Find the two lines intersect - appropriate rule chain

Question

Assignment 1:

Question 1

Evaluate each of the following expressions to get a whole number:

(a) 3⁶

(b) 64^3/5

(c) 125^2/5

(d) 2^-3 (-2)⁶

(e) 27^2/3 x 81^5/4

(f) (64)^5/6 ÷16^-5/4

Question 2

Simplify each of the following expressions:

(a) 6t³ x t x 3t⁹ x 4t⁷

(b) 10a²b⁴ x a⁹ x a⁷b³ x 4b⁵

(c) s^5/2(2s)⁵(4sy)²y^3/2

(d) (5x2)³ ÷ (625x^-8)^1/4

(e) (2x3 )^1/2/(4x)² x (3x²)³/(8x)^1/2

(f) (1/16x¹²)^-3/4 ÷ (x⁶/27)^1/3

Question 3

Simplify each of the following Logarithmic expressions, in some cases, to get a whole number:

(a) log₇ (2401)

(b) log₂ ((64) ÷16 )

(c) log₃ (27^2/3 x 81^5/4 )

(d) log₂ (32x³)

(e) log₃ (27^2/3x³ )

(f) log(x³y) - 2log(x^1/2) + log(1/ y)

Question 4

Solve each of the following equations for the prescribed variable(s):

(a) x⁴ = 65x - x

(b) 5^4x+125^x+3 = 125^2x+5

(c) log₃y + 2log₃(9y) = 4

(d) 4^x = 1024

(e) 252^x+4 = 15

(f) 5a + 3/2a -10 = 4

(g) 2n +1/4n - 3 = n - 7/ 2n + 5

(h) Solve the following two equations and hence find where the two lines intersect:

4x - 6 y = 5
3x + 5 y = 8

Question 5

Solve each of the below equations, or show that there is no solution:

(a) 10x² + 20x -150 = 0

(b) 9x² -12x = -4

(c) 6x + 5 = √(36x² + 25)

(d) 4x² - 7x +15 = 0

(e) Factorise the quadratic in part (a) and (b).

Question 6

Differentiate the following functions with respect to x:

(a) f (x) = 5x⁶ - 3x⁴ +10x - 4

(b) g(x) = 5x^3/7 + 4x^6/5

(c) h(x) = 6x³√x

(d) q(x) = 6/x³

(e) p(x) = √1/x⁵

Question 7

Find the value of the following derivatives if f (3) = 6, f'(3) = 13, g(3) = 7, g'(3) = 5, f '(7) = 8 and g'(6) = 10 :

(a) h'(3) when h(x) = f (x)g(x)

(b) q'(3) when q(x) = f (x)/g(x)

(c) p'(3) when p(x) = f (g(x))

(d) s'(3) when s(x) = g( f (x))

Question 8

Use the appropriate rule (Chain, Product or Quotient) to differentiate the following expressions with respect to x:

(a) 6(4x³ + 3)⁴

(b) 3√x³ + 10

(c) 5(x - 4)(x² + 5)³

(d) 3t + 1/5t + 2

(e) 2x² + 6x + 5/√(x+1)

Question 9

Consider Q to be the number of laptops sold which depends on their price P in dollars, so we write Q = f(P):

(a) What does f (1100) = 2250 mean?

(b) What does f '(1100) = -250 mean?

(c) If the total revenue, R, is given by R = PQ , calculate dR/dp.

(d) Calculate dR/dP for a laptop price of P = $1100.

(e) Would the total revenue increase or decrease if the price was increased to $1101?

Question 10

Find the normal and tangent lines to the following functions at the given point:

(a) f (x) = 5x⁶ - 3x⁴ +10x - 4 when x = 2

(b) 5(x - 4)(x² + 5)³ when x = -1

(c) 2x² + 6x + 5/√x+1 when x = 3