1. A particle moves along the x-axis in such a way that it's position in time t for t is greater or equal to 0 is given by x= 1/3t^3 - 3t^2 +8t
A. Find the position of the particle at time t = 3.
B. Show that at time t = 0, the particle is moving to the right.
C. Find all values of t for which the particle is moving to the left.
D. What is the total distance the particle travels from t = 0 to t = 4?
2. A spherical balloon is being inflated at a rate of 10 cubic centimeters per second.
A. Find an expression for dr/dt, the rate at which the radius of the balloon is increasing.
B. How fast is the radius of the balloon increasing when the diameter is 40 cm?
C. How fast is the surface area of the balloon increasing when the radius is 5 cm?
3. The function f(x) has the value f(1) = 5. The slope of the curve y = f(x) at any point is given by the expression dy/dx= (4x-2)(y+1)
A. Write an equation for the line tangent to the curve y = f(x) at x = 1.
B. Use separation of variables to find an explicit formula for y = f(x), with no integrals remaining.
C. Calculate the slope of the tangent line to the curve at x = 0.
4.The derivative of a function f is given by f'(x)=(-2x-2)e^x, and f(0) = 3.
A. The function f has a critical point at x = -1. At this point, does f have a relative minimum, a relative maximum, or neither? Justify your answer.
B. On what intervals, if any, is the graph of f both increasing and concave down? Explain your reasoning.
C. Find the value of f(-1).
5. The region R is bounded by the x-axis, x = 0, x=2pi/3, and y= 3sin(x/2) .
A. Find the area of R. (2 points)
B. Find the value of k such that the vertical line x = k divides the region R into two regions of equal area. (3 points)
C. Find the volume of the solid generated when R is revolved about the x-axis.
D. Find the volume of the solid generated when R is revolved about the line.
6. Two curious calculus students recorded the speedometer readings every 2 minutes on
their way to school one morning. Their data is given in the table below:
t (minutes) 0 2 4 6 8 10 12 14 16
v(t) (mph) 50 52 51 48 46 45 48 55 60
A. Approximate the average speed over the time interval using a trapezoidal approximation with subintervals of length minutes.
B. If the function f(x) = .03x^3-.5x^2+1.4x+51 is used to model the velocity at time t, find the average value of f over the time interval minutes.
C. Using function f(t) from part (B), find . Round your answer to the nearest
tenth.
D. Explain the meaning of your answer in part (C) in terms of the problem.
Please show all your work I really need help on these six questions
Hi please someone answer I really need this
#3
(A) The point on the curve is (1,5)
y'(1) = 4*6=24
so the tangent line is
y-5 = 24(x-1)
(B) dy/(y+1)= (4x-2)dx
ln(y+1) = 2x^2-2x+c
y+1 = c*e^(2x^2-2)
(C) at x=1, 6=c, so
y = 6e^(2x^2-2) - 1
y' = 24x e^(2x^2-2)
so y'(0) = 0
Now you try the others. Post your work if you get stuck
Certainly! Let's work through each question step by step.
1. A particle moves along the x-axis in such a way that its position in time t for t ≥ 0 is given by x = (1/3)t^3 - 3t^2 + 8t.
A. To find the position of the particle at time t = 3, substitute t = 3 into the equation: x = (1/3)(3^3) - 3(3^2) + 8(3). Simplifying this expression will give you the position at time t = 3.
B. To show that at time t = 0, the particle is moving to the right, we need to calculate the velocity of the particle at t = 0. Velocity is the derivative of position with respect to time. Find the derivative of x with respect to t and substitute t = 0. If the velocity is positive, the particle is moving to the right.
C. To find all values of t for which the particle is moving to the left, find the intervals in which the velocity is negative. Again, the velocity is the derivative of x with respect to t.
D. To find the total distance the particle travels from t = 0 to t = 4, calculate the definite integral of the absolute value of the velocity function from 0 to 4.
2. A spherical balloon is being inflated at a rate of 10 cubic centimeters per second.
A. To find the expression for dr/dt (the rate at which the radius of the balloon is increasing), use the formula for the volume of a sphere: V = (4/3)πr^3. Differentiate this volume formula with respect to t, and you'll get an expression in terms of dr/dt and r.
B. To find how fast the radius of the balloon is increasing when the diameter is 40 cm, substitute the given values into the derived expression from part A and solve for dr/dt.
C. To find how fast the surface area of the balloon is increasing when the radius is 5 cm, use the formula for the surface area of a sphere: A = 4πr^2. Differentiate this formula with respect to t, and you'll get an expression in terms of dA/dt and r.
3. The function f(x) has the value f(1) = 5. The slope of the curve y = f(x) at any point is given by the expression dy/dx = (4x - 2)(y + 1).
A. To write an equation for the line tangent to the curve y = f(x) at x = 1, you need to find the slope of the tangent line at that point using the given expression for dy/dx. Then substitute the point (1, 5) and the slope into the point-slope form of the equation for a line.
B. To find an explicit formula for y = f(x) without any remaining integrals, use separation of variables on the differential equation dy/dx = (4x - 2)(y + 1) and solve for y.
C. To calculate the slope of the tangent line to the curve at x = 0, substitute x = 0 into the given expression for dy/dx and evaluate the result.
4. The derivative of a function f is given by f'(x) = (-2x - 2)e^x, and f(0) = 3.
A. To determine the nature of the critical point at x = -1, you need to analyze the behavior of the derivative around that point. Use the first derivative test or second derivative test to determine if it is a relative minimum, relative maximum, or neither.
B. To identify the intervals on which the graph of f is both increasing and concave down, you'll need to analyze the behavior of the derivative and second derivative. Determine the intervals where both f'(x) > 0 and f''(x) < 0.
C. To find the value of f(-1), substitute x = -1 into the expression for f.