Math Test #2 Info and Review Exercises

Transcription

1 Math Test #2 Info an Review Exercises Spring 2019, Prof. Beyler Test Info Date: Will cover packets #7 through #16. You ll have the entire class to finish the test. This will be a 2-part test. Part 1 will be no calculator. Part 2 will be scientific calculator only. No notes, no books, no phones, no smart watches uring the test. There will be a seating chart for the test. Where to get help as you re stuying: o Office hours o TMARC, LAC, or other tutoring centers o me at beyler@mtsac.eu Formulas an stuff (Note: Know all of these except for the ones with next to them, which I ll give you. This list is not meant to inclue everything you ll nee to know on the test.) x (c) = 0 x (xn ) = nx n 1 (cf) = cf (f ± g) = f ± g (fg) = fg + gf ( f g ) = gf fg x x (sin x) = cos x (sec x) = sec x tan x x (sin 1 x) = 1 1 x 2 x (csc 1 x) = x (ex ) = e x x (ln x) = 1 x 1 x x 2 1 (x > 0) x (cos x) = sin x x (cot x) = csc2 x x (cos 1 x) = x (sec 1 x) = x (ax ) = a x ln a 1 1 x 2 (log x a x) = 1 x ln a 1 x x 2 1 g 2 x (tan x) = sec2 x x (csc x) = csc x cot x x (tan 1 x) = 1 1+x 2 x (cot 1 x) = 1 1+x 2 x x (sinh x) = cosh x (csch x) = csch x coth x x (sinh 1 x) = 1 1+x 2 x (csch 1 x) = 1 x x 2 +1 x x (cosh x) = sinh x (sech x) = sech x tanh x x (cosh 1 x) = 1 x 2 1 x (sech 1 x) = 1 x 1 x 2 x (tanh x) = sech2 x x (coth x) = csch2 x x (tanh 1 x) = 1 1 x 2 x (coth 1 x) = 1 1 x 2

2 Page 2 of 16 sinh x = ex e x coth x = 2 cosh x sinh x cosh 2 x sinh 2 x = 1 cosh x = ex +e x 2 sech x = 1 cosh x tanh x = sinh x cosh x csch x = 1 sinh x Here are some helpful formulas to know for relate rates problems: Distance/rate/time formula: = rt Pythagorean Theorem: a 2 + b 2 = c 2 (or (leg) 2 + (leg) 2 = (hypotenuse) 2 ) Area of rectangle: A = lw Area of circle: A = πr 2 Area of triangle: A = 1 2 bh Circumference of circle: C = 2πr = π How to get perimeter of any polygon (just a the lengths of the sies). How to get the surface area of a 3-D surface (just a the areas of the faces/sies). Volume of a box (also calle a rectangular prism): V = lwh Volume of circular cyliner: V = πr 2 h Surface area of sphere: S = 4πr 2 Volume of sphere: V = 4 3 πr3 Volume of cone: V = 1 3 πr2 h Newton s metho: x n+1 = x n f(x n) f (x n ) L Hospital s Rule Suppose that f(a) = g(a) = 0, that f an g are ifferentiable on an open interval I containing a, an that g (x) 0 on I if x a. Then, f(x) lim x a g(x) = lim f (x) x a g (x)

3 Page 3 of 16 Review Exercises Note: If you write up the answers to all of the review exercises liste below, an han them in at the test, you can earn up to 2% extra creit towars your test! It is important to unerstan that these review exercises are not guarantee to cover all of the potential problems on the test. Please review the notes an homework problems to fully prepare for the test. Types of problems that will appear on Part 1 are labele NC (for No Calculator). 1. Fin y x. (NC) a) x 2 sin y 3x y = x5 b) x 2 e y = y

4 Page 4 of 16 c) cot(x + y) = 1 ln(y 2 + 3) 2. Fin an equation for the tangent line at the given point. (NC) a) x 2 + 4xy + y 2 = 13, (2, 1) b) y = x 2 cos 1 (3x + 3), ( 1, 1)

5 Page 5 of Use logarithmic ifferentiation to fin y. (NC) x a) y = e 5x x 2 4 sin x x (x 3 +1) 5 x+5 b) y = xsec x c) y = (csc x) 1 x

6 Page 6 of The position of a particle is given by the equation s(t) = t 3 12t + 3 (where t 0 is measure in secons an s is measure in meters). a) What is the velocity after 1 secon? b) When is the particle at rest? c) When is the particle moving in the positive irection? ) Sketch a iagram to represent the motion of the particle. e) Fin the total istance travele uring the first 3 secons. f) Fin the acceleration at time t an after 5 secons. g) When is the particle speeing up? When is it slowing own?

7 5. How fast is the volume of a sphere changing with respect to the raius when the raius is 3 inches? Page 7 of The mass of a thin ro from the left en to a point x mm to the right is 3 ln(x + 2) grams. Fin the linear ensity when x is 5 mm. 7. The with of a rectangle is increasing at a rate of 5 cm/s an its length is ecreasing by 1 cm/s. a) When the with is 12 cm an the length is 16 cm, is the area of the rectangle increasing or ecreasing? How fast is the area increasing or ecreasing? b) Is the perimeter of the rectangle increasing or ecreasing? How fast is the perimeter increasing or ecreasing? 8. A laer 20-ft long rests against a vertical wall. If the bottom of the laer slies away at 2 ft/s, how fast is the top sliing own the wall when the bottom is 12 ft from the wall?

8 Page 8 of At 3pm, plane A is 50 miles north of plane B. Plane A is flying east at 120 mph an plane B is flying west at 80 mph. How fast is the istance between the planes changing at 5pm? 10. A trough is 5 meters long, an has cross sections that are isosceles triangles with base 50 cm an height 60 cm (as shown below). If the trough is being fille with water at a rate of 300 cm 3 /min, how fast is the water level rising when the water is 20 cm eep?

9 Page 9 of Prove that the erivative of y = cot x is y x = csc2 x by using the erivatives of sin x an cos x. (NC) 12. Prove that the erivative of y = ln x is y x = 1 x. (NC) 13. Prove that the erivative of y = cos 1 x is y = 1. (NC) x 1 x Fin the ifferential (y) of y = e x cos x. (NC) 15. Fin the ifferential (y) of y = sec 1 (3x). (NC)

10 Page 10 of Use a linear approximation (or ifferentials) to estimate 1 (2.999) The raius of a circle was measure to be 5 ft with a possible error of 0.1 ft. a) Use ifferentials to estimate the maximum error in the calculate area of the circle. What is the relative error? What is the percentage error? b) Use ifferentials to estimate the maximum error in the calculate circumference of the circle. What is the relative error? What is the percentage error? 18. Use Newton s metho to estimate the positive root of 1 x = 1 + x3 correct to six ecimal places. Start with x 1 = 0.8.

11 Page 11 of Fin the following limits. (NC) sin x x a) lim x 0 x 2 e b) lim x e x 2 sin x x 0 3x 3 c) lim x x(e 1/x 1)

12 Page 12 of 16 ) lim sin x ln(sin x) x 0 + e) lim x 0 ( 1 x 1 e x 1 ) x2 f) lim x x 0 + g) lim x (x + 1) e x

13 Page 13 of Let f(x) = x x a) Fin the omain of f. b) Fin the x-intercept(s) an y-intercept of f (if any). c) Fin vertical asymptote(s) an horizontal asymptote(s) (if any). ) Fin f an f, an etermine where each are 0 an/or o not exist (DNE). e) Do a sign analysis on f an f. f) Fin the intervals on which f is increasing an ecreasing. g) Fin the intervals on which f is concave up an concave own. h) Fin all local maxima, local minima, an inflection points of f. i) Sketch the graph of f.

14 Page 14 of Let f(x) = x 1 x 2. a) Fin the omain of f. b) Fin the x-intercept(s) an y-intercept of f (if any). c) Fin vertical asymptote(s) an horizontal asymptote(s) (if any). ) Fin f an f, an etermine where each are 0 an/or o not exist (DNE). e) Do a sign analysis on f an f. f) Fin the intervals on which f is increasing an ecreasing. g) Fin the intervals on which f is concave up an concave own. h) Fin all local maxima, local minima, an inflection points of f. i) Sketch the graph of f.

15 Page 15 of Let f(x) = 4x 1/3 x 4/3. a) Fin the omain of f. b) Fin the x-intercept(s) an y-intercept of f (if any). c) Fin vertical asymptote(s) an horizontal asymptote(s) (if any). ) Fin f an f, an etermine where each are 0 an/or o not exist (DNE). e) Do a sign analysis on f an f. f) Fin the intervals on which f is increasing an ecreasing. g) Fin the intervals on which f is concave up an concave own. h) Fin all local maxima, local minima, an inflection points of f. i) Sketch the graph of f.

16 Page 16 of Let f(x) = e x (x 2 3). a) Fin the omain of f. b) Fin the x-intercept(s) an y-intercept of f (if any). c) Fin vertical asymptote(s) an horizontal asymptote(s) (if any). ) Fin f an f, an etermine where each are 0 an/or o not exist (DNE). e) Do a sign analysis on f an f. f) Fin the intervals on which f is increasing an ecreasing. g) Fin the intervals on which f is concave up an concave own. h) Fin all local maxima, local minima, an inflection points of f. i) Sketch the graph of f.