Calculus Help

3.1 Defining the Derivative

Vocabulary Examples

Difference Quotient	For a function f , the difference quotient Q is: Q = Alternately, for h * 0, Q =
Slope of a Tangent Line	mtan = Alternately, for h * 0, mtan =
Derivative of a Function at a Point	The derivative of f (x) at a, denoted , is defined: f ,(a) = Or f ,(a) =
Instantaneous Rate of Change	The instantaneous rate of change of a function f (x) at a is its

1. For each of the following functions, determine the slope of the secant line between x1 and x2. (a) f (x) = 4x + 7, x1 = 2, x2 = 5

Name:

Defining the Derivative

Section:

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25

(b) f (x) =

x x+3

, x1 = 0, x2 = 3

2. For each of the following functions, determine the f ,(a)

(a) f (x) = 2x2 − x, a = 4

(b) f (x) = √x − 7, a = 10

3. For each of the following functions f , write the equation for the line tangent to f at x = a

(a) f (x) = 1 x5 + 2x at a = 1

3

(b) f (x) =

x

√4 at a = 2

(c) f (x) = 54 + 5 at a = −3

x3

4. Recall that the velocity of a moving object is instantaneous rate of change of its position. A projectile’s position d at time t is given by the function d(t) = −4.9t2 + 20.1x + 24.3.

(a) Determine the velocity of the object after 2 seconds.

(b) Determine the velocity of the object after 3 seconds.

3.2 Derivative as a Function

Vocabulary Examples

Derivative Function	For a function f , the derivative function, denoted , is the function whose domain consists of values of x such that the following limit exists: f ,(x) =	Notations:
Theorem on Differentiabil- ity and Continuity	If a function f is differentiable at a, then f is at a.
Higher-Order Derivative	The of a

1. Use the definition of a derivative to determine the derivative of the following functions. (a) f (x) = 3x2 − 2

(b) f (x) = x−2

(c) f (x) = √3x − 7

(d) f (x) = 3

√

x

2. Use the graph of each of the following functions to sketch the graph of its derivative.

Name:

Derivative as a Function

Section:

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27

(a)

(b)

(c)

4

2

−2

−2

−4

2

4

2

−4

−2

2

4

−2

−4

4

2

−2

−2

−4

2

3. The second derivative f ,,(x) = lim

h→0

h

f ,(x+h)− f ,(x). Determine f ,,(x) for each of the following

functions.

(a) f (x) = 1 x + 7

7

(b) f (x) = −4.9x2 − 7x + 121.9

(c) f (x) = (x + 1)3

4. Velocity is the first derivative of the position (or displacement) function. Acceleration is the second derivative of the position (or displacement) function. Consider a particle whose position can be de- scribed by the function d(t) = 11.2t2 + 3t 10. Using the definition of the derivative, determine

−

(a) the function that models the velocity of the particle and (b) the acceleration of the particle.

3.3 Differentiation Rules

Vocabulary Examples

Constant Rule	For any constant c, d (c) = dx
Power Rule	d (xn) = dx
Constant Multiple Rule	For any constant c and differentiable function f , d (c · f (x)) = c· dx
Sum Rule	d ( f (x) + g(x)) = dx
Difference Rule	d ( f (x) − g(x)) = dx
Product Rule	d ( f (x) · g(x)) = dx
Quotient Rule	d f (x) = dx g(x)

1. Determime f ,(x) for each of the following functions.

x2

Name:

Differentiation Rules

Section:

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29

(a) f (x) = 1 x6 − 3x1/3 + 10

3

x3

(b) f (x) = (x + 2)(2x2 − 3) (c) f (x) = 4x3−2x+1

2. The following graph shows f (x) and g(x). h(x) = f (x) + g(x).

(a) Determine h,(1)

(b) Determine h,(3)

4

f (

x)

2

g(x

)

2

4

(c) Determine h,(4)

3. Assume, for each of the following, that f and g are both differentiable. Determine h,(x).

2

(a) h(x) = 4 f (x) + g (x)

7

(b) h(x) = x3 f (x) (c) h(x) = f (x)g(x)

4. Find the equation of the line tangent to the graph of f (x) = x2 + 4 − 10 at x = 8

x

5. Find the equation of the line tangent to the graph of f (x) = 2x7/3−3x6+x at x = −1

x2

6. Find the equation of the line tangent to the graph 7. Find the equation of the line tangent to the graph of

of f (x) = (3x − x2)(3 − x − x2) at x = 1 f (x) = x61 at and containing the point (1, −6)

−

8. A car driving along a freeway with traffic has 9. The concentration of antibiotic in the bloodstream

traveled d(t) = t3 − 6t2 + 9t meters in t seconds.

(a) Determine the time, in seconds, when the velocity of the car is 0.

(b) Determine the acceleration of the car when the velocity is 0.

t hours after being injected is given by the function

C(t) = 2t2+t , where C is measured in miligrams

t3 50

+

per litre of blood.

(a) Find the rate of change of C(t).

(b) Determine the rate of change for t = 8,

t = 12, t = 24.

(c) Describe what is happening as the number of hours increases.

10. Determine a quadratic function for which f (1) = 5, f ,(1) = 3, and f ,,(1) = −6