Mth Review Problems for Test 2 Stewart 8e Chapter 3. For Test #2 study these problems, the examples in your notes, and the homework.

Transcription

1 For Test # study these problems, the examples in your notes, and the homework. Derivative Rules D [u n ] = nu n 1 du D [ln u] = du u D [log b u] = du u ln b D [e u ] = e u du D [a u ] = a u ln a du D [sin u] = cos u du D [cos u] = sin u du D [tan u] = sec u du D [cot u] = csc u du D [sec u] = sec u tan u du D [csc u] = csc u cot u du D [ sin 1 u ] = du 1 u D [ tan 1 u ] = du 1 + u 1

2 1. Find the derivative for each of the following functions. You do not need to simplify your answers. (a) y = x x + x e + e x (b) y = x4 5x + 1 x + x (c) y = e 4 x + 5 4x3 (d) y = x ln(x 3 + 4x) (e) y = cos(sin(x)) (f) y = e 5x sin 1 (x) (g) r = 6 tan(3t) + t t + 1 (h) y =. Find dy dt 3 (5x + sec(x)) 3/ for each of the following functions. (a) y = t 3 cos 3 (t 3 ) (b) sin (5t) + cos (3y) = t (c) y = 5 10t 5 (d) y = (t + 5) 10 (1 t) (e) y = t 3 tan(t 1) (f) y = sin(π t ) (g) r = sin(cos t) (h) y = sin 3 t + sin t 3 (i) sin y = sin 1 t ( ) 3π sin + h lim in the derivative of the function at x =. h 0 h

3 4. Use the function f(x) = x 1 x. (a) Find the equation of the tangent line for the graph of f(x) when x = 1. (b) For what value of x is the tangent line horizontal? 5. Find the equation of the tangent line for the graph of x y 3 4y = 7x 4 at the point (1, -1). 6. Find the first and second derivatives for f(x) = x sin x. 7. If x = r and r = cos θ find dθ. 8. Use the table below to find h (1) if h(x) = f(g(x)). x f(x) g(x) f (x) g (x) Suppose r(x) is a function describing the amount of money (in dollars) a company makes by selling x clocks. (a) What are the units for the function r (x)? (b) What is the meaning of the statement r (15) = 1? 10. The position function of a particle moving on a line is given by s(t) = t 4 8t + 4, t 0 where t is in seconds and s is in meters. (a) For what values of t (t 0) is the particle at rest? (b) Over what intervals is the particle moving int he negative direction? (c) Find the acceleration at t = seconds. (d) Find the total distance traveled by the particle during the first 3 seconds. 11. If the tangent line for some function y = f(x) at x = 1 is y = 16x 4 then f (x) = and f(1) =. 3

4 x + 1 if x 1. Use g(x) = x 1 if x > (a) Is g(x) continuous for all x? (b) Is g(x) differentiable for all x? How did you decide? (c) Sketch the graphs of g(x) and g (x). 13. The graph of a function is given below. Roughly sketch the graph of the derivative of the function The three graphs below represent the position, velocity and acceleration function for a body moving on a coordinate line. Label each one, then sketch the graph of the speed function. 8 y t 4

5 15. Related Rates: Strategy (a) Draw picture; label constants and variables. (b) indicate what you are given and what you want to find (c) set up an equation relating variables and constants for all time (d) find D t (e) substitute and solve (f) See page 48 in text for problems 16. Find the equation of the tangent lines at the specific point for the following functions: (a) f(x) = x 64 x 8x at x = (b) f(x) = x cos x at x = π Evaluate the following limits: sin(6t) (a) lim t 0 t sin(6t) (b) lim t 0 5t cos(6x) 1 (c) lim x 0 6x 18. Find the velocity, acceleration, and jerk for the following position function. s(t) = cos(t) + 5t 4 + 7t + 1t Find dy (a) y = tan x x of the following: (b) x y + xy = 6 (c) y = ( x x3) 100 (d) y = sec (x )(3x 3 + x) 3 (e) y = tan 1 (x ) + ln(1 + x 4 ) (f) x x = y 5

6 0. Given that u(1) =, u (1) = 0, v(1) = 1, and v (1) = 5, evaluate each of the following when x = 1: (a) (b) (c) (d) (e) d [u(x)v(x)] [ ( ) ] d v(x) u(x) d [u(v(x))] d [cos(u(x))] d [ u(x) ] + v(x) 6

7 Answers 1. Find the derivative for each of the following functions. You do not need to simplify your answers. (a) dy = 3x + 3 x ln 3 + ex e 1 + e x (b) dy = (x + x)(4x 3 10x) (x 4 5x + 1)(x + ) (x + x) (c) dy = ( x)e4 x + (1x )5 4x3 ln 5 (d) dy = ( 3x ) ( ) x x 3 + ln(x 3 + 4x) + 4x x (e) dy (f) dy = e5x = sin(sin(x)) cos(x)() ( 1 4x ) + sin 1 (x)e 5x (10x) (g) dr dt = 18 sec (3t) + 1 (t t + 1) 1/ (t ) (h) dy = 9 (5x + sec(x)) 5/ [10x + sec(x) tan(x)()]. Find dy dt for each of the following functions. (a) dy dt = t3 (3) cos (t 3 )( sin(t 3 ))(3t ) + cos 3 (t 3 )3t (b) sin(5t)(cos(5t))(5) dt dy dt cos(3y) sin(3y)(3) dt = dt dt = dy dt = sin(5t)(cos(5t))(5) cos(3y) sin(3y)(3) (c) dy dt = 1 5 (10t 5) 4/5 (10) = (10t 5) 4/5 (d) dy dt = (t + 5) 10 ( 1) + (1 t) ( 10(t + 5) 9 (t) ) (e) dy dt = t3 ( (t 1) (f) dy dt = cos(π t )(π t) (g) dr = cos(cos t) sin t dt (h) dy dt = 3 sin t cos t + cos t 3 (3t ) ) + tan(t 1)(3t ) (i) dy dt = 1 cos y 1 t ( ) 3π sin + h lim in the derivative of the function f(x) = sin(x) at x = 3π h 0 h. 7

8 4. Use the function f(x) = x 1 x. (a) Find the equation of the tangent line for the graph of f(x) when x = 1. f x x (x) = x 4 m = f (1) = 1 Line: y = x 1 (b) For what value of x is the tangent line horizontal? x x = 0 x = 0 and x = 5. Find the equation of the tangent line for the graph of x y 3 4y = 7x 4 at the point (1, -1). By implicit differentiation: dy = 7 xy3 3x y 4 m = 9 Line: y = 8 9x 6. f (x) = x sin x (sin x + x cos x) f (x) = x sin x (cos x x sin x + cos x) + (sin x + x cos x) (x cos x + sin x) 7. If x = r and r = cos θ find dθ. dθ = dr = (r)( sin θ) = cos θ sin θ dr dθ 8. Use the table below to find h (1) if h(x) = f(g(x)). h (1) = f (g(1)) g (1) = f (3)( ) = ()( ) = 4 9. Suppose r(x) is a function describing the amount of money (in dollars) a company makes by selling x clocks. (a) What are the units for the function r (x)? dr = dollars per clock (b) What is the meaning of the statement r (15) = 1? It means the company lost a dollar for selling the 15th clock. 10. The position function of a particle moving on a line is given by s(t) = t 4 8t + 4, t 0 where t is in seconds and s is in meters. (a) s (t) = 0 = 4t 3 16t = 4t(t 4). At t = 0, 8

9 (b) s (t) < 0 on (0, ) (c) s () = 3 (d) Total distance s(0) s() = 0 ( 1) = 1 (distance traveling backward) + s(3) s() = 13 ( 1) = 5 (distance traveling forward) TD = If the tangent line for some function y = f(x) at x = 1 is y = 16x 4 then f (x) = 16 and f(1) = 1. x + 1 if x 1. Use g(x) = x 1 if x > (a) Is g(x) continuous for all x? Yes (b) Is g(x) differentiable for all x? How did you decide? No because the slope on the left is 1 and the slope on the right is. (c) Sketch the graphs of g(x) and g (x). Make sure g (x) has an open circle where the lines meet at x = 13. The graph of a function is given below. Roughly sketch the graph of the derivative of the function. There should be open circles at the endpoints The three graphs below represent the position, velocity and acceleration function for a body moving on a coordinate line. Label each one, then sketch the graph of the speed function. Red: f(x) 9

10 Blue: f (x) Black: f (x) 8 y t 15. Related Rates: Strategy See book. 16. Find the equation of the tangent lines at the specific point for the following functions: (a) y = x + 9 (b) y = ( ) 1 π 4 x + 3π Evaluate the following limits: (a) 6 (b) 6 5 (c) v(t) = sin(t) + 0t t + 1, a(t) = 4 cos(t) + 60t + 14, j(t) = 8 sin(t) + 10t 19. Find dy of the following: (a) dy = x sec x tan x x 3/ (b) dy xy + y = xy + x 10

11 (c) dy ( ) 99 ( ) = 100 x x3 x 3x (d) dy = 3 sec (x )(3x 3 + x) (7x + ) + (3x 3 + x) 3 sec (x ) tan(x )(4x) (e) dy = x 1 + x 4 + 4x3 1 + x 4 (f) dy = see notes 0. Given that u(1) =, u (1) = 0, v(1) = 1, and v (1) = 5, evaluate each of the following when x = 1: (a) 10 (b) 5 (c) 0 (d) 0 (e) 5 11