Calculus Midterm Version 1 Nov. 15, 2006

Name:

Student ID number:

Guidelines for the test:

• Put your name or student ID number on every page.

• There are 14 problems: 10 problems in Part I and 4 problems in Part II.

• The exam is closed book; calculators are not allowed.

• There is no partial credit for the Problems in the Part I (multiple-choice (選擇) and fill-in (填充) problems).

• For problems in the Part II (calculation (計算題) problems), please show all work, unless instructed otherwise. Partial credit will be given only for work shown. Print as legibly as possible - correct answers may have points taken off, if they’re illegible.

• Mark the final answer.

Part I: (6 points for each problem)

Multiple Choice (Single Choice)

(1) Which of the following pairs of functions are inverse functions of each other on the implied domains?

A) f (x) =|x|; g(x) = |x| B) f (x) = 2x− 1, g(x) = ¹₂x + 1 c) f (x) = _x¹; g(x) = ^x₁, D) f (x) =√³

x; g(x) = x³.

(2) Which of the following curves is NOT the graph of a function?

(A) (B) (C) (D)

(3) Find the graph corresponding to the derivative of the given function?

f(x) (A) (B) (C)

(5) _dx^d(x^x) =?

A) x^x B) x^x(ln x + 1), C) x^xln x, D) x^x−1

Fill-In Problems

(6) Let f (x) =





2x− 3, x < 2

2, x = 2

x²− 3x, x > 2 . lim

x→2⁻f (x) + f (2) + 3 lim

x→2⁺f (x) = .

(7) Let f (x) =

{ x³, x < 2

Ax− 2, x ≥ 2 . Find A given that f is continuous at 2.

A =

(8) lim

x→∞

x² x²− 1 =

(9) lim

x→1⁻

x x²− 1 =

(10) d

dx(2e^x3) = .

Part II: (10 points for each problem)

Calculation Problems (Show all work)

(11) Compute f⁰(x) by definition (f⁰(x) = lim_h→0^{f (x+h)}_h^−f(x)).

f (x) =√ 3 + x

(12) If x² + y² = 4, use implicit differentiation to obtain dy

dx in term of x and y. Find the equation of the tangent line at the point (√

2,√ 2).

(13) Find d dx

(√x²+ 4 x + 1

(14) Given that f (x) = x³ − x, find the critical number of f(x). Find the absolute maximum and absolute minimum values of the function f (x) on the interval [0, 2].

sin 2θ = 2 sin θ cos θ cos 2θ = 2 cos²θ− 1 = 1 − 2 sin²θ

• Rule of exponents

For any integers m and n, x^m/n = √ⁿ

x^m= (√ⁿ

x)^m For any real p, x^−p = _x¹p

For any real p and q, (x^p)^q= x^pq For any real p and q, x^p· x^q = x^p+q

• properties of logarithm function

For any positive base b6= 1 and positive numbers x and y, we have log_b(xy) = log_bx + log_by log_b(x/y) = log_bx− logby log_b(x^y) = y log_bx log_b(x) = ^{ln x}_{ln b}

• Derivative formulas_d

dxsin x = cos x, _dx^d cos x =− sin x,

d

dxsin⁻¹x = √ ¹

1−x², for −1 < x < 1 _dx^d cos⁻¹x =−^√₁¹_−x2,for−1 < x < 1

d

dxtan⁻¹x = _1+x¹2, _dx^d cot⁻¹x =−_1−x^{1 2},

d

dxsec⁻¹x = ¹

|x|√

x²−1,for |x| > 1 _dx^d csc⁻¹x =−_|x|^√1_x2−1for |x| > 1

d

dxe^x = e^x _dx^d ln x = ¹_x