Things need to memorize the Exam 2

Question 1

Problem: Using the limit definition to calculate the derivative (Formula)

Answer 1

f’(x) = lim x -> 0 f(x + h) - f(x) / h

Question 2

Steps to use the limit definition problem to calculate the derivative

Answer 2

Step 1: Plug the f(x + h) into the expression given then combine like term
Step 2: Subtract f(x + h) - f(x) = (the original expression)
Step 3: Divide by h: f(x + h) - f(x) / h
Step 4: Cancel the h terms
Step 5: Take limit as h -> 0: Ex:
f’(x) = lim x->0 = (_____)

Question 3

Problem: Use the alternate form of the limit definition of the derivative to evaluate the derivative at the specified value of x (Formula)

Answer 3

f’(a) = lim x->a f(x) - f(a) / x - a

Question 4

Steps to use the Alternate Form limit definition problem to calculate the derivative

Answer 4

Step 1: Evaluate the given number to plug the original equation
Step 2: Use the lim x->(x=?) f(x) - f(given) / x - (given)
Step 3: Simplify/Perform the operation
Step 4: Substitute back to evaluate x = ?

Question 5

In case the alternate form have absolute value expression using one side limit

Answer 5

Ex: f(x) = |x + 5| at x = -5
Step 1: Set this expression into two different one: {x + 5; x >= -5; -(x + 5); x < -5}
Step 2: Divide them into two different column then label right hand limit and left hand limit
Step 3: Using lim x-> +/- (given num) using formula for alternate form
Step 4: Check to see if it both the same number or not

Question 6

In case the alternate form have piecewise function using one side limit

Answer 6

Step 1: Divide into two column for x-> (given number) with ^-/+
Ex: f’(1^-1) = lim x->1^- f(x) - f(1) / x - 1
Step 2: Calculate the f(given number) (basically plug it in)
Step 3: Substitute back to the lim expression then then simplify
Step 4: Cancel the common (x - ?) term and remember to evaluate back them
(continuing with the remaining part)
Step 5: take limit of it

Question 7

A particle is thrown upward with an initial velocity of _ ft/sec from a height of_ feet. Use the position function s(t) = - 16t^2 + vot + so. Include appropriate units in all answers

Answer 7

a) Determine the position function and velocity function of the particle.
b) Find the average velocity of the particle over the interval [a,b]
c) What is the maximum height the object reaches?
d) What is its instantaneous velocity as it hits the grounds?

Question 8

a) Determine the position function and velocity function of the particle.

Answer 8

Position: s(t) = -16t^2 + vot + so
Velocity: v(t) = d/dt(-16t^2 + vot + so)

Question 9

b) Find the average velocity of the particle over the interval [a,b]

Answer 9

avg velocity = s(b) - s(a) / b - a
(In this case, a = ?, b = ?)
Calculate s(?) and s(?) for example with avg velocity is [0,5]
then plug it in

Question 10

c) What is the maximum height the object reaches?

Answer 10

Use velocity function to set it into 0 and equal to t = ? then plug it in back to the position function.

Question 11

d) What is its instantaneous velocity as it hits the ground?

Answer 11

Step 1: -16t^2 + vot + so = 0
Step 2: Divide by -16
Step 3: Quadratic formula
Step 4: Use the positive result to plug the velocity function to have result

Question 12

Suppose the function (in cm) of a function is given by s(t) = 1 + 2 sin(t). Time is in minutes

Answer 12

a) What is the initial position of the particle?
b) When is the first time the particle’s position is 0?
c) Find the acceleration of the particle at t = ?
d) When is the first time the particle’s velocity is 0?
e) Find the average velocity of the particle over the interval [s1, s2]

Question 13

a) What is the initial position of the particle?

Answer 13

Step 1: Plug 0 into the given function
Step 2: Evaluate it

Question 14

b) When is the first time the particle’s position is 0?

Answer 14

Step 1: Set the function into 0
Step 2: Evaluate
Step 3: Find where zero position is at t = ?

Question 15

c) Find the acceleration of the particle at t = ?

Answer 15

Step 1: Find the velocity function v(t) => d/dt(?)
Step 2: Find the acceleration function a(t) => d/dt(?)
Step 3: Plug the time into the acceleration function

Question 16

d) When is the first time the particle’s velocity is 0?

Answer 16

Step 1: Set the velocity function equal to 0
Step 2: Find where the value of the v(t) is

Question 17

e) Find the average velocity of the particle over the interval [a,b]

Answer 17

Step 1: Use the formula (s(b) - s(a) / b - a)
Step 2: Find s^1(?) and plug it in the position function
Step 3: do the same thing with the step 2 but different # => s^2(?)
Step 4: Compute everything into the average velocity by simplify the denominator then see which value of the s^1 and s^2 value then use for numerator

Question 18

1. Determine if the function has a vertical asymptote at the given value or explain why it does not. (only use the rational expression)

Answer 18

Step 1: Find where the denominator is equal to 0
Step 2: Factor the numerator
Step 3: Rewrite everything to factored form but cancel the common term (same term) leaving the remaining one with this: f(x) = x +/- (#), then rewrite it lim x-> when the cancel part the term is then plug it back

Question 19

2. Determine if the function has a vertical asymptote at the given value or explain why it does not. (in case of sine and secant)

Answer 19

f(x) = sin(x - 8) / (x - 8) at x = 8
Step 1: Identify when is f(x) is undefined
Step 2: Check the limits as x approaches 8 by plug the given number into limit form
Step 3: If the limit exist there is no vertical asymptote instead of removable discontinuity