( x )< 0  over a certain interval, then  f(x) is _______________ over that interval.

Let  g be a function defined for all real numbers except for x=2 Also let g ‘ , the derivative of g    On which intervals is g is increasing

Use the sign chart for f ‘ (x) . There is/are…

Let f be a polynomial function and let f ‘ , its derivative, be defined as f ‘ ( x )=x 4 ( x−2)(x +3) . At how many points does the graph of f have a relative maximum?

What is the maximum value of f ( x )=x ³ −3 x ² −1 on the interval [−3, 2] ?

Let g ( x ) =2 x 3 −21 x 2 + 60 x . What is the absolute maximum value of  g over the closed interval [ 0,6] .

Function f is graphed. Select all the intervals where f ‘ (x)<0 and f ‘ ‘ ( x)>0 . Choose all answers that apply.

The concavity of a function is described by its ____________________.

Let h be a twice differentiable function, and let h (−4 )=−3,h ‘ (−4 )=0, and h ‘ ‘ (−4 )=0 . What occurs in the graph of h at the point (−4,−3) ?

 Let f be a twice differentiable function, and let f ( 2 )=3, f ‘ ( 2 ) =0 and f ‘ ‘ ( 2 )=5 . What occurs in the graph of g at the point (2,3) ?

The twice differentiable function  g and its second derivative g” are graphed What is an appropriate calculus-based justification for the fact that g has an inflection point at x=c ?

Consider the function f ( x )=9−x 2 for f (x) ≥0 only. The shaded region is an isosceles triangle formed by joining the points ( 0, 0 ) , ( x , f ( x ) ) , and ( −x , f ( x ) ) , where 0 ≤ x ≤3 . What is the area of the largest triangle that satisfies the stated conditions?

Kendall tried to find all the equations of vertical lines tangent to the curve given by x ² +2 x y ² =25 . This is her solution: Is Kendall’s solution correct? If not, at which step did she make a mistake?

The derivative of a function f is given by f ‘ ( x )=ln∨x∨∙ x . On which interval is the graph of f concave up? (Use a graphing calculator)

The derivative of a function
g is given by
‘
g ( x )=3 sin ( x ) +ln( x ) .