Next question Find the derivative of y with respect to x. y = In(10x) + 9x dy dxNext question Use logarithmic differentiation to find the derivative of y with respect to t. V11t y = t . . . dy its dt = rary cesNext question Find the derivative of y = cos * 1 01 with respect to x. The derivative of y = cos - 1 with respect to x isNext question Find the derivative of y with respect to t. y = sin ~ (11t) dy (Simplify your answer. Type an exact answer, using radicals as needed.)Next question A 25-ft ladder is leaning against a house when its base starts to slide away. By the time the base is 7 ft from the house, the base is moving away at the rate of 72 ft/sec. 25-ft ladder a. What is the rate of change of the height of the top of the ladder? y(t) b. At what rate is the area of the triangle formed by the ladder, wall, and ground changing then? c. At what rate is the angle between the ladder and the ground changing then? x(t) a. The rate of change of the height of the top of the ladder is ft/sec. (Simplify your answer.)Next question A spherical balloon is inflating with helium at a rate of 48x ft" / min. How fast is the balloon's radius increasing at the instant the radius is 2 ft? How fast is the surface area increasing? The balloon's radius is increasing at a rate of ft / min at the instant the radius is 2 ft. (Simplify your answer.)\fThe accompanying graph shows the total distance s traveled by a bicyclist after t (a) Which of the following is the bicyclist's average speed, in mph, over hours. the time interval [0, 1]? O O A. - 62 mph O B. - 12 mph O C. 62 mph O D. 12 mph Distance traveled (mi) Elapsed time (hr) Using the graph, answer parts (a) through (c).For the function g(x) graphed here, find the following limits or explain why they a. What is lim g(x)? Choose the correct answer below and, if necessary, do not exist. x-1 a. lim g(xx) fill in the answer box to complete your choice. x-1 b. lim g(x) x-+2 OA, lim g(x) = X-1 c. lim g(x) X-+3 O B. Im g(x) does not exist d. lim g(x) X-1 X-2.5 Ay 6- 4 2 XExplain why the limit does not exist. X lim x-0 [x| Fill in the blanks in the following statement, and then answer the multiple choice below. X X As x approaches 0 from the left, approaches As x approaches 0 from the right, approaches X O A. Since the function is not defined at x = 0, there is no way of knowing the limit as x-0. O B. There is no single number L that the function values all get arbitrarily close to as x-0.Next question Find the following limit. lim (-x2 + 8x - 7) x- -3 .. . lim (- x2 + 8x - 7) = (Simplify your answer.) x- - 3\f/15h + 1 -1 Find lim h-0 h V15h + 1 - 1 lim h h-0 (Type an integer or a simplified fraction.)Find the limit. 3 + 2x + sin x lim X-+0 4 cos x 3 + 2x + sin x lim (Type an integer or a simplified fraction.) X-+0 4 cos x\ff(x + h) - f(x) Limits of the form lim occur frequently in calculus. Evaluate this limit for the given value of x and function f. h-+0 h f ( x ) =x, x=8 The value of the limit is . (Simplify your answer.)Use the graph below to determine whether the statements about the function True or false: 'rn x] = 9. y=f(x} are true orfalse. x_,_3+ 0 True 0 False Use the following function and its graph to answer (a) through (d) below. 10 Ty 8-x, x4. 5 - A 10 a. Find lim f(x) and lim f(x). Select the correct choice below and fill in any answer boxes in your choice. X-4+ x-4- O A. lim f(x) = lim f(x) = x-+4+ x-4- (Simplify your answers.) O B. The limit does not exist.a. Graph f(x) = 9-x x# - 3 9. x = - 3. b. Find lim f(x) and lim f(x). x- - 3- X- - 3+ c. Does lim f(x) exist? If so, what is it? If not, why not? x- - 3 a. Choose the correct graph below. O A. OB. O C. OD. O Ay TOState whether the function graphed is continuous on [- 1,3]. If not, where does it fail to be continuous and why? A? a a Is the function continuous on the interval [13]? If not, why? 0 A. The function is not continuous at x = 0 because of an oscillating discontinuity. O B. The function is not continuous at K: 0 because of a removable discontinuity. O C. The function is not continuous at x: 0 because of a jump discontinuity. O D. The function is continuous on the interval [ 1,3]. Next question x + 6 At what points is the function y = continuous? 333tz IE) Describe the set oi xvalues where the function is continuous, using interval notation. D (Simplify your answer. Type your answer in interval notation.) Next question For what value of a is the following function continuous at every x? f(x) = X - 27, x Next question Find the slope of The function's graph at the given point. Then nd an equation for the line tangent to the graph there. f[x)=4x2 +4x, i 3.24} The slope of the function's graph at l: 3.24} is :l. (Simplify your answer.) Next question Using the definition, calculate the derivative of the function. Then find the values f' (x) = of the derivative as specified. f ( x ) = 7 +x"; f'(- 5), f'(0). f'(1)Graph the derivative of the function graphed on the right. O X Choose the correct graph below. O A. O B. OC. OD. O AY Ay Ay A O AGraph The derivative of the function graphed an the right 3: Choose the correct graph below. DA. GB. DC. DD. [V a. [Y 91 [V a [V a. x Q x G. X Q x Q I27 l3 I3 I27 Graph the derivative of the function graphed on the right. O X Choose the correct graph below. O A. O B. O C. OD. Ay O AY Ay(a) The graph in the figure given below is made of line segments joined end to (a) Choose the correct answer below. end. At which points of the interval [ - 2,6] is f' not defined? Give reasons for your answer. O A. f' is not defined at x = - 2, x= 1, and x = 4 because at these (b) Graph the derivative of f. points the graph has comers. O B. f' is not defined at x = - 2, x= 1, and x = 4 because at these points the graph has cusps. O C. f' is not defined at x = 0, x = 1, and x = 4 because at these points -4- the graph has corners. O D. f' is not defined at x = 0, x = 1, and x = 6 because at these points (0,2) (6,2) (-2,0) the graph has cusps. 4 -21 (1, - 4) (4, - 4 )N ext question Find the first and second derivatives of the given function. w: 11322 2 Next question Suppose u and v are functions of x that are differentiable at x = 0 and that u(0) = - 9, u'(0) =7, v(0) =8, and v'(0) =3. Find the values of the following dy (uv) = derivatives at x = 0. a uv) b C. d. -(-7v- 4u)Next question Find all points (x,y) on the graph of f(x) = 2x- - 5x with tangent lines parallel to the line y = 7x + 3. The point(s) is/are (Type an ordered pair. Use a comma to separate answers as needed.)The function 5: t3 - 9t2 +2?t, [I S 154, gives the position of a body moving on a coordinate line, with sin meters and t in seconds. a. Find the body's displacement and average velocity for the given time interval. b. Find the body's speed and acceleration at the endpoints of the interval. c. When, it" ever, during the interval does the body change direction? E> a. What is the body's displacement tor the given time interval? D m (Simplify your answer.) Next question When a model rocket is launched, the propellant burns for a few seconds, accelerating the rocket upward. After burnout, the rocket coasts upward for a while and then begins to fall. A small explosive charge pops out a 2007 parachute shortly after the rocket starts down. The parachute slows the rocket to keep it from breaking when it 150 lands. The figure here shows the velocity data from the flight of the model rocket. Use the data to complete parts (a) through (9). 100 velocity (ft/sec) 50 -50 100-# N- 10 12 Time after launch (sec) (a) How fast was the rocket climbing when the engine stopped? |ft/sec (Round to the nearest integer as needed.)\" I I 1When a bactericide is added to a nutrient broth in which bacteria are growing, The bacterium population continues to grow for a while, but then stops growing and begins to decline. The size of the population at lime t {hours} is b=12T +124t12312. Find the growth rates at t: 0 hours, t: 5 hours, and t=12 hours. The growth rate att= 0 hours is D bacteria per hour. Next question Write the function in the form y = f(u) and u = g(x). Then find - dx as a function of x. 3x - 2 y = 4- 2 U= (Type an expression using x as the variable.)Next question Write the function below in the form y = f(u) and u = g(x). Then find - as a function of x. dy y= v2x- - 7x+6 Write the function in the form y = f(u) and u = g(x). Choose the correct answer below. O A. y = u and u = 2x- - 7x+6 O B. y= Vu and u = 2x - 7x+6 O C. y= 2u- - 7u +6 and u=x O D. y= 2u- - 7u + 6 and u = vxNext question Find the derivative of the function. y= 2xe - 4% - 4ex dy dx\fSuppose that functions f and g and their derivatives with respect to x have the values shown X f(x) g(x) f' (x) g (x) to the right at x = 2 and x = 3. Find the derivatives with respect to x of the combinations below at the given values of x. 2 5 2 UN - 2 3 1 - 2 5 a. 2f(x), x = 2 b. f(x) + g(x), x=3 f(x) c. f(x) . g(x). x = 3 d. g(x) ' x = 2 e. f(g(x)), x=2 f. f(x), x= 2 g. 2 x = 3 g (x) h. of(x)+g? ( x). x=2 a. The derivative of 2f(x) with respect to x at x = 2 is. (Simplify your answer. Type an exact answer, using x as needed.)Next question Use implicit differentiation to find dy dx 9x y + 5xy- = - 3 dy dxNext question If x3 +y =54, find the value of at the point (3,3). dx The value of- at the point (3,3) is dx (Type a simplified fraction.)Next question Let f(x) = 5x - 9x- - 6, x 2 1.5. Find the value of dx at the point x = 170 = f(4). The value of df -1 dx at the point x = 170 = f(4) is (Type a simplified fraction.)