Name Date Miterm Score Overall Grae Math A Miterm 2 Fall 205 Riversie City College (Use this as a Review) Instructions: All work is to be shown, legible, simplifie an answers are to be boxe in the space provie. You are to work alone an any stuent caught cheating will receive a zero. You will be allotte 2 hours an may only use a pencil for this exam. Answers to wor problems are to be written in a complete sentence with the correct units. Stuents are not allowe to
leave an return. Thus, if you nee to use the restroom, o so now! 2
Cheat Sheet A. Basic rules of ifferentiations (Know your power, prouct, quotient, an chain rule of ifferentiations!) B. Derivatives of exponential an logarithmic functions. ( ) = a x lna 2. x ax (You shoul be able to erive #2) C. Derivative of trigonometric functions (You are accountable to memorizing all of them!) ( x log x) = a x lna D. Derivatives of inverse trigonometric functions. x sin x ( ) = x 2 2. x cos x ( ) = x 2 3. ( x tan x) = + x 2 4. ( x csc x) = x x 2 5. ( x sec x) = x x 2 6. ( x cot x) = + x 2 E. Hyperbolic Functions. sinh x = 2 ex e x ( ) 2. cosh x = ( 2 ex + e x ) (You shoul be able to erive tanh x, cschx, sechx, an coth x ) F. Inverse Hyperbolic Functions. sinh x = ln x + x 2 + ( ) 2. cosh x = ln( x + x 2 ) 3. tanh x = 2 ln + x (You shoul know how to erive each one!) G. Derivatives of Hyperbolic Functions (You are accountable to memorizing all of them!) x H. Derivatives of Inverse Hyperbolic Functions. x sinh x ( ) = + x 2 2. x cosh x ( ) = x 2 3. ( x tanh x) = x 2 4. ( x csch x) = ( x coth x) = x 2 x + x 5. ( 2 x sech x) = x x 2 6. 3
. (2pts each) Suppose f an g are ifferentiable functions of x an that f () = 2, f '() = 0, g() = 5, an g'() =. Fin the values of the following erivatives at x =. a) x fg ( ) b) f x g c) x 7g 2 f ( ) 4
2. (2pts each) Fin y x for the following equations. a) y = + xe 2 x b) 2 2 y= 2tan x sec x 5
c) y = coth ( sec x) 2 tan x y x e ) = ( + ) 6
e) y = sin x e 2 x + π (hint: Use natural logs an implicit ifferentiation) f) y = (ln x) ln x (hint: Use natural logs an implicit ifferentiation) 7
2. (6pts each) The point (0,4) is not on the graph of f (x) = x + x, but it is containe in exactly one tangent line to the graph. a) Fin the one value of a for which the tangent line to the graph of f (x) = x + x ( ) contains (0,4). at a,a + a b) Write the equation of the corresponing tangent line. 8
3. (6pts) In an automobile crash-test, a car is accelerate from rest at 2m / s 2 for 5 secons an then ecelerate at 4m / s 2 until it strikes a barrier. The position function is given by s(t) = t 2, 0 t < 5. 2t 2 + At + B, t 5 a) Assuming that both s(t) an s'(t) are continuous at t = 5, etermine A an B. b) If the barrier is locate 33 meters away, fin the velocity of the car when it strikes the barrier. Answer must be in a complete sentence with the correct units. 9
4. (4pts) The point ( 4 0 5, 0 ) lies on the ellipse with equation 9(x + y) 2 + (x y) 2 = 36. Using implicit ifferentiation, etermine the slope of the tangent line to the ellipse at the given point. 5. (4pts) The parabola f (x) = x 2 + C is to be tangent to the line y = x. Fin C. 0
6. (3pts each) Fin the points on the curve tangent is a) perpenicular to the line y = x+ 24 y x x x 3 2 = 2 3 2 + 20 where the b) parallel to the line y = 2x+ 2
7. (4pts) Fin the values for the constants a, b, an c that will make 2 f( x) = cosx an gx ( ) = a+ bx+ cx satisfy the conitions f(0) = g(0), f '(0) = g'(0), an f ''(0) = g"(0). 8. (4pts) Fin the equation of the tangent line to the graph of when x =. f( x) = xtan x 2
9. (3pts each) Show that x cot x ( ) = + x 2 3
0. (4pts) A police cruiser, approaching a right-angle intersection from the north, is chasing a speeing car that has turne the corner an is now moving straight east. When the cruiser is 0.6 mi north of the intersection an the car is 0.8 mi to the east, the police etermine with raar that the istance between them an the car is increasing at 20 mph. If the police cruiser is moving 60 mph at the instant of measurement, what is the spee of the car? Write your answer in a complete sentence with correct units! 4
Extra Creit: (2pts) Show that 2 2 sin x cos x + = cos 2x x + cotx + tanx 5