Less78Derivatives Flashcards
What is a derivative?
It is the slope of the tangent line.
What notations are used for derivatives?
y’, dy/dx, d/dx[f(x)], D[y]
What is the derivative of a constant function line y = 2?
The derivative is 0, because it is a horizontal line.
What is a power function and how do you calculate its derivative?
f(x) = xn,
f’(x) = n*xn-1
What is the derivative of f(x) = x⅔ ?
f’(x) = ⅔*x-⅓ = 2 /3x⅓ ,
What is the derivative of f(x) = x-2?
f’(x) = -2*x-3 = -2/x³
not differentiable if x=0
What can derivatives tell you about graphs?
Where the slope of the graph is negative and positive.
What is the derivative constant multiple rule?
f(x) = 3*x²
f’(x) = 3*2x = 6x
What is the derivative of f(x) = 2/x?
f(x) = 2/x = 2*x-1,
f’(x) = 2*-x-2
f’(x) = -2/x²
What is the derivative of 7/(3x)-2?
f(x) = 7*(3x)² = 7*9x² = 63x²,
f’(x) = 126x
What is the derivative of a sum or difference of 2 functions?
It is the sum or difference of their derivatives
What is the derivative of f(x) = x³ - 4x + 5?
3x² - 4
What is the derivative of 3t5 - 14π?
15t4
Is the product (quotient) of 2 derivatives the product (quotient) of their derivatives?
No.
What is the Product Rule of derivatives?
d/dx[f(x)*g(x)] =
f(x)*g’(x) + g(x)*f’(x)
What is the Quotient Rule of derivatives?
d/dx[f(x)/g(x)] = [g(x)*f’(x) - f(x)*g’(x)]/[g(x)]²
What are the derivatives of sinx and cosx?
d/dxsinx = cosx
d/dxcosx = - sinx
When x is near 0, what line approximates sinx?
y = x
???
Show that y = x approximates sinx near 0.
sin(.2) = .19867
sin(.01) = .009999
x and y are getting closer to each other
What is d/dx[3sinx + π]?
3cosx
What is d/dx[5x² + ⅓cosx]?
10x - ⅓sinx
In the formula for a freefalling object, what do the variables stand for?
s(t) = ½gt² + vot + so
s(t) = ½gt² + vot + so
s(t) = position at time t
s0 = starting position
g = the gravitational constant = -32ft./sec²
v0t = the initial velocity
In s(t) = ½gt² + vot + so
what is the average velocity at any given time v(t)?
The average velcity at time t is the change in position/change in from time 0 to time t.
What is the instantaneous velocity?
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
What is the meaning of the second derivative of the position function?
It measures acceleration at any point.
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?
f’(x) = limx→0[f(x) - f(0)]/(x - 0) = ▲y/▲x
= x|x|/x = |x|
limx→0 |x| = 0
So x*|x| does have a derivative. It is a curved graph.
Where does the graph of f(x)= x³ + 3x have a horizontal tangent?
Where the derivative is 0.
3x² + 3 = 0
3x² = -3
no solution
What if you change it to f(x) = x³ - 3x?
3x² - 3 = 0
x = ±1
What is the derivative of f(x) = 5√x?
f(x) = x1/5
f’(x) = 1/5*x-4/5
f’(x) = 1 /(5*x4/5)
What is the derivative of
s(t) = t³ + 5t² - 3t + 8?
3t² + 10t - 3
What is the derivative of 1/x5?
f(x) = x-5
f’(x) = -5/x6
What is the derivative of f(x) = √x/x?
f(x) = x½*x-1
f(x) = x-½
f’(x) = -½*x-3/2
f’(x) = -1/(2*x3/2)
What is the derivative of f(x) = 6√x + 5cosx?
f(x) = 6*x½ + 5cosx
f’(x) = 3/x½ - 5sinx
Find the equation of the line tangent to f(x) = 3x³ - 10 at (2,14)
f’(x) = 9x²
At x = 2, the slope is 36
:: (y - 14) = 36(x - 2)
y = 36x + 58
Find the equation of the line tangent to f(Θ) = 4sinΘ - Θ at (0,0)
f’(Θ) = 4cosΘ - 1
At Θ = 0, slope = 3
(y - 0) = 3(Θ - 0)
y = 3Θ
Find the equation of the line tangent to
f(x) = (x² + 2)(x + 1) at (1,6)
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
Find where f(x)= x4 - 2x² + 3 has a horizontal tangent.
f(x)= x4 - 2x² + 3
f’(x) = 4x³ - 4x
x³ = x
x = 0, ±1
(0,3) (1,2) (-1,2)
A coin is dropped from 1,362 feet.
When will it hit the ground and at what velocity?
s(t) = ½gt² + vot + so,
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
