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 onetoone function is... (b) The hyperbolic sine function is... 2. (14 pts.) Theorems: Complete the statements of the following theorems. (a) If f is a onetoone 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 = . . . . . . ...
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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.
 Winter '98
 MathDep
 Calculus

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