Bob Brown Math 251 Calculus 1 Chapter 3, Section 2 Completed 1

CCBC Dundalk

Review of the Limit Definition of the Derivative

Write the limit definition of the derivative function: =

Derivative of a Constant Multiple of a Function

Theorem: If f is a differentiable function and if c is a constant, then

= We also use the notation =

Exercise 1: Prove this Theorem using the definition of the derivative.

= =

=

Exercise 2: Illustrate the Theorem above with a geometric example.

y = f(x)
/ Estimate the derivative of f at x = -2.

Estimate the derivative of f at x = 3.
y = g(x) = 4f(x)
/ Estimate the derivative of g at x = -2.


Estimate the derivative of g at x = 3.

Derivative of the Sum or Difference of Functions

Theorem: If f and g are differentiable functions, then

= =

Exercise 3: Prove the Difference Rule using the definition of the derivative. (The proof for the Sum Rule is found on page 130 in the text.)

= =

Derivative of a Power Function (The Power Rule)

Def: A power function is a function of the form , where

Theorem: The derivative function of is

Note: The proof and/or hints for a proof of the Power Rule for various conditions on n can be found on pages 127, 143, and 173 in the text.

Derivative of y = c and of y = x

The derivative of the constant function f(x) = c is
=
That is, = / The derivative of the function g(x) = x is
=
That is, =
Use the Power Rule to show this fact.

f(x) = c =

= / Use the Power Rule to show this fact.

g(x) = x =

=
Use the limit definition to prove this fact.


/ Use the limit definition to prove this fact.

Exercise 4: Use the Power Rule to determine each derivative function.

(i) y = x123 (ii) y = x12.3

(iii) (iv)

(v) y = x3 (vi) y = 3x

(vii) y = xπ (viii) y = xln(2)

Derivative Functions of Polynomials and of Some Other Functions

We can now determine the derivative function of a function expressed (or expressable) as the sum or difference of power functions, including polynomials. (Note that only the function in (i) below is a polynomial.)

Exercise 5: Determine the derivative function.

(i) (ii)

(iii) (iv)


Exercise 6: What is derivative function of f(x) = sin(x)? To answer this question, consider the graph of y = sin(x) on the interval drawn below. As we did in Handout 3.1, draw the tangent line at various points, estimate the slope, and sketch .

/ x / 0 / / / /

Draw below.

/
Conclusion: If f(x) = sin(x), then =

Exercise 7: What is derivative function of g(x) = cos(x)? To answer this question, consider the graph of y = cos(x) on the interval drawn below. As we did in Handout 3.1, draw the tangent line at various points, estimate the slope, and sketch .

/ x / 0 / / / /

Draw below.

/
Conclusion: If g(x) = cos(x), then =


The Derivative of an Exponential Function

Exercise 8: Consider f(x) = 2x. Write the limit definition of each of the following, and use the table feature of the TI to estimate the value of each limit to four decimal places.

= =

=

=

= =

= =

= =

So, =

We recognize the decimal 0.6931 as an approximation of

We conclude that if f(x) = 2x, then =

Note that the derivative function of f(x) = 2x is merely a constant multiple of f(x) = 2x.

The constant multiple, in this case, is

If f(x) = 3x, then = . Note that the derivative function of f(x) = 3x is

merely a constant multiple of f(x) = 3x. The constant multiple, in this case, is

Recall: If one function is a constant multiple times another function, then its graph is a vertical stretch or shrink (depending on whether the constant multiple is greater than 1 or less than 1) of the graph of the other function.


Question: Is there an exponential function such that the constant multiple in the formula for the derivative function is 1? That is to say, is there an exponential function that is equal to its derivative function?

Exercise 9: Let f(x) = ax. Write the limit definition of the derivative function, and compute the value of a for which f(x) = .

= =

Derivative Function of an Exponential Function y = ax (where a > 0)

= “Special”case worth nothing: =

(see p.158 in the text)


Exercise 10: Determine the derivative function of each of the following functions.

(i) y = 4x (ii)

(iii) y = -6πx + 3ex (iv)

(v) y = (ln(2))x (vi) y = 3x + x3

(vii) y = (2e)x (viii) y = e3x


Average Velocity and Instantaneous Velocity of a Rocket

Exercise 11: A rocket is launched vertically from a submarine 200 feet below the surface of the ocean with an initial velocity of 1500 feet per second. (For simplicity, we will ignore water friction and air friction. In reality, neither can be ignored.) The following table and graph represent y = the height in feet of the rocket above the level of the water as a function of

x = time in seconds since the beginning of the launch. (The graph does not indicate the path of the rocket.) The equation which relates y and x is y = -16x2 + 1500 x – 200. Note that the

x-axis is scaled in increments of 10, and the y-axis is scaled in increments of 5000.

(i) Using the table, compute the average velocity of the rocket between x = 20 sec. and x = 50 sec. Draw the corresponding secant line.



/ (ii) Compute the average velocity of the rocket between x = 20 sec. and x = 25 sec. Draw the corresponding secant line.
(iii) The instantaneous velocity of the rocket at x = 20 seconds is the slope of the tangent line to the graph at x = 20. Compute this value exactly (using derivative rules, not by estimating a slope geometrically.)

(iv) What is the velocity (“velocity” means “instantaneous velocity” unless otherwise qualified) of the rocket at the moment in its descent that it is 10,000 feet above the surface of the water?