Name: CALCULUS 1 TEST #1 Choose at most one bonus question. No partial credit will be given without showing work. 1. (30pt) Find the limits 10x3 − x8 = x→2 1 + 2x8 (a) lim √ (b) lim t→0 (c) t+1−1 = t lim − x→−1 5x = (x + 1)4 z (d) lim sec 1 + cos(z) + z→π 3  1 Name: CALCULUS 1 TEST #1 √ (e) lim x→a √ x− a a−x
(f) lim ln z 2 = z→0 2. (6pt) Find all values of z where the tangent lines to the graph of f (z) = z 2 ez are horizontal. 3. (8pt) Find the value of k that makes the function ( x2 − kx, x< g(x) = cos(πx) + k, x ≥ continuous on R. 2 1 2 1 2 Name: CALCULUS 1 TEST #1 4. (8pt) Find the derivative of y = cot(x) using the quotient rule. 5. (12pt) Find s0 (π/4) and s00 (t) where s(t) = cot(t) − 6. (5pt) Find dy dx where y= 7. (6pt) Find the derivative √ 3 1 1 − 5t x4 − ex + xe + π e y = cos((tan(x) + 1)3 ) 3 Name: CALCULUS 1 TEST #1 8. (a) (5pt) Write the definition of the derivative of a function f (x). (b) (9pt) Given f (x) = 4x − 3x2 , find f 0 (x) using the definition of the derivative. (c) (6pt) Find the equation of the tangent line to 4x − 3x3 at x = −3. 9. (5pt) Suppose that f 0 (2) = 3 and f (2) = 10. Calculate  d  5 x f (x) x=2 dx 4 Name: CALCULUS 1 TEST #1 10. (5pt, Bonus) Give separate, practical applications of the instantaneous and average rates of change of a function to real-world phenomena, and show, by example, that they may be different. 11. (5pt, Bonus) Use the product rule and chain rule to prove the quotient rule:   d f (x) = dx g(x) + f (x) dg(x) dx g 2 (x) df (x) g(x) dx 5 
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