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espwksht17

# espwksht17 - ESP Kouba Worksheet l 7 1 Let y =(sin(XI2)x...

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Unformatted text preview: ESP Kouba Worksheet l 7 1. Let y = (sin(XI2))x + 5". Compute y' at x=1c. 2- Assume that y isafunction of x and y3 + xy = 3y3‘. Compute y" at the point (0, 3) . 3. Differentiate. a. y = tanx + arctanx b. y = sinW —— arooosxrxﬁ c. y=cot(sin(5x))+arcsec(cscx) d. y=ln(arctan(lnx)) e. y=log4(x-53X) f. y=log3(x2+e“x) g. y=(x+1)5‘“< 3x-2 h. logxy=eX i. (xy)x2=(tany)xy3 4. A rectangle is to be inscribed in the first quadrantvbelow the graph of y = \l 4 - _x . Determine the dimensions of the rectangle of a. maximum area . b. maximum perimeter. 0. maximum sum of area and perimeter . 5. Evaluate the following limits. a. lim “+1 7” b. lim “3 " n—v—oo n+2 “+4.00 1+n3 6. A baseball is fired horizontally from the top of a cliff, which is one mile high, at 100 miles per hour. See diagram. a. How long does it take for the baseball to reach the ground ? b. How far away from the base of the cliff does the baseball land ? c. What is the " vertical velocity " of the baseball as it strikes the ground ? 7. Consider the function f(x) = x3 - 2x2 + 3/2. a. Sketch the graph of f. b. Use the Intermediate-Value Theorem to prove that f(x) = O has a solution r. 8. Prove that there is some number c , 3 < c < 4 , satisfying 403’ = In (9.57/91) C4+1 ' HlNT: Consider the function f(x) = In (x4+1 ). 7 . For each of the following functions determine the x-values for which fis increasing, decreasing, concave up, and concave down. lndicate all maximum, minimum, and inflection points and intercepts. Neatly sketch the graph off. 9) < H X8 b- y = xlnx O ‘< ll (D + (D ‘0. Use L'Hopital‘s rule to evaluate the following limits. a. lim sinx X-?O x c lim X4'16 x41 ﬁ—wfi‘ e lim 9 “1‘2".1 9' “m xlnx+1—x x—~1 (x—1)2 1. lim 2" +2X Xq+oo 5x . x 2 k. lim xe oos 6x ' X—ro €2X_1 m. "m arcsin x X‘» o arctan 2x sin2x— x2 q. lim {sinx}“" x—v 0* 8. lim (1+5/n)5" Y1~§+oo u. lim x2 lnx X—v o+ . 2 b. llm X ‘1 x91 x-—1 d. “m. tanx X-90 x+sinx f_ "m xzsinx+xsinx X+° x + 1— oosx h. “m tanx X- E 1+secx j. lim X3 Xar+oo 10x L "m eX—1/x X—"+°° 9" + 1/x n. \im 1 2. Xa’o 1-oosx x2 . 1/ p. lim {lnx} x Xv‘r'irOO . 1/ r. hm (1+x) x X90 . 1/ t. hm (1+n) n h~—)+oo . \lx/3 v. hm {tanx} Xao ll . With each of the following functions are given numbers x 1 (the initial 'x-value) and x 2 (the ﬁnal x-value). Compute the associated exact change in functional value, A f, and the differential of f (approximate change in functional value), d f. a. f(x)=x3+x-1 i. x1=1,x2=4 ii. x1=1,x2=2 ii. x1=1,x2=1.1 iv. x1=1,x2=1.01 b. f(x)=lnx i. x1=e,x2=e+2 ii. x1=e,x2=e+.1 ii. x1=e,x2=e+.001 C. f(x) = esinx i. x1=0,x2=-1 ii. x1=0,x2=-.001 iii. x1=0,x2=-.00001 19.. Use differentials to estimate the following quantities. a. 4229 b- (250.1)5 c. sin(1r-—.O4) d. log10(9.9) 13 . Assume that the radius of a sphere is measured with a percentage error of at most 2% . With what percentage error will the following quantities be computed ? a. diameter of the sphere b. volume of the sphere 0. surface area of the sphere ...
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