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mathematics
calculus

Questions and Answers of Calculus

Suppose f is differentiable on R. Let F(x) = f(ex) and G(x) = ef(x). Find expressions for?(a) F’ (x) and(b) G’ (x)
Suppose f is differentiable on R and a is a real number. Let F(x) = f(xa) and G(x) = [f(x)]a, Find expressions for.(a) F’(x) and (b) G’ (x)
Suppose L is a function such that L’ (x) = 1/x for x > 0. Find an expression for the derivative of each function.(a) f(x) = L(x4) (b) g(x) = L(4x)(c) F(x) = [L(x)]4(d) G(x) = L(1/x)
Let r(x) = F(g(h(x))), where h(1) = 2, g(2) = 3, h’(1) = 4, g’(2) = 5, and f’(3) = 6. Find r’ (1).
An equation of motion is given, where is in meters and t in seconds. Find(a) The times at which the acceleration is 0 and(b) The displacement and velocity at these times.
If the equation of motion of a particle is given by s = A cos (wt + δ). , the particle is said to undergo simple harmonic motion. (a) Find the velocity of the particle at time t. (b) When is
A Cepheid variable star is a star whose brightness alternately increases and decreases. The most easily visible such star is Delta Cepheid, for which the interval between times of maximum brightness
In Example 4 in Section 1.3 we arrived at a model for the length of daylight (in hours) in Philadelphia on the day of the year:Use this model to compare how the number of hours of daylight is
The motion of a spring that is subject to a frictional force or a damping force (such as a shock absorber in a car) is often modeled by the product of an exponential function and a sine or cosine
Under certain circumstances a rumor spreads according to the equation where p(t) is the proportion ofthe population that knows the rumor at time t and a and are positive constants.[In Section 9.5 we
(a) Use a graphing calculator or computer to find an exponential model for the charge. (See Section 1.5)(b) The derivative Q€™(t) represents the electric current (measured in
The table gives the U.S. population from 1790 to 1860.(a) Use a graphing calculator or computer to fit an exponential function to the data. Graph the data points and the exponential model. How good
Computer algebra systems have commands that differentiate functions, but the form of the answer may not be convenient and so further commands may be necessary to simplify the answer.(a) Use a CAS to
(a) Use a CAS to differentiate the function and to simplify the result.(b) Where does the graph of f have horizontal tangents?(c) Graph f and f' on the same screen. Are the graphs consistent with
Use the Chain Rule to prove the following.(a) The derivative of an even function is an odd function.(b) The derivative of an odd function is an even function.
(a) Use differentials to find a formula for the approximate volume of a thin cylindrical shell with height , inner radius , and thickness ∆r. (b) What is the error involved in using the
(a) If is a positive integer, prove that(b) Find a formula for the derivative of y = cosnx cos nx that is similar to the one in part (a).
Suppose y = f(x) is a curve that always lies above the –axis and never er has a horizontal tangent, where f is differentiable everywhere. For what value of is the rate of change of y5 with respect
Use the Chain Rule to show that if is measured in degrees, then(This gives one reason for the convention that radian measure is always used when dealing with trigonometric functions in calculus: The
(a) Write | x | = √x2 and use the Chain Rule to show that(b)If f(x) = | sin x |, find f€™(x) and sketch the graphs of f and f€™. Where is f not differentiable?(c)If g(x) =
Suppose P and Q are polynomials and is a positive integer. Use mathematical induction to prove that the nth derivative of the rational function f(x) = P(x) / Q(x) can be written as a rational
(a) Find y' by implicit differentiation.(b) Solve the equation explicitly for and differentiate to get y' in terms of x.(c) Check that your solutions to parts (a) and (b) are consistent by
Regard as the independent variable and as the dependent variable and use implicit differentiation to find dx/dy.
Use implicit differentiation to find an equation of the tangent line to the curve at the given point.
(a) The curve with equation y2 = 5x4 – x2 is called a kampyle of Eudoxus. Find an equation of the tangent line to this curve at the point (1, 2).(b) Illustrate part (a) by graphing the curve and
(a) The curve with equation y2 = x3 – 3x2 is called the Tschirnhausen cubic. Find an equation of the tangent line to this curve at the point (1, – 2).(b) At what points does this curve have a
Fanciful shapes can be created by using the implicit plotting capabilities of computer algebra systems.(a) Graph the curve with equation y(y2 – 1) (y – 2) = x(x – 1) (x – 2)At how many points
(a) The curve with equation 2y3 + y2 – y5 = x4 – 2x3 + x2 has been likened to a bouncing wagon. Use a computer algebra system to graph this curve and discover why.(b) At how many points does this
Find the points on the lemniscates in Exercise 29 where the tangent is horizontal.
Show by implicit differentiation that the tangent to the ellipse
Find an equation of the tangent line to the hyperbola at the point (x0, y0).
Show that the sum of the - and -intercepts of any tangent line to the curve √x + √y = √c is equal to c.
Show, using implicit differentiation, that any tangent line at a point P to a circle with center O is perpendicular to the radius OP.
Find the derivative of the function. Simplify where possible.
Find f'(x) Check that your answer is reasonable by comparing the graphs of f and f'.
Prove the formula for (d/dx)(cos-1x) by the same method as for (d/dx)(sin-1x).
(a) One way of defining sec-1 x is to say that y = sec-1x ↔ sec y = x and 0 < y < π/2 or π < y < 3π/2. Show that, with this definition, d/dx (sec-1 x) = 1 / x√x2 – 1
Show that the given curves are orthogonal.
TV meteorologists often present maps showing pressure fronts. Such maps display isobars—curves along which the air pressure is constant. Consider the family of isobars shown in the figure. Sketch
Show that the given families of curves are orthogonal trajectories of each other. Sketch both families of curves on the same axes.
The equation x2 – xy + y2 = 3 represents a “rotated ellipse,” that is, an ellipse whose axes are not parallel to the coordinate axes. Find the points at which this ellipse crosses the –axis
(a) Where does the normal line to the ellipse x2 – xy + y2 = 3 at the point (– 1, 1) intersect the ellipsea second time? (See page 192 for the definition of a normal line.) (b) Illustrate part
Find all points on the curve x2 – xy + y2 = 3 where the slope of the tangent line is – 1.
Find equations of both the tangent lines to the ellipses x2 + 4y2 = 36 that pass through the point (12, 3).
(a) Suppose f is a one-to-one differentiable function and its inverse function f -1 is also differentiable. Use implicit differentiation to show thatProvided that the denominator is not O(b) If f(4)
(a) Show that f(x) = 2x + cos x is one-to-one.(b) What is the value of f –1(1)?(c) Use the formula from Exercise 67(a) to find (f –1)’ (1)
The figure shows a lamp located three units to the right of the -axis and a shadow created by the elliptical region x2 + 4y2
The figure shows the graphs of f, f€™, and f€™€™. Identify each curve, and explain your choices.
The figure shows graphs of f, f€™, f€™€™ and f€™€™€™. Identify each curve, and explain your choices.
The figure shows the graphs of three functions. One is the position function of a car, one is the velocity of the car, and one is its acceleration. Identify each curve, and explain your choices.
The figure shows the graphs of four functions. One is the position function of a car, one is the velocity of the car, one is its acceleration, and one is its jerk. Identify each curve, and explain
(a) If f(x) = 2 cos x + sin2x, find f’(x) and f’’(x).(b) Check to see that your answers to part (a) are reasonable by comparing the graphs of f, f’ and f’’
Find the given derivative by finding the first few derivatives and observing the pattern that occurs.
A car starts from rest and the graph of its position function is shown in the figure, where is measured in feet and in seconds. Use it to graph the velocity and estimate the acceleration at t = 2
(a) The graph of a position function of a car is shown, where is measured in feet and in seconds. Use it to graph the velocity and acceleration of the car. What is the acceleration at t = 10
The equation of motion is given for a particle, where is in meters and is in seconds. Find(a) The velocity and acceleration as functions of t,(b) The acceleration after 1 second, and(c) The
An equation of motion is given, where is in meters and t in seconds. Find(a) The times at which the acceleration is 0 and(b) The displacement and velocity at these times.
A particle moves according to a law of motion s = f(t) = t3 – 12t2 + 36t, t > 0 , where is measured in seconds and in meters.(a) Find the acceleration at time and after 3 s.(b) Graph the position,
A particle moves along the -axis, its position at time given by x(t) =t/(1 + t2), t > 0, where is measured in seconds and in meters.(a) Find the acceleration at time t. When is it 0?(b) Graph the
A mass attached to a vertical spring has position function given by y(t) = A sin wt, where A is the amplitude of its oscillations and is a constant.(a) Find the velocity and acceleration as functions
A particle moves along a straight line with displacement s(t), velocity v(t), and acceleration a(t). Show thatExplain the difference between the meanings of the derivatives dv/dt and dv/ds.
Find a second-degree polynomial P such that P(2) = 5, P’(2) = 3, and P’’(2) = 2.
Find a third-degree polynomial Q such that Q(1) = 1, Q’(1) = 3, Q’’(1) = 6, and Q’’’(1) = 12.
The equation y’’ + y’ – 2y = sin x is called a differential equation because it involves an unknown function and its derivatives y’ and y’’. Find constants A and B such that the
Find constants A, B, and C such that the function y = Ax2 + Bx + C satisfies the differential equation y’’ + y’ – 2y = x2.
For what values of r does the function y = erx satisfy the equation y’’ + 5y’ – 6y = 0?
Find the values of λ for which y = erx satisfies the equation y + y’ = y’’?
The function is a twice differentiable function. Find f€™€™ in terms of g, g€™, and g€™€™
If f(x) = 3x3 – 10x3 + 5, graph both f and f’’. On what intervals is f’’(x) > 0? On those intervals, how is the graph of f related to its tangent lines? What about the intervals where
(a) Compute the first few derivatives of the function f(x) = 1/ (x2 + x) until you see that the computations are becoming algebraically unmanageable.(b) Use the identity to compute the derivatives
(a) Use a computer algebra system to compute f€™€™, where(b) Find a much simpler expression for f€™€™ by first splitting f into partial fractions. [In Maple,
Find expressions for the first five derivatives of f(x) = x2ex. Do you see a pattern in these expressions? Guess a formula for f(n)(x) and prove it using mathematical induction.
(a) If F(x) = f(x) g(x), where f and have derivatives of all orders, show that(b) Find similar formulas for F€™€™€™ and F(4).(c) Guess a formula for F(n).
If y = f(u) and u = g(x), where f and are twice differentiable functions, show that
If y = f(u) and u = g(x) , where f and possess third derivatives, find a formula for similar to the one given in Exercise 67.
Suppose is a positive integer such that the function f is p-times differentiable and f(p) = f. Using mathematical induction, show that f is in fact n-times differentiable for every positive integer
Explain why the natural logarithmic function y = In x is used much more frequently in calculus than the other logarithmic functions y = log a x.
Find an equation of the tangent line to the curve at the given point.
If f(x) = sin x + In x, find f’(x). Check that your answer is reasonable by comparing the graphs of f and f’.
Find equations of the tangent lines to the curve y = (in x)/x at the points (1, 0) and (e, 1/e). Illustrate by graphing the curve and its tangent lines.
Use logarithmic differentiation to find the derivative of the function.
If sin h x = 3/4, find the values of the other hyperbolic functions at x.
If tan h x = 4/5, find the values of the other hyperbolic functions at x.
(a) Use the graphs of sinh, cosh, and tanh in Figures 1–3 to draw the graphs of csch, sech, and coth.(b) Check the graphs that you sketched in part (a) by using a graphing device to produce them.
Use the definitions of the hyperbolic functions to find each of the following limits.
Prove the formulas given in Table 1 for the derivatives of the functions (a) Cosh, (b) Tanh, (c) Csch, (d) Sech, and (e) Coth.
Give an alternative solution to Example 3 by letting y = sinh–1 x and then using Exercise 9 and Example 1(a) with replaced by y.
Prove Equation 5 using?(a) Te method of Example 3 and(b) Exercise 18 with replaced by y
For each of the following functions (i) give a definition like those in (2), (ii) sketch the graph, and (iii) find a formula similar to Equation 3.(a) Csch–1(b) Sech–1 (c) Coth–1
Prove the formulas given in Table 6 for the derivatives of the following functions.(a) Csch–1(b) Tanh–1(c) Csch–1 (d) Sech–1(e) Coth–1
A flexible cable always hangs in the shape of a catenary’s y = c + a cosh (x/a), where and are constants and a > 0 (see Figure 4 and Exercise 50). Graph several members of the family of functions y
A telephone line hangs between two poles 14 m apart in the shape of the catenary€™s y = 20 cosh(x/20) €“ 15, where and y are measured in meters.(a) Find the slope of this curve
Using principles from physics it can be shown that when a cable is hung between two poles, it takes the shape of a curve y = f(x) that satisfies the differential equation where p is the linear
(a) Show that any function of the form y = A sinh mx + B cosh mx satisfies the differential equation y’’ = m2y.(b) Find y = y(x) such that y’’ = 9y, y(0) = – 4, and y’(0) = 6.
At what point of the curve y = cosh x does the tangent have slope 1?
If x = in (sec θ + tan θ), show that sec θ = cosh x.
Show that if a ≠ 0 and b ≠ 0, then there exit numbers a and β such that aex + be – x equals either a sinh(x + β) or a cosh (x + β). In other words, almost every function
If x2 + y2 + 25 and dy/dt = 6, find dx/dt when y = 4.
If z2 + x2 + y2 dx/dt = 2, find dy/dt = 3, find dz/dt when y = 5 and y = 12.
Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate is the distance between the cars increasing two hours later?

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