Calc BC

Grass clippings are placed in a bin, where they decompose. For 0 ≤ t ≤ 30 , the amount of grass clippings remaining in the bin is modeled by A(t) = 6.687(0.931)t, where A(t) is measured in pounds and t is measured in days. Find the average rate of change of A(t) over the interval 0 ≤ t ≤ 30 . Indicate units of measure.

(A(30) - A(0))/(30-0) = -0.197 lbs/day

A cylindrical can of radius 10 millimeters is used to measure rainfall in Stormville. The can is initially empty, and rain enters the can during a 60-day period. The height of water in the can is modeled by the function S, where S(t) is measured in millimeters and t is measured in days for . The rate at which the height of the water is rising in the can is given by S'(t)=2sin(0.03t)+1.5.

(S(60)-S(0)_/60 = 2.863 mm/day

Traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per minute. The traffic flow at a particular intersection is modeled by the function F defined by f(t) = +4sin(t/2) for, 0<t<30, where F(t) is measured in cars per minute and t is measured in minutes.

(f(15) - f(10))/(15-10) = 1.517 cars/min^2

A continuous function f is defined on the closed interval -4<x<6. The graph of f consists of a line segment and a curve that is tangent to the x-axis at x=3, as shown in the figure above. On the interval 0<<em>x<6, the function f is twice differentiable, with f''(x)>0. For how many values of a, -4<x<6, is the average rate of change of f on the interval [a, 6] equal to 0? Give a reason for your answer.

(f(6)-f(a))/(6-a) = 0 when f(a) = f(6) There are two values of a for which this is true.

Let g be a continuous function with g(2) = 5. The graph of the piecewise-linear function g', the derivative of g, is shown above for −3 ≤ x ≤ 7. Find the average rate of change of g(x) on the interval −3 ≤ x ≤ 7.

(g(7)-g(3))/(7+3) = (3/2 - 15/2)/10 = -3/5

In the xy-plane, the line x + y = k , where k is a constant, is tangent to the graph of y = x^2+ 3x + 1 . What is the value of k ?

A - -3

what is lim h-->0 cos(3pi/2 + h) - cos(3pi/2) / h

A - 1

If f(x)=x + sin x, then f'(x)=

A - 1+cosx

if f(x) = e^x, then ln(f'(2)) =

A - 2

Let f be the function defined above. Which of the following statements about f are true? I. f has a limit at x=2 II. f is continuous at x=2 III. f is differentiable at x=2

A - I only

The graph of a function f is shown above. At which value of x is f continuous, but not differentiable?

A - a

lim h-->0 ln(e+h) - 1 / h

A - f(e), where f(x) = lnx

Find the value of A'(15) . Using correct units, interpret the meaning of the value in the context of the problem.

A'(15)=-0.164 The amount of grass clippings in the bin is decreasing at a rate of 0.164 (or 0.163) lbs/day at time t=15 days.

Let f be the function defined above, where c and d are constants. If f is differentiable at what is the value of c+d

B - -2

d/dx(1/x^3 − 1/x + x^2) at is

B - -4

The graph of the function f shown in the figure above has a vertical tangent at the point (2,0) and horizontal tangents at the points (1, -1) and (3,1) . For what values of x, -2<<em>x<4, is f not differentiable?

B - 0 and 2 only

if f(x) = sinx, then f'(pi/3)

B - 1/2

lim h-->0 ln(4+h)-ln(4) / h

B - 1/4

An equation of the line tangent to the graph of y=2x+3 / 3x-2 at the point (1,5) is

B - 13x+7=18

An equation of the line tangent to the graph of y=x+cos x at the point (0,1) is

B - y=x+l

The function f is defined by f(x)=x/x+2 What points (x,) on the graph of f have the property that the line tangent to f at (x,) has slope 1/2?

C - (0,0) and (-4,2)

d/dx (2^x)

C - (2^x)ln2

if f(x) = x, then f'(5) =

C - 1

What is the average rate of change of the function f given by f (x) = x^4 - 5x on the closed interval [0, 3]?

C - 22

If the line tangent to the graph of the function f at the point passes through the point (1,7) passes through the point (-2,-2), then f'(1) is

C - 3

if f(x)=x^3.2, then f'(4)

C - 3

If y=x^2e^x, then dy/dx

C - xe^x (x+2)

The function f is defined on the closed interval [0,8]. The graph of its derivative f' is shown above. The point (3,5) is on the graph of y=f(x) . An equation of the line tangent to the graph of f at (3, 5) is

C - y-5 = 2(x-3)

Let f be the function defined by f(x) = 4x^3 - 5x + 3. Which of the following is an equation of the line tangent to the graph of f at the point where x = -1 ?

C - y=7x+11

If y=lnx/x, then dy/dx =

D - 1-lnx/x^2

What is the instantaneous rate of change at x=2 of the function f given by

D - 2

The graph of y=e^tanx - 2 crosses the x-axis at one point in the interval [0,1]. What is the slope of the graph at this point?

D - 2.961

Let f be the function given above. Which of the following statements are true about f? I. lim x--> 3 f(x) exists II. f is continuous at x = 3. III. f is differentiable at x = 3.

D - I and II only

Let f be a differentiable function with f(2) = 3 and f'(2) = -5, and let g be the function defined by g(x) = xf(x). Which of the following is an equation of the line tangent to the graph of g at the point where x = 2?

D - y-6 = -7(x-2)

Which of the following is an equation of a curve that intersects at right angles every curve of the family y = 1/x + k (where k takes all real values)?

D - y=1/3x^3

Which of the following is an equation of the line tangent to the graph of f(x)=x^4+2x ^2 at the point where f'(x)=1?

D - y=x-0.122

If y=2x+3/3x+2, then dy/dx =

D = -5/(3x+2)^2

If f(x) = (x-1)^2 sinx, the f'(0)=

D-1

Let f and g be differentiable functions with the following properties (i) g(x)>0 for all x (ii) f(0)=1If h(x)=f(x)g(x) and h'(x)=f(x)g'(x) , then f(x)= :

E - 1

If y=x^2 sin2x, then dy/dx =

E - 2x(sin2x + xcos2x)

If f(x) = 2+|x-3| for all x, then the value of the derivative f'(x) at x = 3 is

E - Nonexistant

At x = 3, the function given by

E - both continuous and differentiable

If y=tanx−cotx, then dy/dx =

E - sec^2x+csc^2x

A zoo sponsored a one-day contest to name a new baby elephant. Zoo visitors deposited entries in a special box between noon (t=0) and 8 P.M. (t=8) . The number of entries in the box t hours after noon is modeled by a differentiable function E for 0 ≤ t ≤ 8 Values of E(t) , in hundreds of entries, at various times t are shown in the table above. Use the data in the table to approximate the rate, in hundreds of entries per hour, at which entries were being deposited at time t=6. Show the computations that lead to your answer.

E'(6) = (E(7) - E(5))/(7-5) = 4 hundred entries per hour

The figure above shows the graph of f′ , the derivative of a function f. The domain of f is the set of all real numbers The figure above shows the graph of f′ , the derivative of a function f. The domain of f is the set of all real numbers

For what values of x does the graph of f have a horizontal tangent? horizontal tangent <-> f'(x) = 0 x = -7,-1,4,8

Line l is tangent to the graph of y= x-x^2/500 at the point Q, as shown in the figure above. Find the x-coordinate of point Q.

Let Q be (a, a - (a^2)/500) dy/dx = 1 - x/250 1 - a/250 = ((a-(a^2)/500)) -20)/a a = 100 y=(1-a/250)x+20

The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a twice-differentiable and strictly increasing function R of time t. The graph of R and a table of selected values of R(t), for the time interval 0 ≤ t ≤ 90 minutes, are shown above. Use data from the table to find an approximation for R'(45). Show the computations that lead to your answer. Indicate units of measure.

R'(45)= (R(50) - R(40))/(50-40) = 1.5 gal/min^2

Let f be the function defined by Suppose the function g is defined by where k and m are constants. If g is differentiable at x = 3, what are the values of k and m?

Since g is continuous at x = 3, 2k = 3m + 2. k/4 = m Since these two limits exist and g is differentiable at x = 3, the two limits are equal. Thus k/4 = m 8m = 3m+2 m = 2/5 k = 6/5

Let f be the function defined by f(x) = (e^x)cos(x). Find the average rate of change of f on the interval

The average rate of change of f on the interval 0 ≤ x ≤ π is (f(pi) - f(0))/(pi-0) = ((-e^pi) - 1)/pi

The function g is defined for x>0 with g(1)=2, g'(x)= sin(x+1/x ) , and g''(x)=(1-1/x^2)cos(x+1/x) Find all values of x in the interval at which the graph of g has a horizontal tangent

The graph of g has a horizontal tangent line when g'(x)=0. This occurs at x=0.163 and x=0.359.

Train A runs back and forth on an east-west section of railroad track. Train A's velocity, measured in meters per minute, is given by a differentiable function v^(t) , where time t is measured in minutes. Selected values for v^ (t) are given in the table above. Do the data in the table support the conclusion that train A's velocity is -100 meters per minute at some time t with 5 < t < 8 ? Give a reason for your answer.

V_A (8) < -100 < V_A (5) V_A is differentiable and continuous Therefore, by the Intermediate Value Theorem, there is a time t, 5<<em>t<8, such that V_A (t) = -100

Let f be the function defined byf(x)=3x^5−5x^3+2 Write the equation of each horizontal tangent line to the graph of f.

f ′ ( x )=0 when x =−1,0,1 x = −1 ⇒ f( x ) = 4 ; y = 4 f( 0 ) = 2 ; y = 2 f( 1 )= 0 ; y = 0

Let f be a function defined on the closed interval -3 ≤ x ≤ 4 with f(0) = 3. The graph of f', the derivative of f, consists of one line segment and a semicircle, as shown above. Find an equation for the line tangent to the graph of f at the point (0, 3).

f'(0) = -2 Tangent line is y=-2x+3

Let h be a function defined for all such that h(4) = -3 and the derivative of h is given by h'(x) = x^2-2.x for all x=/0 Write an equation for the line tangent to the graph of h at x = 4.

f'(4) = (16-2)/4 = 7/2 y+3=(7/2)(x-4)

Let f be a function that is twice differentiable for all real numbers. The table above gives values of f for selected points in the closed interval 2 ≤ x ≤ 13. Estimate f ′(4). Show the work that leads to your answer.

f'(4) = (f(5)-f(4))/(5-3) = -3

Let f be the function given by f(x) = 3 cos x . As shown above, the graph of f crosses the y-axis at point P and the x-axis at point Q.

f'(x) = -3sinx f'(pi/2) = -3sin(pi/2) = -3 y-0=-3(x-pi/2)

Let f be the function given byf(x)=3x^4 + x^3−21x^2 Write an equation of the line tangent to the graph of f at the point (2,-28)

f'(x) = 12x^3 + 3x^2 - 42x f'(2) = 24 y+28 = 24(x-2)

Let f be the function given by f(x)=x^3−7x+6 . Write an equation of the line tangent to the graph of f at x=−1 .

f'(x) = 3x^2 - 7 f'(-1) = -4 f(-1) = -12 y-12 = -4(x+1)

Find the x-coordinate of the point on the graph of f, between points P and Q, at which the line tangent to the graph of f is parallel to line PQ.

f'(x)=-3sinx=-6/pi sinx = 2/pi

Let f be the function that is given by f(x) = ax + b / x^2 − c and that has the following properties. (i) The graph of f is symmetric with respect to the y-axis. (ii) lim ... (iii) f'(1)=2

f(-x) = f(x) --> meaning a=0 lim x--> 2^+ f(x) = +infinity --> meaning c=4 f(x) = b/(x^2-4) f'(x) = (-2bx)/(x^2-4)^2 -2 = f'(1) = -2b/9 --> meaning b=9

Let R be the region in the first quadrant enclosed by the graphs of f(x) = 8x^3 and g(x) = sin (π x) , as shown in the figure above. Write an equation for the line tangent to the graph of f at=1/2 .

f(1/2) = 1 f'(x) = 24x^2 f'(1/2) = 6 y = 1+6(x-1/2)

Let f(x)=12−x^2 for x ≥ 0 and f(x) ≥ 0 . The line tangent to the graph of f at the point(k,f(k)) intercepts the x-axis at x=4 . What is the value of k?

f(x) = 12-x^2 f'(x) = -2x (k,f(k)) = 2k through (4,0) and (k,f(k)) -2k = (12-k^2)/(k-4) -> k^2-8k+12=0 k=2 or k=6 but f(6)=−24 so 6 is not in the domain. k=2

Let f be the function given by \(f\left(x\right)=\frac{\abs{x}-2}{x-2}\). Find f'(1)

for x>0 , x=/2 f(x) = x-2/x-2 = 1 Therefore f'(1) = 0

The figure above shows the graph of f', the derivative of the function f, on the closed interval . The graph of f' has horizontal tangent lines at x=1 and x=3. The function f is twice differentiable with f(2)=6. Let g be the function defined by g(x) = xf(x). Find an equation for the line tangent to the graph of g at x=2.

g'(x) = f(x) + xf'(x) g'(2) = f(2) + 2f'(2) = 6+2(-1) = 4 g(2) = 2f(2) = 12 Tangent line is y=4(x-2) + 12

The functions f and g are given by \(f\left(x\right)=\sqrt{x}\) and g(x) = 6 - x. Let R be the region bounded by the x-axis and the graphs of f and g, as shown in the figure above. There is a point P on the graph of f at which the line tangent to the graph of f is perpendicular to the graph of g. Find the coordinates of point P.

g'(x)=-1 Thus a line perpendicular to the graph of g has slope 1. f'(x) = 1/(2(sqrt(x))) 1/(2(sqrt(x))) = 1 x = 1/4 The point P has coordinated (1/4, 1/2)

Let f and g be the functions defined by f(x) = 1+x+e^x^2-2x and g(x)=x^4-6.5x^2 +6x+2. Let R and S be the two regions enclosed by the graphs of f and g shown in the figure above. Let h be the vertical distance between the graphs of f and g in region S. Find the rate at which h changes with respect to x when x=1.8.

h(x) = f(x) - g(x) f'(x) = f'(x) - g'(x) h'(1.8) = f'(1.8) - g'(1.8) = -3.812

Is f differentiable at x=0? Use the definition of the derivative with one-sided limits to justify your answer.

lim h-->0^- (f(h)-f(0))/h = 2/3 lim h-->0^+ (f(h)-f(0))/h < 0 Since the one-sided limits do not agree, f is not differentiable at x=0.

A car is traveling on a straight road. For 0<t<24 seconds, the car's velocity v(t), in meters per second, is modeled by the piecewise-linear function defined by the graph above. For each of v'(4) and v'(20), find the value or explain why it does not exist. Indicate units of measure.

lim t-->4^-(v(t) - v(4))/(t-4) = 5 =/ 0 = lim t --> 4^+ (v(t) - v(4))/(t-4) v'(20) = (20-0)/(16-24) = -5/2 m/sec^2

Let f be a function defined by f(x)={2x−x^2 for x ≤ 1, x^2+ kx +p for x>1. For what values of k and p will f be continuous and differentiable at x=1 ?

lim x--> 1^- (2x - x^2) = f(1) = lim x--> 1^+ (x^2+kx+p) Therefore 1=1+k+p Since f is continuous at x=1 and is piecewise polynomial, left and right derivatives exist. f(1) = 0, 2+k For differentiability at x=1, 0= 2+k. Therefore k= −2, p= 2

Write an equation for line l.

y = (3/5)x+20