1AB Calculus Non – Calculator – 15 minutesUnit 3 Online Test

Let f be the function defined by for .

(a) Find the x-intercepts of the graph of f.(b) Find the intervals on which f is decreasing.(c) Find the absolute maximum value and the absolute minimum value of f. Justify your answer.

(a)

(b)

(c)

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AB Calculus Calculator – 15 minutesUnit 3 Online Test Consider the curve given by .

(a) Show that .

(b) Find all points on the curve whose x-coordinate is 1, and write and equation for the tangent line at each of these points.(c) Find the x-coordinate of each point on the curve where the tangent line is vertical.(a)

(b)

(c)

3AB Calculus Calculator – 21 minutesUnit 3 Online Test

1. Given the following information about f(x), state the x values where the graph has a maximum and a point of inflection. X 0 < x < 2 2 2 < x < 4 4 4 < x < 6 6 6 < x < 8F(x) POS 1 POS 3 POS 6 POSF’(x) POS 0 POS POS POS 0 NEGF’’(x) NEG 0 POS 0 NEG NEG NEG

(A) x = 6 maximum (B) x = 6 maximum (C) x = 6 maximum x = 2 point of inflection x = 2, 4 point of inflection

(D) x = 8 maximum (E) no maximums or points of inflectionx = 2, 4 point of inflection

2. The first derivative of the function f is given by . How many critical values does f

have on the open interval (0,10)?

(A) One (B) Three (C) Four (D) Five (E) Seven

3. The graphs of are shown below. Which of the functions f, g, or h have a relative maximum on the open interval ?

(A) f only (B) g only (C) h only (D) f and g only (E) f, g, and h

4. If f and g are twice differentiable, and if , then

5. If and exist and for all real x, then the graph of and the graph of :

4(A) intersect exactly once.(B) intersect no more than once.(C) do not intersect.(D) could intersect more than once.(E) have a common tangent at each point of intersection.

6. At x = 3 , the function given by is:

(A) undefined(B) continuous but not differentiable(C) differentiable but not continuous(D) neither continuous nor differentiable(E) both continuous and differentiable

7. The is:

AB Calculus Non – Calculator – 22 minutesUnit 3 Online Test1. What are all the values of x for which the function f defined by is increasing?

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2. If f is the function , what are all the x-coordinates of points of inflection for the graph of f?

(A) –1 (B) 0 (C) 1 (D) 0 and 1 (E) –1,0, and 1

3. The graph of a twice-differentiable function f is shown in the figure. Which of the following is true?

4. Let f be a function defined and continuous on the closed interval [a,b]. If f has a relative maximum at c and , which of the following statements must be true?

I. exists.II. If exists, then = 0.III. If exists, then .

(A) II only (B) III only (C) I and II only (D) I and III only (E) II and III only

5. How many critical points does the function f(x) = (x - 2)2(x + 3)4 have?

(A) One (B) Two (C) Three (D) Fine (E) Nine

6. The slope of the tangent line to the curve at (2,-1) is:

7. If =

8. The graph of is shown. Which of the following could be the

6graph of ?

(A) (B) (C)

(D) (E)

9. If g is a differentiable function such that for all real numbers x, and if , which of the following is true?

(A) f has a relative maximum at x = -2 and a relative minimum at x = 2(B) f has a relative minimum at x = -2 and a relative maximum at x = 2(C) f has relative minima at x = -2 and x = 2(D) f has relative maxima at x = -2 and x = 2(E) It cannot be determined if f has any relative extrema

10. Let f be a function that is differentiable on the open interval (1,10). If f(2) = -5, f(5)=5, and f(9)= -5, which of the following must be true?

I. f has at least 2 zeros.II. The graph f has at least one horizontal tangent.III. For some c, , .

(A) None (B) I only (C) I and II only (D) I and III only (E) I, II, and III

11. The graph of f is shown:Which of the following statements about f is false?

(A) f is continuous at (B) f has a relative maximum at (C) is in the domain of f

(D)

(E)