Calculus I Practice Test 2 PDF

PRACTICE TEST 2 Calculus I - MATH 1220 Name (print): Test 2 will cover sections 3.3-3.9, and 4.1-4.4. The test is closed book. Calculators are not permitted. Show all work, clearly and in order. I reserve the right to take off points if...

PRACTICE TEST 2 Calculus I - MATH 1220 Name (print): Test 2 will cover sections 3.3-3.9, and 4.1-4.4. The test is closed book. Calculators are not permitted. Show all work, clearly and in order. I reserve the right to take off points if I cannot see how you arrived at your answer (even if your final answer is correct). 1. Find the derivatives of the following functions. (a) f (x) = 2x3 + 5x + 1 (b) √ f (x) = x2 + 1 (c) 2x + 1 f (x) = x−2 (d) f (x) = arcsin(2x) · arctan(x3 ) (e) x · ln(x) f (x) = cos(x) + 1 (f) f (x) = ln(x · sin(x)) + tan(e2x ) 2. I throw a ball straight upward, and the height in feet after t seconds is s(t) = 48t − 16t2. (a) Find the velocity function v(t) and the acceleration function a(t). (b) Find the velocity of the ball when it hits the ground. 3. Use implicit differentiation to find y ′ : x2 y + x3 y 2 = 7 4. Use implicit differentiation to find the tangent line to the curve at the point (0, 1). yexy + y 2 = 2 5. Use logarithmic differentiation to find the derivatives of the following functions: (a) (x2 + 2)3 (x − 1)2 f (x) = (2x + 3)4 (b) f (x) = xsin(x) 6. A bird flies in a straight line over your head. It flies horizontally at a constant rate of 20 feet per second, at a constant height of 30 feet. In the picture below, the bird is flying to the right. (a) What is the rate of change of z (the distance between you and the bird) 2 seconds after the bird flies over your head? dθ (b) Find dt 2 seconds after the bird flies over your head. 7. A box has a square base. The dimensions are changing over time, but the base is always square. Say the base is x feet by x feet, and the height is y feet. Let’s say there is an instant when the volume of the box is 12,000 cubic feet, the height is 30 feet, x is increasing at 1 foot per second, and the volume is increasing at 400 cubic feet per second. What is the rate of change of the height? 8. Use linear approximations or differentials to estimate the following:  √ 3 (a) 8.3 0.007 (b) e 9. Find the absolute maximum and minimum values and where they occur for the follow- ing functions on the following intervals: (a) f (x) = 2x2 − 6x, [2, 5] (b) 2 f (x) = 2x + 3x 3 , [−8, 1] 10. Recall that the Mean Value Theorem says the following. If f is continuous on [a, b] and differentiable on (a, b) then there is at least one value c in the interval (a, b) where we have f (b) − f (a) f ′ (c) =. b−a Find the value c guaranteed by the Mean Value Theorem for the following function on the following interval: x2 + 3x + 1, [1, 4]