Find the derivative of the functions and find the domain of

Question

Calculus Assignment Questions:

1. Below is the graph of f(x). For each of the given points determine the value of f(a) and lim_x→a f(x). If any of the quantities do not exist clearly explain why.

(a) a = -3 (b) a = -1 (c) a = 2 (d) a = 4

2. Below is the graph of f(x). For each of the given points determine the value of f(a) and lim_x→a f(x), lim_x→a+ f(x) and lim_x→a- f(x) if any of the quantities do not exist clearly explain why.

(a) a = -2 (b) a =1 ( c) a = 3 (d) a = 5

3. Evaluate lim_x→-3 (√(2x +22)) - 4)/(x+3), If it exists.

4. For f(x) = 9/(x -3)^s evaluate the indicated limits, if they exist

a. lim_x→3-f(x) b.lim_x→3+f(x) c. lim_x→3f(x)

5. For (8 + x -4x²)/(√(6 + x² + 7x⁴))answer each of the following questions.
a. Evaluate limx→-∞ f(x)
b. Evaluate limx→∞ f(x)
c. Write down the equation(s) of any horizontal asymptotes for the function.

6. Find the derivative of the following functions
a. f(x) = (3x² - 4)⁵

b. f(x) = (e^x - e^-x)/(e^x + e^-x)

c. f(x) = x²tan y + y¹⁰ sec x = 2x

7. Find the tangent line to g(x)= 16/x - 4√x at x = 4

8. Find the domain of the functions

a. f(x) = 1 / |x² - 4|

b. g(x) = 1 / (x² + 4x + 3)

c. h(x) = √(x² + 5x - 6)

d. k(x) = 1 / √(x - 2)²

e. j(x) = 1 / (x - √(x + 2))

f. l(x) = ln|x + 3| - 5

9. Find the range of the functions

a. f(x) = -x² + 6x + 5

b. g(x) = | x + 3 | - 2

c. h(x) = (x - 2) / (x + 3)

d. k(x) = |x³ + 4|

e. j(x) = | (x + 4) (x - 2) |

f. l(x) = | 1/(x - 3) |

10. Find the integral of the following

1. Find ∫xsin(x²)dx.

2. Find ∫e^x cot(e^x)dx.

3. Find ∫sin^-1 xdx.

A. Integration by partial fractions:

1. Find ∫(x + 3)/(x² + 2x + 5) dx.

2. Find ∫(x² + 3)/(x² + 5x + 6) dx.

3. Find ∫(sin x)/(cos²x + 5cosx + 6) dx.

B. Integration by trignometric substitution:

1. Find ∫dx/(4 + x²)^3/2

2. Find ∫dx/(x + 5√x + 6)

3. Find ∫tan⁴θdθ.

11. Find the inverse function of

a. f(x) = 4x/(5-x)

b. f(x) = log5(2x -1)

c. f(x) = e^5x-1

12. Find the inverse of the relation:

{(0, -2), (1, 0), (3, 4), (6, 10)}

13. Find the inverse of a given relation and determine whether the inverse is a function.

{(7, 4), (8, 4)}

14. Find the inverse power function.

f(x) = -16/9 x², x ≤ 0

15. Jason is 21 years older than his sister Sarah. In 5 years he will be twice as old as Sarah. How old is each of them now?