Chapter 2 Test Flashcards
Tangent Line
Goes through 1 point
Secant Line
Goes through 2 points
When asked to find the slope of a curve what are you really looking for?
The slope of a line that resembles that curve when you zoom in enough
What is the formula for the slope of a secant line?
The limit as x approaches 0 is f(c + deltax)- f(c) over delta x
What is the ideal value for delta x?
0
How do you graph the derivative of f by just looking at the graph of f?
Any curves will have a slope of 0 and everything else you follow the direction it is going in.
Ex: F is a parabola. At x=3, you are at the max of the parabola, so on f’ at x=3, it is 0, then to the right of 3 on graph f it is decreasing, so do the same on graph f’ and then to the left of graph f it is increasing so do the same on graph f’
How do you find the slope of a tangent line using the slope of a secant?
The slope of a secant will always have 0 on the denominator so we have to simplify until we cannot anymore
Differentiation
Process of finding the derivative of a function
What does the limit as x approaches 0 is f(c + deltax)- f(c) over delta x tell you about a derivative?
If this limit exists it gives us a function whose outputs are the slopes of the original function(f)
If a function is continuous everywhere does that mean it is differentiable everywhere?
NO
What does it mean if a function is differentiable?
It has a derivative/slope
How are differentiability and continuity related?
If f is continuous, it MAY or MAY NOT be differentiable at x=c. But if f is differentiable then f is continuous at x=c. If f is not continuous at x=c, then f’(c) doesn’t exist.
What are times that f’ will not exist?
- If there is a hole at x=c. Because f is not continuous at x=c, so f’(c) cannot exist.
- If there is an asymptote at x=c. Because f is not continuous at x=c, so f’(c) cannot exist.
- If there is a jump discontinuity. Because f is not continuous at x=c, so f’(c) cannot exist.
- If there is a sharp turn at x=c.
- If there is something like a sharp turn but the lines are becoming more vertical.
If you are asked to find the slope of a tangent line to the graph of the function at a given point how do you solve this?
First, find the derivative of the function. Then, plug the x value into your derivative.
Sum and Difference Rules
Ex: x^3 + 3x^2- 3
Find the derivative of every term and add it all together or subtract every term
If you are asked to find the equation of a tangent line to the graph of the function at a given point how do you solve this?
First find the derivative. Then plug the x value into the derivative to get the slope. Then, use y=mx+b and plug the point into the equation to get the x intercept.
Shortcut rule for derivative of a constant function
f= 5
f=7
f’=??
0 ALWAYS
What is the one trig identity we need to know?
sin^2x + cos^2x =1
Power Rule
Ex: F= 3x^2
f’=?
X^n= nx^(n-1)
f’= 3(2x^(2-1))
f’=6x
Constant Multiple Rule
ex: f=3x^2
f’=
c* f(x) = c * f’(x)
f’= 3(2x^(2-1))
f’=6x
Derivative of sin x
cos x
Derivative of cos x
- sin x
If asked to find velocity, what do you do?
Find the derivative of the function
If asked to find acceleration, what do you do?
Find the second derivative of the function
How do you interpret derivatives?
Ex: Let N(t) be the number of vehicles a particular valet parks t hours into their shift. Interpret N’(3)=7.5
After 3 hours have elapsed since the valet began their shift, they are parking vehicles at a rate of 7.5 cars/hr.
IMPORTANT PARTS:
- Cars/hr
- Rate
How do you solve this question: Determine if the function is differentiable at x=2, and the function is a piecewise.
You have to plug 2 into all parts of the piecewise and if they equal the same thing that means the function is continuous. So then turn each piece of the function into its derivative and plug in 2 again and if it’s the same then yes it is differentiable.
How do you determine the points where a graph of a function has a horizontal tangent line?
Find the derivative of the graph and then set that derivative equal to 0 and find its zeroes(prolly by factoring). IF YOU CANNOT SET THAT EQUATION EQUAL TO 0 OR FIND ANY ZEROES THEN IT HAS NO TANGENT. Then, plug those x values into the ORIGINAL FUNCTION, to get the y-value.
Practice 65 and 67 in 2.1/2.2 PS A
What does it mean if a question says find out when the object is at rest?
The object is at rest when the velocity function equals 0
Product Rule
fg’+ gf’
Quotient Rule
B= Bottom
T=Top
BT’- TB’
—————
B^2
Derivative of tan x
Sec ^2 x
Derivative of cot x
- csc^2 x
Derivative of sec x
(sec x) (tan x)
Derivative of csc x
- (csc x) (cot x)
What is a jerk function?
The 3rd derivative of a function
How do you find a higher order derivative?
Just do the derivative of the derivative
What is the notation for second derivative? Third?
d^2y/ dx^2; d^3y/ dx^3
Chain Rule
f’(g(x)) times g’(x)
What does it mean if a question asks for the instantaneous rate of change of f’?
Find the derivative of F’
