InsallW98T1 - Insall WS98 Calculus II Exam#1 1(14 pts...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Insall - WS98 Calculus II Exam #1 January 29, 1998 1. (14 pts.) Definitions: Write the definition of each of the following concepts. (a) A one-to-one function is... (b) The hyperbolic sine function is... 2. (14 pts.) Theorems: Complete the statements of the following theorems. (a) If f is a one-to-one differentiable function and f (f −1 (a)) = 0, then... (b) Suppose f and g are differentiable and g (x) = 0 on an open interval that contains the number a, except possibly at a. Suppose that lim (f (x)) = 0 = lim (g (x)), or that x→a x→a x→a lim (f (x)), lim (g (x)) ∈ {∞, −∞}. If lim x→a x→a f (x) g (x) ∈ R ∪ {−∞, ∞}, then... 3. (14 pts.) Limits: Compute the following limits: (a) lim e3x − e−3x = x→∞ e3x + e−3x (b) lim 1 + sin(x) − cos(x) = 1 − sin(x) − cos(x) x→0 4. (14 pts.) Derivatives: Solve the following: (a) If f (x) = arctan x − √ 1 + x2 , then compute f . (b) If g (x) = ex 4 +sin(x) , then compute g . 5. (14 pts.) Integrals: Solve the following: (a) √ e2x dx = 1 − e4x (b) cos(ln(x)) dx = x 6. (14 pts.) Proofs: Prove the following: (a) cosh2 − sinh2 = 1 (b) The only solutions of the differential equation y = ky are given by y (t) = y (0)ekt , where y(0) is some arbitrary initial condition. 7. (16 pts.) Problems: Solve the following: (a) A thermometer is taken from a room where the temperature is 20◦ C to the outdoors where the temperature is 5◦ C. After one minute the thermometer reads 12◦ C. Use Newton’s Law of Cooling to answer the following i. What will the reading on the thermometer be after one more minute? ii. When will the thermometer read 6◦ C? After one minute, the temperature reading is The time when the thermometer will read 6◦ C is x x+a (b) Given a number a, let f (a) = lim . x→∞ x − a i. ii. iii. iv. Compute f (0). Compute f (1). Find an expression for f (a) in terms of elementary functions. Find a so that f (a) = e. f (0) = f (1) = f (a) = If f (a) = e, then a = . . . . . . ...
View Full Document

This note was uploaded on 01/24/2011 for the course MATH 21 taught by Professor Mathdep during the Winter '98 term at Missouri S&T.

Ask a homework question - tutors are online