Study Flashcards
How is this implicit equation differentiated?
x^2 +y^2 = 25
2x + 2y(dy/dx) = 0 since differentiating with respect to x so (dy/dx) is written
How is the tangent line to a point on a function?
By first differentiating to find the derivative, and then plugging in the x and y values at the point to find slope and using point slope formula to complete
Differentiating to find the derivative, and then plugging in the x and y values at the point to find slope and using point slope formula to complete
Finding the tangent line to a point on a function
Derivative of arcsin
1 / √1-x^2
1 / √1-x^2
Derivative of arcsin
Derivative of arccos
-1 / √1-x^2
-1 / √1-x^2
Derivative of arccos
Derivative of arctan
1 / 1+x^2
1 / 1+x^2
Derivative of arctan
Derivative of arccot
-1 / 1+x^2
-1 / 1+x^2
Derivative of arccot
Derivative of arcsec
1 / |x|√x^2 - 1
1 / |x|√x^2 - 1
Derivative of arcsec
Derivative of arccsc
-1 / |x|√x^2 - 1
-1 / |x|√x^2 - 1
Derivative of arccsc
Differentiate:
xe^y = x - y
dy/dx = 1-e^y / e^y(x)+1
If f(x) + x^2[f(x)]^4 = 18 and f(1)=2, find f’(1)
Differentiate both sides with respect to x to get -32/33
How would you determine if two curves intersect orthogonally?
Take derivative of both curves and plug in the values of intersection into both derivatives. Then, set the derivatives with the values equal and if you get -1, the curves intersect orthogonally.
How would you differentiate this equation?:
y=x^6cos(x)
Since a^b = e^b•ln(a), this is how you would solve it.
Is this true or false:
If f’’(7)=0, then (7, f(7)) is an inflection point of the curve y=f(x)
False
Is this true of false:
There exists a function f such that f(x)>0, f’(x)<0, and f’’(x)>0 for all x
True
Is this true or false:
There exists a function f such that f(x)>0, f’(x)>0, and f’’(x)<0 for all x
False
Is this true or false:
If f is even and f is differentiable, then f’ is even.
False
Is this true or false:
If f is periodic and f is differentiable, then f’ is periodic
True
What is the derivative of e^9x?
9e^9x
What is ln(9e^9x)?
ln(9) + ln(e^9x)
If f’(x)=g’(x) for 0
False
What does L’Hopital’s Rule say?
If we have indeterminate types or 0/0 or ∞/∞ then differentiate the numerator and denominator and take the limit
If we have indeterminate types or 0/0 or ∞/∞ then differentiate the numerator and denominator and take the limit
L’Hopital’s Rule
Evaluate lim xln(x^2)
x—>0+
Indeterminate so use L’Hopital’s Rule to get limit = 0.
Find the limit with either L’Hopital’s Rule or some other method.
lim (1-8x)^1/x
x—>0
Use the rule that a^b = e^b•ln(a) to get 1/e^8
