Test 1 Review PDF

1) Evaluate the following limits, if they exist. In your answer, specify ∞ or -∞ where appropriate. 𝑥 2 −3𝑥 a) lim 𝑥→3 𝑥 2 −9 𝑥 2 −3𝑥 b) lim + 𝑥 2 −9 𝑥→−3 1−cos 𝑥 c) lim 𝜋 𝑥→ ⁄2...

1) Evaluate the following limits, if they exist. In your answer, specify ∞ or -∞ where appropriate. 𝑥 2 −3𝑥 a) lim 𝑥→3 𝑥 2 −9 𝑥 2 −3𝑥 b) lim + 𝑥 2 −9 𝑥→−3 1−cos 𝑥 c) lim 𝜋 𝑥→ ⁄2 𝑥 sin(4𝑡) tan(3𝑡) d) lim 𝑡2 𝑡→0 𝑥 2 −81 e) lim 𝑥→9 √𝑥−3 𝑥 2 −3𝑥 3 +1 f) lim 4 𝑥→−∞ 𝑥 −9𝑥−5 3𝑥 3 +𝑥−2 g) lim 𝑥→−∞ √𝑥 3 +1+𝑥 6 𝑑𝑦 2) Find 𝑑𝑥 if: 4 a) y = (2𝑥 + 𝑥) cos 𝑥 sec 𝑥+𝑥 b) y = tan 𝑥−𝑥 3 c) y = (3x2 – sin x + 1)( csc x + √𝑥 ) d) y = ex cos x 1 −√𝑥 𝑥 e) y = 1 2+ 3 √𝑥 𝑑2 𝑦 3) Find 𝑑𝑥 2 if: 1 a) y = 3𝑥 4 − √𝑥 + 𝜋 − 𝑥 b) y = ex sin x −2 4) Use the definition of the derivative to obtain f’(x) when f(x) = 2𝑥+1. 5) Find the points where the graph of y = x4 – 8x2 + 7 is horizontal. 1+𝑥 6) Find an equation of the line tangent to the graph of the curve f(x) = 𝑥 at the point (-1, 0) 2 𝑥 − 5𝑥 + 7 𝑖𝑓 𝑥 < 2 7) Is the function 𝑓(𝑥) = { continuous at x = 2? cos(𝑥 − 2) 𝑖𝑓 𝑥 ≥ 2  8) Consider the graph of the function. In parts a – f, evaluate each limit (use ∞ 𝑜𝑟 − ∞ where appropriate). a) lim 𝑓(𝑥) 𝑥→−∞ b) lim 𝑓(𝑥) 𝑥→∞ c) lim 𝑓(𝑥) 𝑥→1− d) lim 𝑓(𝑥) 𝑥→1+ e) lim 𝑓(𝑥) 𝑥→2 f) lim 𝑓(𝑥) 𝑥→−2 g) Identify all horizontal and vertical asymptotes of the graph of f(x). h) Identify all x values at which f is not continuous. i) Identify all x values at which f is not differentiable. 9) A particle moves according to a low of motion s(t) = t3 – 13t2+ 35t – 15, t≥0, where t is measured in seconds and s in feet. a) Find the velocity at time t. b) What is the velocity after 3 seconds. c) When is the particle at rest? d) When is the particle moving in the positive direction? e) Find the total distance traveled in the 1st 8 seconds. f) Find the acceleration at time t.