# (30%) /20 MCV4U Quiz 5C: Chapter 5 1. Determine y  for each of the following. a) y = 9 sinx

(30%) /20 MCV4U Quiz 5C: Chapter 5 1. Determine y  for each of the following. a) y = 9 sinx b) y = extan7x c) y = 66x d) y = –2.7(7𝑐𝑜𝑠 √𝑥 ) e) y = 𝑒 𝑠𝑖𝑛 𝑥 𝑥2 f) y = 8xe8 g) y = sin7(x6) h) y = 𝑡𝑎𝑛 6𝑥 𝑠𝑖𝑛 6𝑥 A (30%) /20 T (20%) /13 Total Mark: C (20%) /13 /66 [K – 14] 2. For each of the following functions, determine an equation for the tangent to the curve at the point with the given x-coordinate. [T – 6] 𝜋 a) f(x) = tan (6x – π), 𝑥 = 10 b) f(x) = –7x, x = π 3. A certain radioactive substance decays exponentially over time. The amount of a sample of the substance that remains, expressed as a percent of the original amount, after t years have passed, is given by A(t) = 100e-0.08t. [A – 3, C – 3] a) Determine the rate of change of the function, dA . dt b) What is the half-life for this substance? c) What is the rate of decay when half of the original sample has decayed? 4. Determine the second derivative for each of the following functions: [K – 4] a) f(x) = –5e-5x b) f(x) = tan6x 5. Consider the function f(x) = –5x. a) Determine f  (x). b) Does this function have any extreme values? Explain. c) Determine f  (x). [T – 4] 6. Suppose that a particle moves along a line so that, at time t measured in seconds, its position in metres is given by s(t) = 9 sin(6t). [A – 4, C – 3] a) Differentiate the function with respect to t. b) When is the first time that the particle changes direction? c) When does the particle reach maximum velocity for the interval 0  x  2 ? d) What is the maximum velocity of the particle? 𝜋 7. Consider the function f(x) = –sin6x on the interval 0 ≤ x ≤ 2 . [T – 3, C – 4] a) Determine f  (x). b) Determine the absolute extreme values on the given interval. 𝜋 c) Determine the equation of the line tangent to the graph at x = 6 . 8. For each of the following functions, determine the slope of the tangent to the curve at the point with the given x-coordinate. [A – 3] 𝜋 a) f(x) = sin (6x) – tan (7x), x = 5 b) f(x) = –x2ex + 2ex, x = 4 9. For the following functions, determine any maximum and/or minimum values that are turning points 𝜋 on the interval 0 ≤ x ≤ 2 , if there are any. [A – 3, C – 3] a) f(x) = 7 tan x sinx b) f(x) = –9e-x +7 2 10. Determine the first and second derivative of the function f(x) = 𝑒 2.7𝑥 . 11. a) Determine the point at which the tangent to the function f(x) = ex is horizontal. x3 b) Is this point a local minimum or a local maximum? How do you know? [K – 2] [A – 7]

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