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mathematics
applied calculus

Questions and Answers of Applied Calculus

Find an antiderivative.p(x) = x2 − 6x + 17
Use the Fundamental Theorem to determine the value of b if the area under the graph of f(x) = 4x between x = 1 and x = b is equal to 240. Assume b > 1.
Find the integrals in Problem. Check your answers by differentiation. t 1+ 3t² dt
Using Figure 6.16, sketch a graph of an antiderivative G(t) of g(t) satisfying G(0) = 5. Label each critical point of G(t) with its coordinates. g(1) 1 Area = 16 2 3 Area = 8 Figure 6.16 Area = 2 4 5
Find an antiderivative.g(z) = √z
During a surge in the demand for electricity, the rate, r, at which energy is used can be approximated byr = te−at,where t is the time in hours and a is a positive constant.(a) Find the total
(a) Between 2005 and 2015, ACME Widgets sold widgets at a continuous rate of R = R0e0.125t widgets per year, where t is time in years since January 1, 2005. Suppose they were selling widgets at a
Find the integrals in Problem. Check your answers by differentiation. [ x sin(4x²)dx
Figure 6.19 shows the derivative F'(t). If F(0) = 3, find the values of F(2), F(5), F(6). Graph F(t). Area = 5 2 F' (t) 5 Area = 16 Figure 6.19 6 t Area = 10
Oil is leaking out of a ruptured tanker at the rate of r(t) = 50e−0.02t thousand liters per minute.(a) At what rate, in liters per minute, is oil leaking out at t = 0? At t = 60?(b) How many liters
Find the integrals in Problems. Check your answers by differentiation. [si sin³ a cos a da
Find an antiderivative.f(x) = 5x − √x
If t is in years, and t = 0 is January 1, 2005, worldwide energy consumption, r, in quadrillion (1015) BTUs per year, is modeled byr = 462e0.019t.(a) Write a definite integral for the total energy
Graph y = 1∕x2 and y = 1∕x3 on the same axes. Which do you think is larger: ∫∞1 1∕x2 dx or ∫∞1 1∕x3 dx? Why?
Give a graph of f'(x). Graph f(x). Mark the points x1, . . . , x4 on your graph and label local maxima, local minima and points of inflection. x tx Ex Tx Tx Τ (2) f
Find the integrals in Problem. Check your answers by differentiation. [si sin’x cos xdx
(a) Estimate ∫40 f(x) dx for f(x) in Figure 6.20.(b) Let F(x) be an antiderivative of f(x). If F(0) = 100, what is F(4)? 0 -4 -8 -12 2 f(x) Figure 6.20 3 4 X
Decide if the improper integral ∫0∞ e−2t dt converges ,and if so, to what value, by the following method.(a) Use a computer or calculator to find ∫b0 e−2t dt forb = 3, 5, 7, 10.What do you
Let F(x) be an antiderivative of f(x) = 1 − x2.(a) On what intervals is F(x) increasing?(b) On what intervals is the graph of F(x) concave up?
A graph of f is given. Let F'(x) = f(x).(a) What are the x-coordinates of the critical points of F(x)?(b) Which critical points are local maxima, which are local minima, and which are neither?(c)
Find the integrals in Problem. Check your answers by differentiation. s e³x-4dx
You will show that the following improper integral converges to 1:(a) Use the Fundamental Theorem to find ∫b1 1∕x2 dx. Your answer will contain b.(b) Now take the limit as b → ∞. What does
Find an antiderivative.f(z) = ez + 3
A graph of f is given. Let F'(x) = f(x).(a) What are the x-coordinates of the critical points of F(x)?(b) Which critical points are local maxima, which are local minima, and which are neither?(c)
Find the integrals in Problem. Check your answers by differentiation. 1 + et 1: √x + ex dx
(a) Graph f(x) = e−x2 and shade the area represented by the improper integral ∫∞−∞ e−x2 dx.(b) Use a calculator or computer to find ∫a−a e−x2 dx for a = 1, a = 2, a = 3, a = 5.(c)
For Problems find the derivative. Assume a, b, c, k are constants.w = 3ab2q
For t ≥ 0 in minutes, the temperature, H, of a pot of soup in degrees Celsius isH = 5 + 95e−0.054t.(a) Is the temperature increasing or decreasing with time?(b) How fast is the temperature
The distance, s, of a moving body from a fixed point is given as a function of time by s = 20e t∕2. Find the velocity, v, of the body as a function of t.
On September 6th, 2017, there was a full moon. If t is the number of days since September 6th, the percent of moon illuminated can be represented by(a) Find H'(t). Explain what this tells about the
Find the equation of the tangent line to the graph of y = 3x at x = 1. Check your work by sketching a graph of the function and the tangent line on the same axes.
Find the derivative. Assume a, b, c, k are constants. h(x) = ax + b с
A company estimates that the total revenue, R, in dollars, received from the sale of q items is R = ln(1 + 1000q2). Calculate and interpret the marginal revenue if q = 10.
Find the equation of the tangent line to the graph of f(x) = x3ex at the point at which x = 2.
Find the equation of the tangent line to the graph of f(x) = x2e−x at x = 0. Check by graphing this function and the tangent line on the same axes.
Paris, France, has a latitude of approximately 49◦ N. If t is the number of days since the start of 2009, the number of hours of daylight in Paris can be approximated by(a) Find D(40) and D'(40).
(a) Use Figure 3.18 to rank the quantities f'(1), f'(2), f'(3) from smallest to largest.(b) Confirm your answer by calculating the quantities using the formula, f(x) = 2ex − 3x2√x.
If you invest P dollars in a bank account at an annual interest rate of r%, then after t years you will have B dollars, where(a) Find dB∕dt, assuming P and r are constant. In terms of money, what
Find the derivative. Assume that a, b, c, and k are constants.f(x) = (ax2 + b)3
Differentiate the functions in Problems. Assume that A and B are constants. Z = +t sin(2t)
Find the derivative. Assume that a, b, c, and k are constants.g(p) = p ln(2p + 1)
Differentiate the functions in Problems. Assume that A, B, and C are constants.P(t) = 12.41(0.94)t
Find the derivative of the functions in Problems.f(x) = ln(ex + 1)
Find the derivative. Assume a, b, c, k are constants. f(z) = 1 Z.6.1
Find the derivative. Assume that a, b, c, and k are constants.f(t) = te5−2t
Differentiate the functions in Problems. Assume that A and B are constants.f(x) = x2 cos x
Differentiate the functions in Problems. Assume that A, B, and C are constants.P = 50e−0.6t
Find the derivative of the functions in Problems.f(x) = ln(1 − x)
Differentiate the functions in Problems. Assume that A, B, and C are constants.y = 2x + 2/x3
Find the derivative of the functions in Problems.f(x) = (x3 + x2)−90
Find the derivative. Assume a, b, c, k are constants.y = x4∕3
Use the definition of the derivative to obtain the following results.If f(x) = 4x2 + 1, then f'(x) = 8x.
Find the derivative. Assume that a, b, c, and k are constants.y = t2(3t + 1)3
Differentiate the functions in Problems. Assume that A and B are constants.y = 5 sin x − 5x + 4
Differentiate the functions in Problems. Assume that A, B, and C are constants.y = 5 ⋅ 5t + 6 ⋅ 6t
Find the derivative of the functions in Problems.w = (5r − 6)3
Find the derivative. Assume a, b, c, k are constants.y = 8t3
Use the definition of the derivative to obtain the following results.If f(x) = x2 + x, then f'(x) = 2x + 1.
Since the 1950s, the carbon dioxide concentration in the air has been recorded at the Mauna Loa Observatory in Hawaii. A graph of this data is called the Keeling Curve, after Charles Keeling, who
Find the derivative. Assume that a, b, c, and k are constants.f(x) = xex
Differentiate the functions in Problems. Assume that A and B are constants.y = t2 + 5 cos t
Differentiate the functions in Problems. Assume that A, B, and C are constants.y = 5t2 + 4et
Find and interpret the value of the expression in practical terms. Let C(t) be the concentration of carbon dioxide in parts per million (ppm) in the air as a function of time, t, in months since
Find the relative rate of change, f'(t)∕f(t), of the function f(t).f(t) = 15t + 12
Find the relative rate of change f'(t)∕f(t) at the given value of t. Assume t is in years and give your answer as a percent.f(t) = ln(t2 + 1); t = 2
Find the derivative. Assume a, b, c, k are constants.ℎ(θ) = θ(θ−1∕2 − θ −2)
Find the derivative. Assume that a, b, c, and k are constants. y = 1+z In z
Find the derivative of the functions in Problems.f(x) = ln(1 − e−x)
Find the derivative. Assume a, b, c, k are constants. g(t) =
Find and interpret the value of the expression in practical terms. Let C(t) be the concentration of carbon dioxide in parts per million (ppm) in the air as a function of time, t, in months since
Find the relative rate of change, f'(t)∕f(t), of the function f(t).f(t) = 10t + 5
Find the relative rate of change of f(t) using the formula d/dt ln f(t).f(t) = 6.8e−0.5t
Find the derivative. Assume a, b, c, k are constants.y = √x(x + 1)
Find the derivative. Assume that a, b, c, and k are constants.f(x) = axe−bx
Find the relative rate of change of f(t) using the formula d/dt ln f(t).f(t) = 4.5t−4
Find the derivative. Assume a, b, c, k are constants.f(x) = kx2
Find the derivative. Assume that a, b, c, and k are constants.g(α) = eαe−2α
Find the relative rate of change, f'(t)∕f(t), of the function f(t).f(t) = 30e−7t
Find the relative rate of change of f(t) using the formula d/dt ln f(t).f(t) = 5e1.5t
Find the relative rate of change of f(t) using the formula d/dt ln f(t).f(t) = 3t2
Find the derivative. Assume a, b, c, k are constants. v = at² + +2
Find the derivative. Assume that a, b, c, and k are constants.f(t) = aebt
Normal human body temperature fluctuates with a rhythm tied to our sleep cycle. If H(t) is body temperature in degrees Celsius at time t in hours since 9 am, then H(t) may be modeled by(a) Calculate
Find the relative rate of change, f'(t)∕f(t), of the function f(t).f(t) = 8e5t
Find the derivative. Assume a, b, c, k are constants.P = a + b √t
Find the equation of the tangent line to the graph of f(x) = 5xex at the point at which x = 0.
The depth of the water, y, in meters, in the Bay of Fundy, Canada, is given as a function of time, t, in hours after midnight, by the functiony = 10 + 7.5 cos(0.507t).How quickly is the tide rising
For f(t) = 4−2et, find f'(−1), f'(0), and f'(1). Graph f(t), and draw tangent lines at t = −1, t = 0, and t = 1. Do the slopes of the lines match the derivatives you found?
Find the equation of the tangent line to f(x) = (x −1)3 at the point where x = 2.
Find the derivative. Assume a, b, c, k are constants. 4 V = r²b
A boat at anchor is bobbing up and down in the sea. The vertical distance, y, in feet, between the sea floor and the boat is given as a function of time, t, in minutes, byy = 15 + sin(2πt).(a) Find
Find the relative rate of change, f¨(t)∕f(t), of the function f(t).f(t) = 6t2
For f(x) = (2x − 3)3, find the equation of the tangent line at(a) x = 0 (b) x = 2
Find the derivative. Assume a, b, c, k are constants.Q = aP2 + bP3
A company’s monthly sales, S(t), are seasonal and given as a function of time, t, in months, by(a) Graph S(t) for t = 0 to t = 12. What is the maximum monthly sales? What is the minimum monthly
If f(x) = (3x + 8)(2x − 5), find f'(x) and f"(x).
Find the relative rate of change, f'(t)∕f(t), of the function f(t).f(t) = 35t−4
Find the derivative. Assume a, b, c, k are constants. f(x) = x3
Differentiate the functions in Problems. Assume that A and B are constants. f(t) = 1² COS
For Problems find the derivative. Assume that a, b, c, and k are constants.w = (t3 + 5t)(t2 − 7t + 2)

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