Sample Problems
Chapter 3 Section 4 Velocity and Other Rates of Change
1.
The graph below shows the motion of a particle that is moving left and right.
a.
When is the particle moving to the right?
b. When is the particle moving to the left?
c. When is the particle standing still?
2.
Below is the graph of the velocity as a function of time for an object.
a. When does the object move forward?
b. When does the object move backward?
c. When does the object speed up?
d. When does the object slow down?
e. When is the object’s acceleration positive?
f.
When is the object’s acceleration negative?
g. When is the object’s acceleration zero?
h. When does the object move at its greatest speed?
i.
3.
When does the object stand still for more than an
instant?
A particle moves along a line so that its position at any time t > 0 is given by the
function s(t) = t3 – 8t + 1, where s is measured in feet and t is measured in
seconds.
a. Find the displacement during the first 3 seconds.
b. Find the average velocity during the first 3 seconds.
c. Find the instantaneous velocity during when t = 3.
d. Find the acceleration of the particle when t = 3.
e. At what value or values of t does the particle change direction?
4.
If a ball is thrown vertically upward with a velocity of 80 ft/s, then its height after
t seconds is s = 80t – 16t2.
a. How long did it take the ball to reach its highest point?
b. How high did the ball go?
c. What is the velocity of the ball when it is 96 ft above the ground on its way
up? on its way down?
d. How long was the ball aloft?
5.
The number of gallons of water in a tank t minutes after the tank has started to
drain is Q(t) = 180(20 – t)2. How fast is the water running out at the end of 15
minutes? What is the average rate at which the water flows out during the first 15
minutes?
6.
A body’s velocity at time t sec is v = t3 – 9t2 + 15t – 15 m/sec. Find the body’s
speed each time the acceleration is zero.
Sample Problems
Chapter 3 Section 5 Derivatives of Trigonometric Functions
Once the students have proved the derivative of the sine and cosine function, they should
now be able to solve for the other four trigonometric derivatives.
1.
Find
2.
Find
3.
Find
4.
Find
Find
𝑑𝑦
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
(𝑠𝑒𝑐𝑥).
(𝑐𝑠𝑐𝑥).
(𝑡𝑎𝑛𝑥).
(𝑐𝑜𝑡𝑥).
.
5.
𝑦 = 3 − 8𝑥 + 𝑠𝑖𝑛𝑥
6.
𝑦 = 𝑥 ∙ 𝑐𝑜𝑠𝑥
7.
𝑦 = 5 − 𝑥 3 𝑐𝑜𝑡𝑥
8.
𝑦=
9.
𝑦=
3
𝑐𝑜𝑡𝑥
3𝑠𝑖𝑛𝑥
𝑠𝑒𝑐𝑥+7𝑥
A weight hanging from a spring bobs up and down with position function s = f(t) (s in
meters, t in seconds). What are its velocity and acceleration at time t.
10.
𝑠 = 8𝑐𝑜𝑠𝑡
𝑣=
𝑎=
11.
𝑠 = 12𝑠𝑖𝑛𝑡
A body is moving in simple harmonic motion with position function s = f(t) (s in meters,
t in seconds).
(a) Find the body’s velocity, speed, acceleration, and jerk at time t.
(b) Find the body’s velocity, speed, acceleration, and jerk at time t = π/4.
12.
𝑠 = 7 + 4𝑠𝑖𝑛𝑡
13.
Find equations for the lines that are tangent and normal to the graph of
𝑦 = 2𝑠𝑖𝑛𝑥 + 5 𝑎𝑡 𝑥 = 𝜋.