MATH260 Quiz Week 5 - VI - KEY Name: _________________________ Logarithmic and Exponential Derivatives, L'Hospital's Rule 1) Find y': y = log4(3x2) a) 2log4 (e) y' x *b) y' c) y' d) y' 6log3 (4) x log4 (e) x log4 (3x) 6x 2) Expand using the rules of logarithms, then differentiate: 7x 2x 3 y ln a) y = 7ln(x) - ln(2x + 3) , y' b) y = ln(7) + ln(x) - ln(2x) + ln(3) 7 x 1 2x 3 , y' *c) y = ln(7) + ln(x) - ln(2x + 3) , y' d) y = ln(7x) - ln(2x + 3) , y' 1 7x 1 x 1 7 1 x 1 2x 3 2 2x 3 1 2x 1 3 3) Find y': y 4 23x 2 *a) y ' 24x 2 3x 2 ln(2) b) y ' 6x 8 3x 2 ln(8) c) y ' 24x 2 3x 2 ln(24) d) y' 6x 2 3x 2 ln(2) 4) Find y' and y'': y e 2x 2 Answer: y ' 4x e 2x 2 2 2 2 , y '' 16x 2e 2x 4e 2x 4e 2x (4x 2 1) 5) Find the limit using L'Hospital's rule. Show the indeterminate form. lim x 0 Answer: 2e3(0) 2 0 6e3x 6 lim , lim 2 x 0 3(0) 5(0) x 0 6x 5 0 5 2e3x 2 3x 2 5x 6) The voltage in a certain circuit is given by V = -10e-t sin(224t), t in milliseconds and V in volts. How fast is the voltage changing at t = 4 ms? Use degree mode; 4 decimal places. Answer: .7271 MATH260 Quiz Week 7 - v3 Name: ______________________________ Indefinite Integrals, Integration With Substitution, Finding C, Riemann Sums, Definite Integral 1) Find the indefinite integral. a) b) 3 6x 2 x e 2 dx 4x 4 6x 3 x 2 e3 C 4 3 x 4 x 3 2x 2 ex C c) x 4 2x 3 d) 4x = 2 x e2 x C 2 12x 2 12x e C 2) Identify u, du, then give the correct answer to the integral. a) u 3x 2 2x , du 6x 2 dx , answer : c) 3x 6x 2 4(3x 2 2x)4 b) u 3x 2 2x , du 6x 2 dx , answer : 1 4(3x 2x)4 2 3x 2 2x 2 3 u 2x 1 , du x x dx , answer : 3 6 d) u 6x 2 , du 3x 2 2x dx , answer : 6x 2 (3x 2 2x)4 4 2 2x = 5 dx C C 6 C C 3) Find the position function s(t) satisfying the given conditions: v(t) = 4t 3 - 2t + 1 , s (-1) = 1 4) a) Use Riemann sums to approximate the area under the curve of f(x) = x 2 + 4 from 0 to 4 using 4 left rectangles. Give answers correct to the 10ths. b) Use Riemann sums to approximate the area under the curve of f(x) = x 2 + 4 from 0 to 4 using 4 right rectangles. c) Using the sum of the left and right rectangles, find the average area. d) Find the value of the exact area using the integral. 5) Which of the following will find the value of =? 1 5x 4 4x 2 dx 0 a) (1) 2(1)2 2(1) (0)5 2(0)2 2(0) 5 b) 5(1)4 c) d) 5(0) 4 2 2 4(1) 1 4(0) (0) 1 4 4 (0) 5 (1)5 5 2 2(0)2 (2)(0) (1)5 2(1)2 (2)(1) (1)2 2 (0) 5 2(1) 5 (0)2 2 2(0) 6) Find the area under the curve for f(x) = bound on the left by x = 0, the right by x = 1, and by x x 8 the x-axis. Give an exact answer, improper fractions or radicals as needed. 2