Differentiation Questions and Answers

11. (20 points) Use Newton's method to find the zero of f(x)=x²-3 with zo = 2. (Perform four iterations.)

A lighthouse is located 720 feet from the nearest point P on a straight shoreline. The revolving beacon in the lighthouse makes one revolution every 6 seconds. Find the rate at which a ray from the light moves along the shore at a point 990 feet from P. feet per minute

Divide. (12x³ + 14x²-22x-8) ÷ (6x² + x) Your answer should give the quotient and the remainder.

At 4:00pm, ship A is 53 kilometers west of ship B. Ship A is sailing north at 41 kilometers per hour whereas ship B is sailing south at 35 kilometers per hour. Determine the rate of change of the distance between the two ships at 1:00am. Round the solution to the nearest ten-thousandth, if necessary. kilometers per hour

Divide. (8x³+4x²+20x-7)+(4x²-2x) Your answer should give the quotient and the remainder.

Let g(x) = log3(x+5) + 3 Determine the derivative of g-¹ at x = 4. (g-¹)'(4) =

Question 16 feet per second < A 6 foot tall woman walks at a rate of 6 feet per second toward a lamppost that is 15 feet tall. Determine the rate of change of the tip of the woman's shadow when she is 9 feet away from the lamppost. feet per second Determine the rate of change of the length of the woman's shadow when she is 9 feet away from the lamppost. dra d > Video ?

Find x where 0 ≤ x ≤ π. 3 tan²x = sec²x - tan²x

Suppose and y are both differentiable functions of t and x²y = − 8 Determine dy dt = dy dt " dx dt given = 6, 7, and y = 6. =

Suppose and y are both differentiable functions of t and y =- 3x² + 6x - 2 Determine dy / dt given dx/dt =2 and x=8 dy/dt=

Let g(x) = 2x³ + 4 Determine the derivative of g-¹ at x = 2. (g-¹) '(2) = -

Let h(v) = arcsin ( - 5v³) Determine the derivative of h. h'(v) = Determine the slope of h at v = 0. h'(0) =

Let f(y) = arctan 6 (5) Determine the derivative of f. Determine the slope of f at y = -7. f'(-7)=

Let g(x) = 9√x - 5 Determine the derivative of g-¹ at x = 18. (g-¹) '(18) =

Let g(v) arccsc (e7v) Determine the derivative of g. = g'(v) = Determine the slope of g at v = 12. g'(12) =

Let g(x) = 2 sin x on π 2 ≤ x ≤ FIN Determine the derivative of g¯¹ at x = − 1. (g-¹) '(-1) =

Let h(x) = x³ + 7x + 6 Determine the derivative of h-¹ at x = 14. (n-¹) '(14) =

Let g(z)= z^2 /3 +10 cos(z) +e^z +10 sin(z) Determine the third derivative of g. d^3 z/dz³=

Let f(x) = (x+8)² +8, for x > - 8. Determine the derivative of f-¹ at x = 12. (f ¹)'(12) =

Divide. (12x² + 30x+12) + (2x+4) Your answer should give the quotient and the remainder.

Let f(w) = 11^w³ + log₁11 (w^8) Determine the derivative of f. f'(w) =

Let g(x) =x + 8/x+2 Determine the derivative of g-¹ at x = 3. (g-¹) '(3) = |

Let f(x) = arcsec (e-2) Determine the derivative of f. f'(x) = Determine the interval(s) on which f is differentiable. Report the solution using interval notation. f is differentiable on

Let h(t) = arccos (9t7) Determine the derivative of h. h'(t) = -63t6 V1-81t14 Determine the slope of h at t = 0. h'(0) = 0

Let g(y) = arctan(- 9y) Determine the derivative of g. g'(y) = Determine the slope of g at y = -6. g'(- 6) =

Let f(x) = 7 3√x-7+2 Determine the derivative of f-¹ at x = - 12. (f-¹) '(- 12) =

Let f(x) = 7+2 Determine the derivative of f¹ at x = 9. (ƒ-¹) '(9) =

Consider Determine the derivative of y with respect to a. y = 42y6 sin (6y7) + 3+ xy = cos cos (6y7)

ª(7) 4 Determine the derivative of f. Let f(v) = - 7arctan ( f'(v) = Determine the slope of f at v= - 2. f'(-2) =

Let f(t)= = arccos (√4t) Determine the derivative of f. f'(t) = Determine the interval(s) on which f is differentiable. Report the solution using interval notation. f is differentiable on

Cliff divers on an exotic island dive off cliffs that are 59 meters above the sea. The position of a diver above the sea at time t is given by s(t) = -4.9t2 + 5t + 59, where t is measured in seconds and s is measured in meters. Determine the average rate of change of a diver over the time interval [1, 2]. Average Rate of Change = Determine the velocity function and the diver's velocity at t = 1 seconds and t = 2 seconds. I v(t) a(t) = Velocity of the diver at t = 1 seconds: Velocity of the diver at t = 2 seconds: = Determine the accelerations function and the diver's acceleration at t = 1 seconds and t = 2 seconds. Acceleration of the diver at t = 1 seconds: Select an answer Acceleration of the diver at t = 2 seconds: Select an answer Select an answer Select an answer Select an answer

Suppose both f and f-¹ are differentiable functions such that • f(5) = 3 9 13 Determine (f¹) (3). (ƒ-¹) '(3) = • f'(5) = =

Let f(x) = 9 Determine the derivative of f. f'(x) = -(9)(x) 3G Determine the interval(s) on which f is differentiable. Report the solution using interval notation. f is differentiable on (-∞,0) u (0,00)

Let h(u) Determine Du[h(u)] 4e" tan(u) D [h(u)]

Let f(x) = 9 Determine the derivative of f. f'(x) = Determine the interval(s) on which f is differentiable. Report the solution using interval notation. f is differentiable on

Let h(x) = x³ + 9x - 3 Determine the derivative of h -¹ at x = 7. (h-¹) '(7) =

3 + xy Determine the derivative of y with respect to x. = os (6y7) COS

Let f(x) 8arctan Min Determine the derivative of f. f'(x) = Determine the interval(s) on which f is differentiable. Report the solution using interval notation. f is differentiable on x

Let f(a) P 9 Determine the derivative of f. f'(x) = Determine the interval(s) on which f is differentiable. Report the solution using interval notation. f is differentiable on

Let g(u) =- 10u/√u²-6 Use logarithmic differentiation to determine the derivative of g. d/du[g(u)]=

Let h(t) = arccos (9t) Determine the derivative of h. [ Determine the slope of h at t = 0. h'(0) = h'(t) =

Let f(x) = e5 + cos s (−117) 4 Determine the derivative of f. f'(x) = Determine the slope of f at x = 6. f'(6) =

Let h(x) = x² - 15x + 4. Use either of the limit definitions of the derivative to differentiate h. h'(x)= Determine the slope of h at x = 7. h'(7) = =

An object's velocity after t seconds is v(t) = 26 – 2t feet per second. (a) How many seconds does it take for the object to come to a stop (velocity = 0)? seconds (b) How far does the car travel during that time? feet (c) How many seconds does it take the car to travel half the distance in part (b)?

Find a function f such that f '(x) = 5x³ and the line 135x + y = 0 is tangent to the graph of f. =

Find the derivative of the function. sin(2x) f(x) cos(9x)

Use Newton's method with the specified initial approximation x₁ to find x3, the third approximation to the solution of the given equation. (Round your answer to four decimal places.) 2-x²+1=0, x₁-2 = = X

Consider the function below. f(x) = -3x² + 6x + 6 Exercise (a) Find the critical numbers of f. Step 1 Differentiate f(x)=-3x² + 6x + 6 with respect to x. f'(x)= = Submit Skip (you cannot come back) Exercise (b) Find the open intervals on which the function is increasing or decreasing. Click here to begin! Exercise (c) Apply the First Derivative Test to identify the relative extrema.

Explain why Newton's method doesn't work for finding the root of the equation x³ 3x + 1 = 0 if the initial approximation is chosen to be x₁ = 1. f(x) = x³ 3x + 1 = f'(x) = - X . If x₁ = 1, then f'(x₁) =

Find all relative extrema of the function. Use the Second Derivative Test where applicable. (If an answer does not exist, enter DNE.) 36 X f(x) = x + relative maximum relative minimum (x, y) = (x, y) =