Name: AP Calculus BC AP Problem Set 1 Non-‐calculator. Show all your work in order to receive full credit. 1. Let f be the function defined below, where b Is a constant. If f is a continuous function, what is the value of b? ⎧ x 2 + bx
x ≤5
⎪
⎛
⎞
π
f (x) = ⎨
x >5
⎪⎩5sin⎜⎝ 2 x ⎟⎠
A) -‐6 B) -‐5 €
C) -‐4 D) 4 E) 5 2. The graph of y = 3x 2 − x 3 has a relative maximum at A) (0, 0) only B) (1, 2) only C) (2, 4) only € D) (4, -‐16) only E) (0, 0) and (2, 4) 3. A particle moves in the xy-‐plane so that its velocity vector at time t is v(t) = t 2 , sin(πt) and the particle’s position vector at time t = 0 is 1, 0 . What is the position vector of the particle when t = 3? Hint: You need to take the integral with +C. Then plug in the initial condition to find the particular solution (position vector). Finally, plug in t = 3. €
€
1
A) 9,
π
2
B) 10,
π
C) 6, − 2π €
D) 10, 2π E) 10, 2 €
€
€
€
€
4. If f (x) =
A)
B)
C)
4 sin x + 2 , then f '(0) = -‐2 0 1 €
2
D)
2
E) 2 €
€
5. An equation of the line tangent to the curve x 2 + y 2 = 169 at the point (5, -‐12) is: A) 12x – 5y = 120 B) 5x – 12y = 119 €
C) 5x – 12y = 169 D) 12x + 5y = 0 E) 12x + 5y = 169 6. For an object moving along a straight line, the graph below represents the velocity of the moving object as a function of time. At which of the marked points is the speed the greatest? A)
B)
C)
D)
E)
A B C D E k
7. If the graph of f (x) = 2x 2 + has a point of inflection at x = -‐1, then the value x
of k is: A) -‐2 B) -‐1 € C) 0 D) 1 E) 2 8. Which of the following is an equation of the line tangent to the curve with parametric equations x = 3e −t , y = 6e t at the point where t = 0? Hint: To find the point (x, y), you will need to plug in t = 0 to your x and y parametric equations. A) 2x €+ y – 12 = 0 B) 2x – y + 12 = 0 C) x – 2y + 9 = 0 D) 2x – y = 0 E) x + 2y – 15 = 0 9. If x = sin t and y = cos2 t , then €
€
A) -‐2 1
B) − 4
€
C) 0 1
D) 4
E) 2 d2y
at t = π equals dx 2
€
10. The circumference of a circle is increasing at the rate of 0.5 meters/minute. € is the rate of change of the area of the circle when the radius is 4 What meters? dA
Hint: You are finding . However you will need the circumference formula dt
dr
for a circle to find . dt
A) 2 €
m2/min B) 6 m2/min C)€ 4π m2/min 1
D)
m2/min 2π
1
E)
m2/min 4π
€
€