c Math 151 WIR, Spring 2013, Benjamin Aurispa Math 151 Week in Review 10 Sections 4.5, 4.6, & 4.8 1. A bacteria culture starts with 2000 bacteria and quadruples every 25 minutes. (a) Find a function that models the number of bacteria after t minutes, assuming the population grows at a rate proportional to the number of bacteria. (b) At what time are there 30,000 bacteria? 2. The half-life of a radioactive substance is 10 days. How much of a 30 g sample remains after 2 weeks? 1 c Math 151 WIR, Spring 2013, Benjamin Aurispa 3. An object with temperature 150◦ F is placed into a room with temperature 80◦ . After 20 minutes, the temperature of the object is 120◦ F. Find a function that models the temperature of the object after t minutes. 4. A curve has the property that at every point the slope of the curve is 5 times the y-coordinate. If the curve passes through the point (2, 4), find the equation of the curve. 2 c Math 151 WIR, Spring 2013, Benjamin Aurispa 5. Evaluate the following. √ (a) arcsin(− 2 2 ) (b) arccos(− 21 ) (c) tan−1 √1 3 (d) sin−1 (sin 5π 6 ) (e) cos(arccos 54 ) 3 c Math 151 WIR, Spring 2013, Benjamin Aurispa (f) cos−1 (cos 5π 4 ) (g) tan(tan−1 18) (h) arctan(tan 6π 7 ) (i) cos(arcsin(− 56 )) 4 c Math 151 WIR, Spring 2013, Benjamin Aurispa (j) sin(2 arctan 5) (k) tan(cos−1 x) " −1 6. Calculate lim sin x→∞ x2 + 3 2x2 − 5 ! + tan −1 x2 4−x 7. What is the domain of f (x) = arcsin(4x − 1)? 5 !# c Math 151 WIR, Spring 2013, Benjamin Aurispa 8. Calculate the derivatives for the following functions. √ (a) f (x) = x arcsin( 5x) (b) g(x) = tan−1 (3x2 ) 5 9. Find the equation of the tangent line to y = cos−1 ( x1 ) at the point where x = 2. 6 c Math 151 WIR, Spring 2013, Benjamin Aurispa 10. Calculate the following limits. x2 + 3 x − 4 x→1 42x + ln x − 16 (a) lim sin x − x x→0 x3 (b) lim (c) lim x→1 1 1 − ln x x − 1 7 c Math 151 WIR, Spring 2013, Benjamin Aurispa (d) lim (xe1/x − x) x→∞ (e) lim e−x (ln x)2 x→∞ (f) lim cot x ln(1 + 3x + 5x2 ) x→0+ 8 c Math 151 WIR, Spring 2013, Benjamin Aurispa (g) lim x→∞ 1+ 2 3 + x3 x4 x3 /5 (h) lim sin xtan x x→0+ (i) lim (4 + e3x )−2/x x→∞ 9