Math

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Recently wolframalpha.comhttp://calculus.sfsu.edu/javascript/icons/pop_out.png became public. This site is an example of an extremely sophisticated computational engine. You could use it to check your differentiation problems and do much more.

1. Find derivatives (and check your answer with the differentiator from Wolfram). /dx (sin x+cos x+e^x)

2. /dx (tan x-sec x)

3. /dx [sec x(tan x+cos x)]

4. /dx (cot x)|_(x = pi/4).

5. Find an equation of the tangent line to = cos xat the point 3 pi/4,-root(2)/2). Check your answer by graphing the function and the tangent line on the same graph.

6. Plot (use technology) = sin x+cos xrestricted to leq x leq pi/2. Locate the points on the graph at which the tangent line appears to have slope zero. Then solve '(x) = 0where in (0, [pi]/2). This way you find http://calculus.sfsu.edu/latexrender?src=%5Cbegin%7Bdisplaymath%7D+%5Ctextstyle%7Bx%7D+%5Cend%7Bdisplaymath%7D&type=png&size=15, for which the graph = sin x+cos xhas a horizontal tangent line.

7. Graph both = sin 3xand = 5xrestricted to 0.2 leq x leq 0.2. Based on the graph what is your guess for imit as (x to 0) [sin 3x]/[5x]? Then use the fact that imit as (h to 0) [sin h]/[h] = 1and the substitution = 3xto find imit as (x to 0) [sin 3x]/[5x].

8. Find (a) d^2]/[dx^2] (tan x)(Hint: To get the second derivative write ec^2 x = sec x sec xand use the Product Rule)

(b) find the fourth derivative of (x) = sin x+3 cos x+2 e^x.