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Name: Derivatives as Rates of Change Section:

3.4 Derivatives as Rates of Change Vocabulary Examples

Average Rate of Change The of a secant line connecting two points on a

function

For any two points (a, f (a)) and (a + h, f (a + h)) on a

function f (continuous over the interval [a, a + h]), ∆y ∆x =

1. Each of the following functions models the displacement d of a particle at time t. For each function, (i) determine the velocity function, (ii) determine the acceleration function, (iii) determine the intervals over which the particle is speeding up, and (iv) determine the invervals over which the particle is slowing down.

(a) d(t) = 2t3 − 3t2 − 12t + 8 (b) d(t) = 2t3 −15t2 + 36t−10 (c) d(t) = t 1+t2

2. The position d fo a particle at time t is shown in the graph below.

2 4 6 8

−4

−2

4 (a) Indicate for which time intervals the velocity is positive, negative, and zero.

(b) Sketch the velocity function.

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Name: Derivatives as Rates of Change Section:

3. The height of a projectile can be modeled by the function h(t) = 248 − 4.9t2.

(a) Determine the average velocity from t = 1 to t = 2.5. (b) Determine the instantaneous velocity at t = 1. (c) Determine the instantaneous velocity at t = 2.5. (d) Determine the time at which the instantaneous velocity is equal to the average velocity from

t = 1 to t = 2.5.

4. A culture of bacteria grows in number according to the function N(t) = 3000 ( 1 + 4t

t2+100

) , where t

is measured in hours.

(a) Find the rate of change function for the number of bacteria.

(b) Find N′(0), N′(10), N′(20). (c) Interpret the results of part (b). (That is, what does those results mean in the context of the

problem?)

(d) Find N′′(0), N′′(10), N′′(20) and interpret these results.

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Name: The Derivatives of Trigonometric Functions Section:

3.5 The Derivatives of Trigonometric Functions Vocabulary Examples

Derivatives of the Trigonometric Functions

d d x sin x =

d d x cos x =

d d x tan x =

d d x cot x =

d d x sec x =

d d x csc x =

Very Important Note: These identities are only true for values of θ measured

1. Use the squeeze theorem to show that lim h→0

sin h h = 1 and limh→0

cos h−1 h = 0

Determine the derivative of the following functions.

2. f (x) = x2 sin x 3. f (x) = sin x cos x 4. f (x) = cos2 x

5. f (x) = x2 − sec x + 1 6. f (x) = 3 csc x + 5 x3

7. f (x) = sec xx

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Name: The Derivatives of Trigonometric Functions Section:

8. f (x) = sec 2 x−cot x sec x

9. f (x) = 12 x + 1 4 sin(2x) + π10. f (x) = cos

3 x

11. Find all x values on the graph of f (x) = x − 2 cos x for 0 < x < 2π where the tangent line has a slope of 2.

12. A mass on a spring oscillates in harmonic motion modeled by the function d(t) = −6 cos t where d is the displacement in meters and t is the time in seconds. What is the velocity of the mass at t = 5 s?

13. Use the quotient rule to verify the following formulas.

(a) dd x cot x = −csc 2 x

(b) dd x sec x = sec x tan x

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Name: Differentiation Rules Practice 1 Section:

A.3 Differentiation Rules Practice 1 Differentiate the following functions.

1. f (x) = x−1 2. f (x) = 321.17−14.179 3. f (x) = x3/7

4. f (x) = 3x−5 + 6x5 − √

x 5. f (x) = 14 5 √

(x2) 6. f (x) = 8−3x 5

x2+1

7. f (x) = x 2−x−20

x−5 8. f (x) = (3x − 1)(1 − x1/2 − x−3)

9. f (x) = x 2−7x+12

x−3 10. f (x) = x3−x−1

x+5 11. f (x) = −2x3+7x1/2−11x

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Name: Differentiation Rules Practice 1 Section:

Both f and g are differentiable. Determine the dd x h(x) for the following functions.

12. h(x) = 4 f (x) + g(x)7 13. h(x) = 2 3 f (x)g(x) 14. h(x) =

1 f (x) − 2x · g(x)

15. The height h, in meters, of a projectile can be modeled by that function h(t) = −4.9t2 +42.7t +1.35 at time t, in seconds.

(a) Determine the initial velocity of the projectile.

(b) Determine the velocity of the projectile after 3.5 seconds.

(d) What does the constant 1.35 represent in terms of the situation that is being modeled?

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Name: Differentiation Rules Practice 2 Section:

A.4 Differentiation Rules Practice 2

1. Determine the equation of the line tangent to the function f (x) = x 2−4

(x−2)2 at x = 3

2. The height of a particle can be modeled by the function h(t) = 4.2t − 0.4t4

(a) Sketch a reasonable graph of the function. Be sure to consider all critical points.

(b) Based on your sketch, what is the derivative of h at the projectile’s highest point? (c) In this real-world context, what does h′(t) represent? (d) Determine the time at which the projectile reaches it’s greatest height.

(e) Determine the maximum height of the projectile.

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Name: Differentiation Rules Practice 2 Section:

3. Determine the equation of a line that is tangent to the function f (x) = 25 − x2 and also contains the point (13, 0).

4. The population, in millions, of arctic flounder in the Atlantic Ocean is modeled by the function P(t) = 8t+3

0.2t2+1

(a) Determine the initial population of arctic flounder.

(b) Determine P′(10). What does this mean in the context of the arctic flounder?

5. Given the function f (x) = −x 2

4 + 8.5x − 60.69, solve the following equations.

(a) f (x) = 0 (b) f ′(x) = 2 (c) f ′(x) = −2 (d) f ′(x) = 0

(e) How does our answer to part (d) relate to the original function f ?

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