Math 165
1
Section 3.1: Introducing the Derivative
Fall 2020
Content
1.1
Introducing the Derivative (3.1)
1. Let s(t) be the position of an object at time t.
(a) Write the expression for the average velocity of the object over the interval
[t, a].
(b) Write the expression for the instantaneous velocity of the object at the
time t = a.
(c) Why do we need limits to calculate the instantaneous velocity?
2. Let f (x) be a function.
(a) Geometrically, what does the quantity f 0 (a) represent?
(b) What does this quantity tell us about the function f (x) at x = a?
Math 165
Section 3.1: Introducing the Derivative
Fall 2020
(c) What does it mean for f to be differentiable at a?
(d) Write down a step-by-step process for finding the equation of a line tangent
to the graph of f (x) at the point (a, f (a)).
(e) In general, what are the units of f 0 (x) in terms of the units of f (x) and
the units of x?
Math 165
2
Section 3.1: Introducing the Derivative
Fall 2020
Problems
1. A dynamite blast launches a rock straight up at a velocity of 160 m/s. The
height of the rock after t seconds is given by
s(t) = −16t2 + 160t
(a) Calculate the velocity of the rock after 2 seconds.
(b) Calculate both the velocity and the speed of the rock when its height is
256 f t.
(c) Find an expression v(t) for the velocity of the rock after t seconds.
(d) Using your velocity equation from part (c), find when the rock reaches its
maximum height. (Hint: what is the velocity of the rock at its maximum
height?)
(e) What is the maximum height of the rock?
Math 165
Section 3.1: Introducing the Derivative
Fall 2020
(f) Find an expression for a(t) the rock’s acceleration after t seconds. (Hint:
Acceleration is the rate of change of velocity with respect to time)
(g) For each velocity calculation above, draw the corresponding tangent line
on the graph of s(t) below
√
2. Let f (x) = 2x + 1. Find f 0 (2) and use this to find the equation of the line
tangent to f (x) at (2, 3).
Math 165
Section 3.1: Introducing the Derivative
Fall 2020
3. Let f (x) = x1 . Find f 0 (2) and use this to find the equation of the line tangent
to f (x) at (2, 1/2).
4. Sound intensity I, measured in watts per square meter (W/m2 ) at a point x
meters from a sound source with acoustic power P is given by
I(x) =
P
4πx2
A typical sound system at a rock concert produces an acoustic power of about
P = 3 W . Compute I 0 (3) and interpret the result in the context of the problem.