Calculus I Unit 4 - 8 Notations

Question 1

What is the Linear Approximation Equation ?

Answer 1

ℓ(x) = ƒ(a) + ƒ’(a)(x - a)

Question 2

Explain the Linear Approximation Process for √(15.9)

Answer 2

Identify the equation we are looking to approximate: √(x)

Then find a number close to our inside value: a = 16

Solve for f(a) using the original function then solve for f’(a) using the derivative of the original function:

ƒ(16) = √(16) = 4
ƒ’(16) = 1/2√(16) = 1/8

Now plug our values into the Linear Approximation equation, then plug in the true value:

ℓ(x) = 4 + 1/8 (15.9 - 16) → ℓ(15.9) 4 + 1/8 (15.9 - 16).

Lastly solve ℓ(15.9) for our approximation:

ℓ(15.9) = 4 + 1/8 (-1/10) → 420/80 - 1/80 = 319/80

Question 3

Explain the Optimization Problem Process

Answer 3

Draw out the system being solved for.

Obj Equation: Identify the Equation being solved for (Ie: Surface Area/Volume Equations).

Constraints: Identify the equation (typically given) that forms our constraints.

Solve: the equation to make it single variable.

Identify: the domain of the variable using the new equation.

Differentiate: the Objective Equation with respect to our singled variable.

Critical Points are identified to find the minimum and maximums.

1st/2nd Derivative Test is performed to find the true minimum and maximum points.

Question 4

What are the assumed intervals of Minimum and Maximum Optimization Problems ?

Answer 4

Maximum Problems: (Closed Intervals).
Minimum Problems: (Open Intervals ie: (0 ,∞)) use RD Test.

Question 5

What does the Rolles Theorem state ?

Answer 5

If ƒ is cont. on [a,b] and ƒ(a) = ƒ(b) then there’s a value c between (a,b) such that ƒ’(c) = 0

Question 6

What is the Equation to Mean Value Theorem in the interval [a,b] and how do you solve for the points guaranteed to exist ?

Answer 6

ƒ(a) - ƒ(b) / a - b = ƒ’(c)

Where ƒ’(c) is the derivative of ƒ(x) and x is set to c. c is then solved for using algebra

Question 7

What does Mean Value Theorem state?

Answer 7

ƒ is cont. on [a,b] and differentiable on (a,b) then theres at least one point c such that:

ƒ(a) - ƒ(b) / a - b = ƒ’(c)

Question 8

Under which conditions should L’hopital’s Rule be applied ?

Answer 8

When the Limit is in an indeterminate form:

0/0 | ∞/∞ | ∞ ⋅ 0 | ∞ - ∞ | 0⁰ | ∞⁰ | 1∞

Question 9

How is L’hopital’s Rule applied ?

Answer 9

By format-ing the limit into a fraction then taking the derivative of the top and bottom separately

Question 10

How is the Average Value of a function found ?

Answer 10

ƒₐ = (1/b - a) ⋅ ∫ ƒ(x) dx

where b is the upper bound and a is th lower bound

Question 11

Explain the difference between Displacement and Distance

Answer 11

Displacement is the amount of distance traveled from the start point.

Distance is the amount traveled total regardless of displacement from starting point.

Question 12

How to find the are bounded by the curve of
y = -x² + 3 and y = -x + 1

Answer 12

Take the intergral of the topmost function and subtract it by the bottomost intergral of the bottomost function at the same bounds:

∫ [(-x² + 3) - (-x + 1)] dx

Question 13

Explain the structure of Sigma Notation:

Answer 13

Question 14

Define the property of:

Answer 14

= n(n + 1) / 2

Question 15

Define the property of:

Answer 15

= n(n + 1)(2n + 1) / 6

Question 16

Define the property of:

Answer 16

n²(n + 1)² / 4

Question 17

Define the propety of:

Answer 17

Question 18

What is the equation for dy ?

Answer 18

dy = ƒ’(x)dx

Where dx is the change in x.

Question 19

Evaluate the limit:

lim h → 0⁺
(1/3) - (1/3+h) / h

Answer 19

Algebriacally manipulate the equation:

lim h → 0⁺
(3 + h - 3) / 3h(3 + h)

Cancel the h in the numerator and solve.

lim h → 0⁺ 1 / 3(3) = 1/9

Question 20

find a if ƒ(x) =

(x² + 4x) / (x) if x ≠ 0
a if x = 0

Answer 20

Find the limit of
(x² + 4x) / (x) using l’hopital’s rule.

limit x → 0
(x² + 4x) / (x)

limit x → 0
= (2x + 4) / 1

a = 4

Question 21

Explain how to solve

Given the velocity 𝑣(𝑡) of an object moving along a line and its initial position 𝑠(0),
which of the following represents the position of the object 𝑠(𝑡) for time 𝑡 ≥ 0.

Answer 21

Since we know that 𝑠(𝑡) is the antiderivative of 𝑣(𝑡) and the initial constant is given by 𝑠(0)

𝑠(𝑡) = 𝑠(0) + ∫₀ᵗ 𝑣(𝑥)d𝑥

Question 22

Find the solution to the initial value problem.

ƒ(𝜋/3) = -3

ƒ’(𝑡) = 1 + 6sin(𝑡)

Answer 22

First find the antiderivative of ƒ’(𝑡) and set it equal to ƒ(𝜋/3)

-3 = ∫ 1 + 6sin(𝑡)d𝑡

Take the antiderivative the right and solve for the constant c at 𝑡 = (𝜋/3)

-3 = 𝜋/3 -6cos(𝜋/3) + c

-3 = 𝜋/3 - 3 + c

-𝜋/3 = c

Question 23

What is the Fundamental Theorem of Calculus ?

Answer 23

The Derivative of the integral of ƒ is just ƒ itself

Ex: d/dx ƒ(𝑡)