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Disc5 - Math 17A Kouba Discussion Sheet 5 1 Use the...

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Unformatted text preview: Math 17A Kouba Discussion Sheet 5 1.) Use the Intermediate Value Theorem to prove that each of the following equations is solvable. a.) 11:3 —- a: = 7 on the interval [2, 3] b.) 1~ m = (/5 x2 2 0') £132 + 1 — _ a: 2.) Use the Bisection Method to estimate the value of the solution to x3 — a: — 1 = O to two decimal places. 3.) Use the Bisection Method to estimate the value of the solution to 11:3 + 1 = m2 — :1: to two decimal places. 4.) Use the limit definition of derivative to compute f’ (.73) for each of the following func- tions. a.) f(z)=3 b.) f($)=3:r+5 c.) f(m)=2$2—3a:+7 d-)f(f€)=3+\/5 €-)f($)= f-)f($)=\/$2+4 5.) Find an equation of the line tangent to the graph of the function in problem 4.)c.) at the point r = 2. 2—510 6.) Find an equation of the line normal (perpendicular) to the graph of the function in problem 4.)d.) at the point a: = 1/4. 7.) Let f(a:) = 2:173 + 3.132 — 1251:. Solve f’(a:) = 0 for :c and set up a sign chart for f’. 8.) Let f(x) = a; — 2\/E. Solve f’(:z:) = 0 for :1: and set up a sign chart for f’. 2 n 9.) Let f (2:) = {ff—3:; : ii: 2 8 . Use the limit definition to show that f is differen- tiable at a: = 0. 10.) Let f (:12) = 561/3. Use the limit definition to show that f is NOT differentiable at a: = 0. $+1 2—93 11.) Derive an equation of the line tangent to the graph of y 2 at :r = —1 . 2 12.) Derive equations of all lines which are tangent to the graph of y = —7 — :c and passing through the point (3, 0) . 13.) After a long day of collecting nectar a tired honeybee is headed to its home beehive (sitting somewhere on the :c-axis). It is flying (left to right) along the curve y = 3 cos 3r. 7r At the point a; = g it sees home and travels straight towards the hive along a‘line tangent 1 to the graph of y = 3cosa: .' At What point :v is the home beehive ? (You may assume that D cosx = — sin st.) 14. Sketch the graph of J" from the graph of f. ‘ .‘1 15. Sketch the graph of f from the graph of f’. Assume that f (0) = 1. .C‘ + +++++++++++++++++++++++++++++++++++++++++++ The following problem is for recreational purposes only. 16.) Assume that you have three boxes labeled and filled with fruit. One box contains APPLES only. One box contains ORANGES only. One box contains APPLES and OR— AN GES. Unfortunately, ALL of the boxes are labeled incorrectly. Explain how to correctly relabel all of the boxes by (without peeking into any box) selecting exactly one fruit from exactly one box. ...
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