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
1. Find the derivative for each of the following functions. You do not need to simplify your answers. (a) y = x 3 + 3 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 + 1 3. lim in the derivative of the function at x =. h 0 h
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) 0 1-1 3 1 1 0 3 1-3 0-1 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)? (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
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. 3 1 3 1 1 1 3 4 5 3 4 5 14. 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 6 4 4 1 3 4 5 6 t 4
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 4 17. 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 + 1 19. 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
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
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 ) ( ) + 4 1 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 ( 1 1 + (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 + 1 3. lim in the derivative of the function f(x) = sin(x) at x = 3π h 0 h. 7
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
(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 = 37 11. 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. 3 1 3 1 1 1 3 4 5 3 4 5 14. 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
Blue: f (x) Black: f (x) 8 y 6 4 4 1 3 4 5 6 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 8 17. Evaluate the following limits: (a) 6 (b) 6 5 (c) 0 18. v(t) = sin(t) + 0t 3 + 14t + 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
(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