MTH100 Paper Questions by VUBuddy

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Short-Answer Questions. Questions 1 and 2 are short-answer questions. Put your answer in the box provided. Full marks will be given for a correct answer placed in the box. Show your work also, for part marks. Each part is worth 2 to 3 marks, and not all parts are of equal difficulty. Simplify your answers as much as possible in Questions 1 and 2. 3 marks 2. (a)

Q3. marks

1. Find all x-coordinates at which the tangent line to the curve x2y2 = 16 has slope −1.

Answer: x = ±2

Solution: Implicit differentiation gives

2xy2 + x2 · 2yy0 = 0.

Plugging in y0 = −1 and factoring gives

2xy(y − x) = 0.

This implies x = y since there are no points on the curve with x = 0 or y = 0. The curve’s equation thus becomes x4 = 16 or x = ±2.

3 marks dp (c) Which of the following functions satisfies = p ?

dt A)

B)

C)

Q3. marks

1. Find the absolute maximum and absolute minimum values of

Answer: C) Simplify sin(sec x Note: sec x function inverse the is sec of x

Answer: x x

Solution:

Let y sec x sec then y x cos or equivalently y x Drawing a triangle with angle y as done in class gives sin y x x

of None D) these.

Solution:

By differentiating each expression for p t we conclude that separately, C) is the correct answer, i.e,

d p d t 2(1 t 2) (1 t t p 3 marks (e) Write down the degree 2 Taylor polynomial of f(x) = tanx at x = π/4.

Full-Solution Problems. In questions 3–5, justify your answers and show all your work. If a box is provided, write your final answer there. Unless otherwise indicated, simplification of answers is not required in these questions.

3 marks 3. (a) Use a suitable linear approximation to estimate the value of 3 262. Express your answer is value max.

Answer: y (0 2) min. value is y (1)

Solution:

by finding numbers by the critical Use closed interval method. Start derivative the where checking defined: isn’t or equals zero

y x

So x 1. Ignore and since they aren’t in [0 4]. We compare

y (0 2) y (1) y (4)

box. the in answer the get to

Answer: T x 2( x π/ 2( 4) x π/ 4)

Solution:

need We to determine

T x X k =0 c k x π/ 2) k

where c k f k π/ 2) k We have

f (0) x tan x f x cos x f x tan x cos x

and so c c c gives which box. the in answer the