Calculus

Question 1

Intermediate Value Theorem

Answer 1

1. Continuity
2. f(a) < L < f(b)
3. Then by I.V.T c E (a,b) such that f(c) = L

Question 2

Prove Vertical Asymptote

Answer 2

lim(f(x)) = +/- infinity
x->a+
OR
lim(f(x)) = +/- infinity
x->a-
then asymptote at x = a

Question 3

Find horizontal asymptote

Answer 3

If
lim(f(x)) = L
x->infinity
Then right end asymptote at y = L
If
lim(f(x)) = M
x-> -infinity
Then left end asymptote at y = M

Question 4

lim(sin(kx)/kx)
x->0
where k =/ 0

Answer 4

1

Question 5

lim(kx/sin(kx))
x->0
where k =/ 0

Answer 5

1

Question 6

lim((cos(x) -1)/x)
x->0

Answer 6

0

Question 7

lim(sin(kx)/wx)
x-> +/- infinity
where k and w =/ 0

Answer 7

0

Question 8

lim(cos(kx)/wx)
x-> +/- infinity
where k and w =/ 0

Answer 8

0

Question 9

cos(2x)

Answer 9

cos^2(x) - sin^2(x)

Question 10

sin(2x)

Answer 10

2sin(x)cos(x)

Question 11

tan(a+b)

Answer 11

(tan(a) + tan(b)) / (1 - tan(a)tan(b))

Question 12

tan(2x)

Answer 12

2tan(x) / (1 - tan^2(x))

Question 13

1 - sin^2(x)

Answer 13

cos^2(x)

Question 14

sin(x/2)

Answer 14

+/- sqrt((1 - cos(x)) / 2)

Question 15

cos(x/2)

Answer 15

+/- sqrt((1 + cos(x)) / 2)

Question 16

Law of Cosines

Answer 16

a^2 = b^2 + c^2 - 2bc(cos(A))

Question 17

sin((pi/2) - x)

Answer 17

cos(x)

Question 18

tan((pi/2) - x)

Answer 18

cot(x)

Question 19

sin^2(x)

Answer 19

(1 - cos(2x)) / 2

Question 20

cos^2(x)

Answer 20

(1 + cos(2x)) / 2

Question 21

Derivative Product Rule

Answer 21

d(uv) / dx = u(dv / dx) + v(du / dx)

Question 22

Derivative Quotient Rule

Answer 22

d(u / v) / dx = (v(du / dx) - u(dv / dx)) / v^2

Question 23

d(sin(x)) / dx

Answer 23

cos(x)

Question 24

d(cos(x)) / dx

Answer 24

-sin(x)

Question 25

d(tan(x)) / dx

Answer 25

sec^2(x)

Question 26

d(sec(x)) / dx

Answer 26

sec(x)tan(x)

Question 27

d(cot(x)) / dx

Answer 27

-csc^2(x)

Question 28

d(csc(x)) / dx

Answer 28

-csc(x)cot(x)

Question 29

Chain Rule

Answer 29

dy / dx = (dy / du)(du / dx)

Question 30

Prove Continuity

Answer 30

If lim(f(x)) x---> a- = lim(f(x)) x---> a+ = f(a) then f(x) is continuous at x = a

Question 31

d(a^x) / dx

Answer 31

(a^x)lna

Question 32

d(lnx) / dx

Answer 32

1 / x

Question 33

d(log(base a) (x)) / dx

Answer 33

1 / xlna

Question 34

Derivative of Inverse Rule

Answer 34

If g(a) = b then (g^-1)'(b) = 1 / g'(a)

Question 35

d(sin^-1(x)) / dx

Answer 35

1 / sqrt(1 - x^2) for |x| < 1

Question 36

d(cos^-1(x)) / dx

Answer 36

-1 / sqrt(1 - x^2) for |x| < 1

Question 37

d(tan^-1(x)) / dx

Answer 37

1 / (x^2 + 1)

Question 38

d(cot^-1(x)) / dx

Answer 38

-1 / (x^2 + 1)

Question 39

d(sec^-1(x)) / dx

Answer 39

1 / (|x|sqrt(x^2 - 1))

Question 40

d(csc^-1(x)) / dx

Answer 40

-1 / (|x|sqrt(x^2 - 1))

Question 41

Extreme Value Theorem

Answer 41

1. Continuity 2. Check bounds, f'(x) = 0 and f'(x) = NDE 3. Largest number is the absolute maximum, smallest is the absolute minimum.

Question 42

Mean Value Theorem

Answer 42

1. Continuity on [a,b] and differentiability on (a,b) 2. By MVT, there is some c E (a,b) such that f'(c) = (f(b) - f(a)) / (b - a) 3. Solve for c

Question 43

Rolle's Theorem

Answer 43

Let f be continuous on [a,b] and differentiable on (a,b). If f(a) = f(b), then there is at least one number c E (a,b) such that f'(c) = 0 (horizontal tangent).

Question 44

The Second Derivative Test

Answer 44

Let f be a function such that f'(c) = 0 and f" exists on an open interval containing c. 1. If f"(c) > 0 (concave up), then local min at c. 2. If f"(c) < 0 (concave down), then local max at c. 3. If f"(c) = 0, then must use First Derivative Test instead.

Question 45

L'Hopital's Rule

Answer 45

If lim(f(x) / g(x)) x->a is indeterminate, then it is equal to lim(f'(x) / g'(x)) x->a

Question 46

Newton's Method

Answer 46

x(n+1) = x(n) - (f(x(n)) / f'(x(n)))

Question 47

1 + tan^2(x)

Answer 47

sec^2(x)

Question 48

1 + cot^2(x)

Answer 48

csc^2(x)

Question 49

N E (k) k=1

Answer 49

(N(N+1)) / 2 = (N^2 + N) / 2

Question 50

N E (k^2) k=1

Answer 50

(N(N+1)(2N+1)) / 6 = (2N^3 + 3N^2 +N) / 6

Question 51

N E (k^3) k=1

Answer 51

((N(N+1)) / 2)^2 = (N^4 + 2N^3 + N^2) / 4

Question 52

N E (a1(r^(i-1))) i=1

Answer 52

(a1(1 - r^N)) / (1 - r)

Question 53

infinity E (a1(r^i) i = 0 for |r| < 1

Answer 53

a1 / (1 - r)

Question 54

Simpson's Rule

Answer 54

DX/3 (f(x0) + 4f(x1) + 2f(x3) + 4f(x4) + ... + 4f(xn-1) + f(xn)) Must have even # of partitions

Question 55

Trapezoidal Rule

Answer 55

DX/2 (f(x0) + 2(f(x1) + f(x2) + ... + f(xn-1)) + f(xn))

Question 56

b S f(x) dx a

Answer 56

lim n--> infinity of N E (DXk)(f(ck)) K=1