Math1036 - calculus - what does the derivative of a

Question

Part A

1. (a) Give the s - δ definition of lim f(x)_x→c = L.

(b) Evaluate the following limits:

(i) lim_x→1+ (x² - 2)/(x - 1)    (ii) lim_x→1- (x² - 2)/(x - 1)   (iii) lim_x→+∞ (x² - x³ + 6x⁵ - x⁶)

(iv) lim_x→+∞ (x⁴ - 2)/3x³ - 1 (v) lim_x→-∞ (x⁴ - 2)/(3x³ - 1) (vi) lim_x→-∞ (x² - x³ + 6x⁵)

(vii) lim_x→+∞ (x⁴ - 2)/(3x⁵ -1)  (viii) lim_x→-∞ (x⁴ - 2)/(3x⁴ -1)  (viii) lim_x→4 (x² - 3x⁷ - 4)/(x - 4)

2. (a) Give a clear and concise definition of the statement: "f (x) is continuous at the point x = c ".

(b) Give a clear and concise definition of the statement: "f (x) is continuous on the interval [a, b] ".

(c) Find a value of K so that the function f (x) is continuous at x = 0 where

           sinx/x + K(x -3)², x < 0,

f (x) =
           x³ + 3x² - 4x + 73 , 0 ≤ x

3. (a) Let ?f (1) = f (1 + h) - f (1) for f(x) = x² - 5x + 6. Calculate lim ?f_h→0(1)/h.

(b) Evaluate lim_x→0 tan(2x)/sin(4x)

(c) Evaluate lim_x→+∞ {ln(2x² + 3) - 2 ln x }.

4. Use L'Hopital's rule or otherwise to find the following limits:

a. lim_x→π/4 (sin x - cos x)/(x - π/4)

b. lim_x→0 (x - tan x) (x - sin x)

Part B

1. (a) Give the limit definition of the derivative of a function. What does the derivative of a function at a single point tell us about its graph at that point?

(b) Use the definition of derivative to find f j(1) for f (x) = x² - 5x + 6.

(c) Find the equation of the tangent line of the graph of the function f (x) = x² - 5x + 6 at the point (1, 2). What is the area of the triangle formed by this tangent line and the co-ordinate axes?

2. (a) Find the equation of the tangent line to the curve y = 2x³ - 3x² - 36x - 82 at its point of inflection.

(b) Let y(x) be defined implicitly by the equation x²y = x + 2y. Find y^j at the point (2, 1).

3. Find the derivatives of the following functions. Simplify your answers.

(a) (4 - 2x - x²)^5/3   b) ex arctan x   c) ln | sec x |

d) cot x/(x + 1)²    (e) (-x^9/2 - 3x^3/2 - x^5/2 + √x)/√x   f) ln √x /(Σx² + 1)

4. (a) Clearly and concisely state the result known as the Mean Value Theorem for differentiable functions.

(b) Find a value of c that satisfies the conclusion of the MVT for f (x) = x² - 5x + 6 on the interval [0, 2].

Part C

1. (a) Clearly and concisely state the result known as the Intermediate Value Theorem for continuous functions.

(b) Show that the equation f(x) = 2x³ - 3x² -36x - 82 = 0 has a solution for some x* ∈ (5, 6).

(c) Apply Newton's method to this equation with an initial guess x₀ = 5 to find x₁. Now use x₀ = 6 to find x₁. Which one of these two approximations would you use for x^∗?

2. Let f (x) = (x² - 1)(x - 3).

(a) What is the domain of the function f (x) ?

(b) Use logarithmic differentiation to find f_j(x) .

(c) For what values of x in the domain of f is f_j(x) = 0 ?

3. Use the 7 - step method to sketch the graph of the curve y = 2x³ - 3x² - 36x - 82.

4. Prove that 1 - Π/2 ≤ x - 2 arctan x ≤ Π/2 - 1 for -1 ≤ x ≤ 1.