Quiz 6 Nov. 14, 2007

Calculus I Name:

TA/classroom: Student ID:

Quiz 6

Nov. 14, 2007

1. (10 pts) Find the derivative of

f (x) = ln√

x¹⁰e^2x(x²+ 1) Ans: Since

f (x) = ln (x¹⁰e^2x(x²+ 1))^1/2

= 1

2ln (x¹⁰e^2x(x²+ 1))

= 1

2[ln x¹⁰+ ln e^2x+ ln (x²+ 1)]

= 1

2[10 ln x + 2x + ln (x²+ 1)], f⁰(x) = 1

2[101

x + 2 + 1

x²+ 1 · (x²+ 1)⁰] = 5

x+ 1 + x x²+ 1.

2. (10 pts) Given the curve y³− x² =−3. Find _dx^dy implicitly. What is the equation of the tangent line at (2, 1).

Ans: Apply _dx^d to the equation, y³− x² =−3, we have 3y²dy

dx− 2x = 0,

or dy

dx = 2x 3y².

Since the point (2, 1) is on the curve, slope of the tangent line at (2, 1) is given by m = dy

dx|(2,1) = 2· 2 3 = 4

3. The equation of the tangent line is given by

y− 1 = 4

3(x− 2).