AP Calculus 12 – Test 1 Outline and Practice Questions
Horton High School and AVRSB Virtual AP Calculus Courses Page 1
You should be able to:1. Given a graph or an equation be able to discuss its continuity or discontinuity.
2. Interpret the derivative
3. Determine the average and the instantaneous velocity given a distance/time equation.
4. Use the intermediate value theorem for continuity
5. Given a graph be able to determine its limit as x
→
various values6. Given an equation be able to determine its limit as x
→
various values both by graphing and algebraically7. Given a graph or an equation be able to discuss its differentiability or non-differentiability.
8. Given a function be able to determine its derivative both by definition (first principles) and using the power, chain, quotient and product rules (whether the function is written explicitly or implicitly) 9. Be able to determine the equation of a tangent line and that of a normal line to a function at a given
point.
10. Given an application that is represented by a function, be able to answer various questions about it using the derivative as well as the original equation. (motion applications, marginal revenue)
11. Use of various derivative notations.
12. Given the graph of a function be able to sketch its derivative.
13. Understand and use the Sandwich Theorem.
14. Be able to differentiate trigonometric functions 15. Be able to differentiate inverse trig functions.
16. Be able to solve related rates application problems.
Some Practice Questions:
1. Use the Sandwich Theorem to find
x x x
x 2
5 ) 2 cos(
lim
3 +∞
→
2. Use the graph below to estimate the limits and the value of the function, or explain why the limits do not exist.
3. Determine the following limits:
a)
4 9
3 6 5
2 3
lim + + −
∞
→
x
x x
x
b)
x x
x 3
) 5 sin(
lim
20
→
c)
2
lim − + 5
∞
−
→
x
x
x
d) 3
3
0
8
) 5 (
lim sin x x
x→
AP Calculus 12 – Test 1 Outline and Practice Questions
Horton High School and AVRSB Virtual AP Calculus Courses Page 2
4. Sketch f(x) if you know the following:.
5. Algebraically determine each of the following limits. Show your work!
x x x
x x d x
x c x
x b x
x a x
x x x x
5 9 2
5 ) 5
1 ) 1
3 2 ) 1
8 ) 4
2 3
2 3
5 8 3 3
3 2
2
lim lim lim lim
−
−
− +
− + +
−
−
−
−
−
→
→
→
→
6. Find a value k so that the function
≤ +
−
>
+
=
1 , 4 2
1 , 5 ) 3
( kx
2x x
x x x
f
is continuous.7. For the function:
g ( x ) = 5 x
2− 3 x + 2
find:a)equation of the tangent at x = 2 b) equation of the normal at x = 2
8. The equation for free fall at the surface of the moon is
d = 31 . 8 t
2 cm. [ time , t, is in seconds]Assume a rock is dropped from the top of a 12000 cm cliff.
a) Find the speed the rock is falling when it has bean falling for four seconds.
b) Find the height of the rock from the moon’s surface when it has been falling for four seconds.
9. The graph of f(x) is shown below, At what values of x in
x ∈ [ − 6 , 6 ]
does f(x) appear to be non-differentiable? Explain10. Sketch the derivative for the following:
0 ) (
2 ) (
) (
6 ) 1 (
lim lim lim
5=
=
−∞
=
=
−
−∞
→
∞
→
→
x f
x f
x f f
x x x
6 ) (
3 ) 2 (
1 ) (
5 ) (
lim lim lim
1 2 2
=
−
=
=
=
−
→
→
→
=
−
x f f
x f
x f
x x x
AP Calculus 12 – Test 1 Outline and Practice Questions
Horton High School and AVRSB Virtual AP Calculus Courses Page 3
11. Use the definition of the derivative to determine the slope of the tangent tog ( x ) = 6 + 3 x − 2 x
2at x = 4
12. Find the points of discontinuity of the function whose graph is shown here For each discontinuity, identify the type (removable, jump, infinite or oscillating)
13. Find the average rate of change of
f ( x ) = x + 2
over the intervalx ∈ [ − 1 , 5 ]
14. A man starts walking north at 1.3 m/s from a point P. Five minutes later a woman starts walking west at 1.8 m/s from a point 200 m due east of P. At what rate are they separating 10 minutes after the woman starts?
15. A ball is thrown directly upward from a cliff. Its height, in meters above the cliff top, after t seconds is given by y = 30t – 4.9t2. The ball hits the ground after 10 seconds.
(a) Determine the height of the cliff.
(b) Determine the maximum height that the ball attains.
(c) Determine the velocity of the ball when it hits the ground.
16. After investing $1000 at an annual interest rate of 7% compounded continuously for t years, you balance is $B, where B = f(t).
(a) What are the units of
dt
dB
?(b) What is the financial interpretation of
dt dB
?17. Let P(t) represent the price of a share of stock of a corporation at time t. What does each of the following statements tell us about the signs of the first and second derivatives of P(t)?
(a) “The price of the stock is rising faster and faster.”
(b) “The price of the stock is close to bottoming out.”
18. Given that m(x) = 3x-4 for x<-4 and m(x) = ax2 + b for x >-4, determine the values of a and b if m(x) is both continuous and differentiable.
19. A boat leaves a port sailing at 20 kph to the north-west. Half an hour later another boat leaves the same port sailing at 25 kph to the north-east. Determine the speed at which these boats are separating 2 hours after the first boat left the port.
20. A conical sand pile is formed as a conveyor belt dumps sand at a rate of 10 m3/min. If the height is always 1.5 times the diameter of the cone, determine the rate at which the height of the cone is increasing when the radius is 4 m.
AP Calculus 12 – Test 1 Outline and Practice Questions
Horton High School and AVRSB Virtual AP Calculus Courses Page 4
21. Determine the rate at which the height of the water is increasing when h= 4 cm if the container shownbelow is being filled at a constant rate of 15 cm3/min.
22. Differentiate each of the following expressions with respect x and then solve for dy/dx.
) 4 3 ( cos )
1 3 )
2 6
3 2 ) )
3 4 ( tan )
) 3 6 sin(
5 ) 3 ) 5
) 3 )(
5 3 ( ) 25
. 0 125 . 0 )
1 2
2
2 2
2
4 2 2
5
−
=
=
−
=
− +
−
=
+
=
−
−
=
+
−
= +
−
=
− x y
h y
x g
xy y x f x
y e
x y
x d x y x c
x x y b x x
x y a
23. Are there any points on the curve
4 2
1
2 −
+
=
x
y x
where the slope is2
− 3
? If so, find them.24. Find the equations for the lines that are tangent and normal to the curve
x + xy = 6
at (4,1).25. Is any real number exactly 1 less than its cube? Support your answer.
26. Determine the following derivatives from first principles.
a)
y = 3 x
2− 5 x + 1
b)3 2
5 +
= y x
27. A particle P moves on the number line shown below. The diagram to the right of the number line shows the position of the particle as a function of time.
a) When is P moving to the left?
b) Graph the particle’s velocity and speed d
28. The position of a body at time t seconds is
s = t
3− 6 t
2+ 9 t
m. Find the body’s acceleration each time the velocity is zero.29. A company estimates that the cost (in dollars) of producing x items is C(x)=2600+2x+0.001x2. Find the cost, average cost and marginal cost of producing 1000 items.
31. A ladder is sliding down a wall at a rate of 1.2 cm/min. Determine how quickly the angle that the ladder makes with the ground is changing when the foot of the 5m ladder is 1 meter from the base of the wall.
32. A light is making 4 revolutions per minute and is 10 metres away from a straight wall. Determine the rate at which the light beam is moving on an object located on the wall if it is 2m from the location that is directly across from the light.
