Less78Derivatives

Question 1

What is a derivative?

Answer 1

It is the slope of the tangent line.

Question 2

What notations are used for derivatives?

Answer 2

y’, ^dy/_dx, ^d/_dx[f(x)], D[y]

Question 3

What is the derivative of a constant function line y = 2?

Answer 3

The derivative is 0, because it is a horizontal line.

Question 4

What is a power function and how do you calculate its derivative?

Answer 4

f(x) = xⁿ,

f’(x) = n*x^n-1

Question 5

What is the derivative of f(x) = x^⅔ ?

Answer 5

f’(x) = ⅔*x^-⅓ = 2 /3x^⅓ ,

Question 6

What is the derivative of f(x) = x^-2?

Answer 6

f’(x) = -2*x^-3 = -2/x³

^{not differentiable if x=0}

Question 7

What can derivatives tell you about graphs?

Answer 7

Where the slope of the graph is negative and positive.

Question 8

What is the derivative constant multiple rule?

Answer 8

f(x) = 3*x²

f’(x) = 3*2x = 6x

Question 9

What is the derivative of f(x) = 2/x?

Answer 9

f(x) = 2/x = 2*x^-1,

f’(x) = 2*-x^-2

f’(x) = -2/x²

Question 10

What is the derivative of 7/(3x)^-2?

Answer 10

f(x) = 7*(3x)² = 7*9x² = 63x²,

f’(x) = 126x

Question 11

What is the derivative of a sum or difference of 2 functions?

Answer 11

It is the sum or difference of their derivatives

Question 12

What is the derivative of f(x) = x³ - 4x + 5?

Answer 12

3x² - 4

Question 13

What is the derivative of 3t⁵ - 14π?

Answer 13

15t⁴

Question 14

Is the product (quotient) of 2 derivatives the product (quotient) of their derivatives?

Answer 14

No.

Question 15

What is the Product Rule of derivatives?

Answer 15

^d/_dx[f(x)*g(x)] =

f(x)*g’(x) + g(x)*f’(x)

Question 16

What is the Quotient Rule of derivatives?

Answer 16

^d/_dx[^f(x)/_g(x)] = [g(x)*f’(x) - f(x)*g’(x)]/[g(x)]²

Question 17

What are the derivatives of sinx and cosx?

Answer 17

^d/_dxsinx = cosx

^d/_dxcosx = - sinx

Question 18

When x is near 0, what line approximates sinx?

Answer 18

y = x

???

Question 19

Show that y = x approximates sinx near 0.

Answer 19

sin(.2) = .19867

sin(.01) = .009999

x and y are getting closer to each other

Question 20

What is ^d/_dx[3sinx + π]?

Answer 20

3cosx

Question 21

What is ^d/_dx[5x² + ⅓cosx]?

Answer 21

10x - ⅓sinx

Question 22

In the formula for a freefalling object, what do the variables stand for?

s(t) = ½gt² + v_ot + s_o

Answer 22

s(t) = ½gt² + v_ot + s_o

s(t) = position at time t

s₀ = starting position

g = the gravitational constant = -32ft./sec²

v₀t = the initial velocity

Question 23

In s(t) = ½gt² + v_ot + s_o

what is the average velocity at any given time v(t)?

Answer 23

The average velcity at time t is the change in position/change in from time 0 to time t.

Question 24

What is the instantaneous velocity?

Answer 24

It is the velocity at any point in time and is measured as the derivative of the original position equation.

v(t) = s’(t) = gt + v₀

Question 25

What is the meaning of the second derivative of the position function?

Answer 25

It measures acceleration at any point.

Question 26

f(x) = |x| does not have a derivative at 0 because it is a sharp point.

Does f(x) = x*|x| have a derivative at 0?

Answer 26

f’(x) = lim_x→0^{[f(x) - f(0)]}/_{(x - 0)} = ^▲y/_▲x

= ^x|x|/_x = |x|

lim_x→0 |x| = 0

So x*|x| does have a derivative. It is a curved graph.

Question 27

Where does the graph of f(x)= x³ + 3x have a horizontal tangent?

Answer 27

Where the derivative is 0.

3x² + 3 = 0

3x² = -3

no solution

Question 28

What if you change it to f(x) = x³ - 3x?

Answer 28

3x² - 3 = 0

x = ±1

Question 29

What is the derivative of f(x) = ⁵√x?

Answer 29

f(x) = x^1/5

f’(x) = 1/5*x^-4/5

f’(x) = 1 /(5*x^4/5)

Question 30

What is the derivative of

s(t) = t³ + 5t² - 3t + 8?

Answer 30

3t² + 10t - 3

Question 31

What is the derivative of 1/x⁵?

Answer 31

f(x) = x^-5

f’(x) = -5/x⁶

Question 32

What is the derivative of f(x) = ^√x/_x?

Answer 32

f(x) = x^½*x^-1

f(x) = x^-½

f’(x) = -½*x^-3/2

f’(x) = -1/(2*x^3/2)

Question 33

What is the derivative of f(x) = 6√x + 5cosx?

Answer 33

f(x) = 6*x^½ + 5cosx

f’(x) = ³/x^½ - 5sinx

Question 34

Find the equation of the line tangent to f(x) = 3x³ - 10 at (2,14)

Answer 34

f’(x) = 9x²

At x = 2, the slope is 36

:: (y - 14) = 36(x - 2)

y = 36x + 58

Question 35

Find the equation of the line tangent to f(Θ) = 4sinΘ - Θ at (0,0)

Answer 35

f’(Θ) = 4cosΘ - 1

At Θ = 0, slope = 3

(y - 0) = 3(Θ - 0)

y = 3Θ

Question 36

Find the equation of the line tangent to

f(x) = (x² + 2)(x + 1) at (1,6)

Answer 36

f(x) = x³ + x² + 2x + 2

f’(x) = 3x² + 2x + 2

at x=1, f’(x) = 7

(y - 6) = 7(x - 1)

y = 7x - 1

Question 37

Find where f(x)= x⁴ - 2x² + 3 has a horizontal tangent.

Answer 37

f(x)= x⁴ - 2x² + 3

f’(x) = 4x³ - 4x

x³ = x

x = 0, ±1

(0,3) (1,2) (-1,2)

Question 38

A coin is dropped from 1,362 feet.

When will it hit the ground and at what velocity?

Answer 38

s(t) = ½gt² + v_ot + s_o,

g = -32

s(t) = -16t² + 1362

It hits the rounds when s(t) = 0

-16t² = 1362; t² = 1362/16

t = √1362/4 ≈ 9.226 seconds

v(t) = s’(t) = -32t

v(t) = -32* 9.226 = 295 ft/sec