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Unformatted text preview: Practic Midterm 2
You will ﬁnd below some questions that are representative of the material we have covered in Chapter 3 (3.1-3.8). These questions are not comprehensive, though. You are responsible for everything in Chapter 1 and 2 (such as limits, continuity, composition of functions, inverse functions). This exam is longer than an actual 50 minute midterm. If you are timing yourself, a reasonable exam may consist of problems 1,2, four parts of 3, 4, 5, 7, 8. Another exam may consist of 1, the other four parts of 3, 5, 6, 8,9, 10. 1 Find the normal line to the curve 2 Find
ds dw √ x + y = e2y at the point (1, 0).
2 √t . (t+ t)1/3 √ if t = Tan−1 (ln w) and s = 3 Find the derivative with respect to the appropriate independent variable. (a) y = cot2 (e
√ x ) (b) r = ln[Cos−1 (x3 − x)] (c) x = (v 2 + 1)sec v (d) r = θtan θ (e) z = 4w3 + ln sec w √ (f) y = sin(x + 2x cos x)
− (g) y = ln( (xe−3)(xx−7x) ) ) x (1−
2 7 (h) y = cos4 (x)x5/7 x8 e4x 27x 4 True or False: If f (x) is diﬀerentiable, then it is continuous. 5 State the formal deﬁnition for the derivative of f (x). 6 Find the normal line to f (x) = 2x2 + 3 that has slope 4. 7 Find lim
f (x+h)−f (x) , h h→0 where f (x) = x . 2x+1 8 Find the following limits or explain why they do not exist. (a) limz→1 (b) limt→0 (c) limh→0
z √ −1 z −1 2 tan t cos(π/2+h)−cos(π/2) h 9 On our moon, the acceleration of gravity is 1.6 m/s2 . If a rock is dropped into a crevasse, how fast will it be going just before it hits the bottom, 30 seconds later? 10 Find the value of
dr dt at t = 0 if r = (θ2 + 7)1/3 and θ2 t + θ = 1. ...
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