Calculus Answers

A function is given by the equation 𝑦 = 2𝑥 3 + 4𝑥 2 − 5𝑥 + 10.

i) Obtain an expression for 𝑑𝑦 𝑑𝑥.

ii) Find the gradient of the tangent to the curve at (2, 6).

iii) Determine the equation of this tangent line which passes through (2, 6)

Differentiate the function 𝑓(𝑥) = 5𝑥 2 + 2 using the first principle

1. Find an infinite series series of positive terms which sums to 3
2. Find the exact value of 1-2+4/2!-8/3!+16/4!-31/5!+64/6!-128/7!..+(-2)ⁿ/n!

The formula for calculating the sum of all natural integers from 1 to n is well-known:

Sn = 1 + 2 + 3 + ... + n =

n

2 + n

2

Similary, we know about the formula for calculating the sum of the first n squares:

Qn = 1 · 1 + 2 · 2 + 3 · 3 + ... + n · n =

n

3

3

+

n

2

2

+

n

6

Now, we reduce one of the two multipliers of each product by one to get the following sum:

Mn = 0 · 1 + 1 · 2 + 2 · 3 + 3 · 4 + ... + (n − 1) · n

Find an explicit formula for calculating the sum Mn.

The formula for calculating the sum of all natural integers from 1 to n is well-known:

Sn = 1 + 2 + 3 + ... + n =

n

2 + n

2

Similary, we know about the formula for calculating the sum of the first n squares:

Qn = 1 · 1 + 2 · 2 + 3 · 3 + ... + n · n =

n

3

3

+

n

2

2

+

n

6

Now, we reduce one of the two multipliers of each product by one to get the following sum:

Mn = 0 · 1 + 1 · 2 + 2 · 3 + 3 · 4 + ... + (n − 1) · n

Find an explicit formula for calculating the sum Mn.

Problem A.1

The graph below is made of three line segments:

-1 1 2 3 4 5 6 7 8 9 10 11 12

1

2

3

4

y

x

f(x)

g(x)

h(x)

The segments correspond to the following three functions:

f(x) = x − 2, g(x) = p

4 − (x − 6)2 + 2, h(x) = x − 6

Find the total length L of the graph between x = 2 and x = 10.

Problem A.2

Let f(x), g(x) and h(x) be the functions from Problem A.1. Find the derivative λ

(x) of the

following function with respect to x:

λ(x) = f(x) · g(x) + f(x) · h(x) − g(x) · h(x)

*Please give specific answers to both Problem A.1 & A.2

Using Weiestrass M-test, show that the following series converges uniformly.

.

3

1

,

3

1

n x ,x

n 1

3 n 

















 

.

3

1

,

3

1

n x ,x

n 1

3 n 

















 

Problem A.1

The graph below is made of three line segments:

-1 1 2 3 4 5 6 7 8 9 10 11 12

1

2

3

4

y

x

f(x)

g(x)

h(x)

The segments correspond to the following three functions:

f(x) = x − 2, g(x) = p

4 − (x − 6)2 + 2, h(x) = x − 6

Find the total length L of the graph between x = 2 and x = 10.

Problem A.2

Let f(x), g(x) and h(x) be the functions from Problem A.1. Find the derivative λ

(x) of the

following function with respect to x:

λ(x) = f(x) · g(x) + f(x) · h(x) − g(x) · h(x)

*Please give specific answers to both Problem A.1 & A.2

Problem A.1

The graph below is made of three line segments:

-1 1 2 3 4 5 6 7 8 9 10 11 12

1

2

3

4

y

x

f(x)

g(x)

h(x)

The segments correspond to the following three functions:

f(x) = x − 2, g(x) = p

4 − (x − 6)2 + 2, h(x) = x − 6

Find the total length L of the graph between x = 2 and x = 10.

Problem A.2

Let f(x), g(x) and h(x) be the functions from Problem A.1. Find the derivative λ

(x) of the

following function with respect to x:

λ(x) = f(x) · g(x) + f(x) · h(x) − g(x) · h(x)

*Please give specific answers to both Problem A.1 & A.2