Unit 3 - Differentiation: Definition and Basic Derivative Rules

Question 1

Equation of a tangent line to a graph at a point

Answer 1

f’(c)(x-c)+f(c)

Question 2

How do you find the normal?

Answer 2

Find the slope of the tangent, and take the negative reciprocal.

Question 3

What is the normal when the tangent line has a slope of 0?

Answer 3

The normal is a vertical line, represented as x=c

Question 4

What is the equation for the average rate of change?

Answer 4

average rate of change = (f(x)-f(c)) / (x-c) where x=/=c

Question 5

What letter represents distance in Calculus?

Answer 5

s

Question 6

What is average velocity?

Answer 6

Change in position with respect to time.

Question 7

Difference between speed and velocity?

Answer 7

Speed is absolute. Velocity is signed (can be positive or negative)

Question 8

What is instantaneous velocity?

Answer 8

The limit of average velocity as Delta t approaches 0.

Question 9

How do you find the derivative when given a table of data?

Answer 9

Use the two closest independent values, and find the secant. (this approximates the tangent)

Question 10

How do continuity and differentiability relate?

Answer 10

If a function is differentiable at c, it is continuous as c (f must be continuous at c for f to be differentiable at c. If a function is not continuous at c, it is not differentiable at c. If a function is continuous at c, it is not necessarily differentiable.

Differentiability implies continuity, but continuity does not imply differentiability.

Question 11

What is the derivative of e^x

Answer 11

e^x

Question 12

Simple power rule

Answer 12

x^n = nx^(n-1) Does not work for exponentials (n^x).

Question 13

Sum/Difference rule.

Answer 13

f(x)=g(x)+-h(x). If g(x) and h(x) are differentiable, then f(x) is differentiable and f’(x)=g’(x)+-h’(x)

Question 14

Notations for derivatives.

Answer 14

f’(x)
y’
dy/dx
(d/dx)y
(d/dx)f(x)
Df(x)

Question 15

Derivative of a constant.

Answer 15

If f is a constant function f(x)=a, then f’(x)=0

Question 16

What is the derivative of the natural logarithm?

Answer 16

y=ln x, y’=1/x

Question 17

What is the product rule?

Answer 17

(uv)’=uv’+u’v

Question 18

What is the quotient law?

Answer 18

(u/v)’=(vu’-uv’)/(v^2)

Question 19

Higher order derivatives

Answer 19

y^(n)
f^(n) (x)
d^(n) y / dx^(n)
d / dx^(n)

Question 20

derivative of sin(x)

Answer 20

cos(x) over all real numbers

Question 21

derivative of cos(x)

Answer 21

-sin(x) over all real numbers

Question 22

derivative of tan(x)

Answer 22

sec^2 (x) where x does not equal (2k+1)/(2) * Pi, k an intiger

Question 23

derivative of cot(x)

Answer 23

-csc^2 (x) where x does not equal nPi, n an intiger

Question 24

derivative of csc(x)

Answer 24

-csc(x) * cot(x), where x does not equal nPi, n an intiger

Question 25

derivative of sec(x)

Answer 25

sec(x) * tan(x), where x does not equal (2k+1)/(2) * Pi, k an intiger

Question 26

Useful note for remembering some derivatives of trigonometric functions?

Answer 26

If the trigonometric function starts with a c, its derivative has a negative sign

Question 27

How do you find the derivative using its definition

Answer 27

find the limit as h approaches 0 of (f(x+h)-f(x)) / h

Question 28

What does it mean of f'(a) (f prime of x at a) does not exist?

Answer 28

The tangent might not exist, might be a vertical line, or might not be unique

Question 29

What does limit as x approaches c of (f(x)-f(c)/(f-x) equal?

Answer 29

The tangent line to the graph of f at a c, provided the limit exists

Question 30

Average rate of change equation

Answer 30

(f(x)-f(c))/(x-c), x =/= c

Question 31

What is average rate of change?

Answer 31

behavior over an interval

Question 32

What is instantaneous rate of change?

Answer 32

behavior at a number (not over a range)

Question 33

Average velocity equation

Answer 33

Delta s/ Delta t = (f(t1))-f(t0)) / (t1-t0)

Question 34

Instantaneous velocity equation

Answer 34

v = limit as delta t approaches 0 of Delta s/ Delta t = limit as t0 approaches t0 of (f(t1))-f(t0)) / (t1-t0)

Question 35

How to use the definition for derivative of a function at a number?

Answer 35

f'(c) = limit as x approaches c of ((f(x)-f(c))/(x-c)

Question 36

How to use the definition for a derivative function f' of a function f?

Answer 36

f'(x) = limit as h approaches 0 of ((f(x+h)-f(x))/h

Question 37

What does it mean when asked to use "the definition" to find the answer?

Answer 37

Longer form. No shortcuts.

Question 38

What are the 3 ways a function cannot be differentiable at a point on a continuous function?

Answer 38

The limit from the left and right exist but are not equal (corner). The one-sided limits are both infinite (+ or -) (vertical tangent). The one-sided limits are infinite but one is positive and the other is negative (cusp).

Question 39

How do you know where the horizontal tangent lines of a graph of f are?

Answer 39

Where f' is 0.

Question 40

What are the common errors you make?

Answer 40

Incorrectly reducing exponents. Forgetting negatives/adding them where they don't exist. Improperly formatting quotient rules. READ THE QUESTION AND KNOW WHAT IT WANTS. Remember that some limits of trigonometric equations involve squares.

Question 41

Memory trick for quotient rules of derivatives.

Answer 41

Low d-high minus high d-low, draw a line and square below

Question 42

What do you need to remember about solving for x of a trigonometric equation?

Answer 42

Remember the domain. If none is given, give the general solution.

Question 43

How would you format a falling object's speed?

Answer 43

Falling at abs(velocity). Falling refers to the direction, so signs are not needed.

Question 44

Position... Velocity... what comes next?

Answer 44

Position... Velocity... then acceleration, jerk and snap.

Question 45

Form 1 and Form 2

Answer 45

todo

Question 46

What should you watch out for when a question asks for a limit?

Answer 46

Check if it is using the definition of a derivative

Question 47

For the current unit, do vertical tangents exist

Answer 47

No.

Question 48

What does m (subscript tangent) represent?

Answer 48

Slope of the tangent. Equal to form 1 of the definition of a derivative

Question 49

What do you need to remember about graphs?

Answer 49

Check the scale of the axi.

Question 50

When asked to match a function’s graph to its derivatives graph, what do you do?

Answer 50

Remember that you don’t know the equations. Describe the behaviour of the graphs over intervals and points.

Question 51

Finding limit of a piece wise function at a point

Answer 51

Use form 1 (x -> c) from the left and right sided limits. Check for continuity as well

Question 52

Do cusps have derivatives?

Answer 52

No. Derivatives cannot be infinite (for now at least)

Question 53

How to determine continuity using differentiability.

Answer 53

Wherever f is differentiable, f is continuous. Where f is not differentiable, you need more information to determine continuity.

Question 54

Difference between at a point and value.

Answer 54

Point are like (x,y). Values are x=… units