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mathematics
calculus

Questions and Answers of Calculus

Find an equation of the tangent line to the curve at the given point. y = 4√x, (1, 1)
Find equations of the tangent line and normal line to the curve at the given point. Y = x4 + 2ex, (0, 2)
Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same screen. Y = 3x2 - x3, (1, 2)
Find f'(x). Compare the graphs of f and f' and use them to explain why your answer is reasonable. F(x) = x4 - 2x3 + x2
(a) Use a graphing calculator or computer to graph the function f(x) = x4 - 3x3 - 6x2 + 7x + 30 in the viewing rectangle [-3, 5] by [-10, 50]. (b) Using the graph in part (a) to estimate slopes, make
Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of f, f', and f". F(x) = 2x - 5x3/4
The equation of motion of a particle is s = t3 - 3t, where is in meters and is in seconds. Find (a) The velocity and acceleration as functions of t, (b) The acceleration after 2 s, and (c) The
Boyle's Law states that when a sample of gas is compressed at a constant pressure, the pressure P of the gas is inversely proportional to the volume V of the gas. (a) Suppose that the pressure of a
Find the points on the curve y = 2x3 + 3x2 - 12x + 1 where the tangent is horizontal.
Show that the curve y = 2ex + 3x + 5x3 has no tangent line with slope 2.
Find equations of both lines that are tangent to the curve y = 1 + x3 and parallel to the line 12x - y = 1.
Find an equation of the normal line to the parabola y = x2 - 5x + 4 that is parallel to the line x - 3y = 5.
Draw a diagram to show that there are two tangent lines to the parabola y = x2 that pass through the point (0, - 4). Find the coordinates of the points where these tangent lines intersect the
Use the definition of a derivative to show that if f(x) = 1/x, then f'(x) = -1/x2. (This proves the Power Rule for the case n = - 1.)
Find a second-degree polynomial P such that P(2) = 5, P'(2) = 3, and P"(2) = 2.
Find a cubic function y = ax3 + bx2 + cx + d whose graph has horizontal tangents at the points (-2, 6) and (2, 0).
LetIs f differentiable at 1? Sketch the graph of f and f'.
(a) For what values of is the function f(x) = |x2 - 9| differentiable? Find a formula for f'.(b) Sketch the graphs of f and f'.
Find the parabola with equation y = ax2 + bx whose tangent line at (1, 1) has equation y = 3x - 2.
For what values of and is the line 2x + y = b tangent to the parabola y = ax2 when x = 2?
LetFind the values of m and that make f differentiable everywhere.
Evaluate x1000 - 1 / x - 1.
If c > 1/2, how many lines through the point (0, c) are normal lines to the parabola y = x2? What if c ≤ 1/2?
Find the linearization L(x) of the function at a. (a) f(x) = x4 + 3x2, a = - 1 (b) f(x) = √x, a = 4
Find the differential of each function. (a) y = x2 sin 2x (b) y = ln√1 + t2
(a) Find the differential dy (b) evaluate dy for the given values of and dx. 1. y = ex/10, x = 0, dx = 0.1 2. y = √3 + x2, x = 1, dx = - 0.1
Compute Î”y and dy for the given values of x and dx = Î”x. Then sketch a diagram like Figure 5 showing the line segments with lengths dx, dy, and Î”y.(a) y = 2x - x2,
Use a linear approximation (or differentials) to estimate the given number. (a) (1.999)4 (b) 3√1001
Explain, in terms of linear approximations or differentials, why the approximation is reasonable. (a) sec 0.08 ≈ 1 (b) In 1.05 ≈ 0.05
The edge of a cube was found to be 30 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum possible error, relative error, and percentage error in
The circumference of a sphere was measured to be 84 cm with a possible error of 0.5 cm. (a) Use differentials to estimate the maximum error in the calculated surface area. What is the relative
(a) Use differentials to find a formula for the approximate volume of a thin cylindrical shell with height h, inner radius r, and thickness Δr. (b) What is the error involved in using the formula
If a current passes through a resistor with resistance R, Ohm's Law states that the voltage drop is V = RI. If V is R constant and is measured with a certain error, use differentials to show that the
Establish the following rules for working with differentials (where denotes a constant and u and v are functions of x). (a) dc = 0 (b) d(cu) = c du (c) d(u +v) = du + dv (d) d((uv) u dv + v du (e)
Suppose that the only information we have about a function is that f(1) = 5 and the graph of its derivative is as shown.(a) Use a linear approximation to estimate f(0, 9)and f(1, 1).(b) Are your
Find the linear approximation of the function f(x) = √1 - x at a = 0 and use it to approximate the numbers √0, 9 and √0.99. Illustrate by graphing f and the tangent line.
Verify the given linear approximation at a = 0. Then determine the values of for which the linear approximation is accurate to within 0.1. (a) ln(1 + x) ≈ x (b) 4√1 + 2x ≈ 1 + 1/2x
Find the numerical value of each expression. 1. (a) sinh 0 (b) cosh 0 2. (a) sinh (In 2) (b) sinh 2 3. (a) sech 0 (b) cosh-1 1
If cosh x = 5/3 and x > 0, find the values of the other hyperbolic functions at x.
Use the definitions of the hyperbolic functions to find each of the following limits.(a)(b) (c) (d) (e) (f) (g) (h) (i)
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. Example 3 Let y - sinh-1 x. Then x = sinh y = ey - e-y/2 Example 1
Prove Equation 5 using (a) the method of Example 3 and (b) Exercise 18 with x replaced by y.
Prove the formulas given in Table 6 for the derivatives of the following functions. (a) cosh-1 (b) tanh-1 (c) csch-1 (d) sech-1 (e) coth-1
Find the derivative. Simplify where possible. (a) f(x) = x sinh x - cosh x (b) h(x) = In (cosh x) (c) y = ecosh 3x
Show that d/dx arctan (tanh x) = sech 2x.
If a water wave with length moves with velocity in a body of water with depth , thenwhere is the acceleration due to gravity. Explain why the approximation is appropriate in deep water.
A telephone line hangs between two poles 14 m apart in the shape of the catenary y = 20 cosh(x/20) - 15, where x and y are measured in meters.(a) Find the slope of this curve where it meets the right
A cable with linear density p = 2 kg/m is strung from the tops of two poles that are 200 m apart. (a) Use Exercise 52 to find the tension so that the cable is 60 m above the ground at its lowest
(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?
Show that if a ≠ 0 and b ≠ 0, then there exist numbers a and β such that aex + be-x equals either or α sinh(x + β) or α cosh(x + β), In other words, almost every function of the form f(x) =
Prove the identity. (a) Sinh (-x) = -sinh x (This shows that sinh is an sod function.) (b) cosh x + sinh x = ex (c) sinh(x + y) = sinh x cosh y + cosh x sinh y
Find the derivative of f(x) = (1 + 2x2) (x - x2) in two ways: by using the Product Rule and by performing the multiplication first. Do your answers agree?
Find f'(x) and f"'(x). (a) f(x) = x4ex (b) f(x) = x2/1 + 2x
Differentiate. (a) f(x) = (x3 + 2x)ex (b) y = x/ ex (c) g(x) = 1 + 2x / 3 - 4x
Find an equation of the tangent line to the given curve at the specified point. y = x2 - 1/x2 + x + 1
Find equations of the tangent line and normal line to the given curve at the specified point. y = 2xex, (0, 0)
(a) The curve y = 1/(1 + x2) is called a witch of Maria Agnesi. Find an equation of the tangent line to this curve at the point (- 1, 1/2). (b) Illustrate part (a) by graphing the curve and the
(a) If f(x) = (x3 - x) ex, find f'(x). (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f'.
(a) If f(x) = (x2 - 1) /(x2 + 1), 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"'.
If f(x) = x2 / (1 + x), find f"'(1)
Suppose that f(5) = 1, f'(5) = 6, g(5) = - 3, and g'(5) = 2. Find the following values. (a) fg'(5) (b) (f/g)'(5) (c) (g/f)'(5)
If f(x) = exg(x), where g(0) = 2 and g'(0) = 5, find f'(0).
If g(x) = xf(x), where f(3) = 4 and f'(3) = - 2, find an equation of the tangent line to the graph of g at the point where x = 3.
If f and g are the function whose graphs are shown. Let u(x) = f(x) g(x) and v(x) = f(x) / g(x),(a) Find u'(1).(b) Find v'(5).
If is a differentiable function, find an expression for the derivative of each of the following functions. (a) y = xg(x) (b) y = x/g(x) (c) y = g(x) /x
How many tangent lines to the curve y = x/(x + 1) pass through the point (1, 2)? At which points do these tangent lines touch the curve?
Find R'(0), where R(x) = x - 3x3 + 5x5 / 1 + 3x3 + 6x6 + 9x9
In this exercise we estimate the rate at which the total personal income is rising in the Richmond-Petersburg, Virginia, metropolitan area. In 1999, the population of this area was 961,400, and the
(a) Use the Product Rule twice to prove that if , , and are differentiable, then (fgh)' = fgh + fg'h + gfh'. (b) Taking f = g = h in part (a), show that d/dx[f(x)]3 = 3[f(x)]2 f'(x) (c) Use part (b)
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.
Differentiate. (a) f(x) = 3x2 - 2 cos x (b) f(x) = sin x + 1/2 cot x (c) y = sec θ tan θ
Prove that d / dx (csc x) = - csc x cot x.
Prove that d / dx (cot x) = csc2x.
Find an equation of the tangent line to the curve at the given point. (a) y = sec x, (π/3, 2) (b) y = cos x - sin x, (π, - 1)
(a) Find an equation of the tangent line to the curve y = 2x sin x at the point (π/2, π). (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.
(a) If f(x) = sec x - x, find f'(x). (b) Check to see that your answer to part (a) is reasonable by graphing both f and f' for |x| < π/2.
If H(θ) = θ sin θ, find H'(θ) and H"(θ)
(a) Use the Quotient Rule to differentiate the functionf(x) = tan x - 1/ sec x(b) Simplify the expression for by writing it in terms of sin x and cos x, and then find f'(x).(c) Show that your answers
For what values of does the graph of have a horizontal tangent? F(x) = x + 2 sin x
A mass on a spring vibrates horizontally on a smooth level surface (see the figure). Its equation of motion is x(t) = 8 sin t, where is in seconds and in centimeters.(a) Find the velocity and
A ladder 10 ft long rests against a vertical wall. Let θ be the angle between the top of the ladder and the wall and let be the distance from the bottom of the ladder to the wall. If the bottom of
Find the limit.(a)(b) (c)
Find the given derivative by finding the first few derivatives and observing the pattern that occurs. d99 / dx99 (sin x)
Find constants A and B such that the function y = A sin x + B cos x satisfies the differential equation y" + y' - 2y = sin x.
Differentiate each trigonometric identity to obtain a new (or familiar) identity. (a) tan x = sin x / cos x (b) sec x = 1/cos x (c) sin x + cos x = 1 + cot x/csc x
The figure shows a circular arc of length and a chord of length d, both subtended by a central angle Î¸. Find
Write the composite function in the form f(g(x)). [Identify the inner function u = g(x) and the outer function y = f(u).] Then find the derivative dy/dx. (a) y = 3√1 + 4x (b) y = tan πx (c) y =
Find y' and y". (a) Y = cos(x2) (b) y = eax sin βx
Find an equation of the tangent line to the curve at the given point. (a) y = (1 + 2x)10, (0, 1) (b) y = sin (sin x), (π, 0)
(a) Find an equation of the tangent line to the curve y = 2/(1 + e-x) at the point (0, 1). (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.
(a) If f(x) = x √2 - x2, find f'(x). (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f'
Find all points on the graph of the function f(x) = 2 sin x + sin2 x at which the tangent line is horizontal.
If f(x) = f(g (x)), where f(-2) = 8, f'(-2) = 4, f'(5) = 3, g(5) = - 2, and g'(5) = 6, find F'(5).
A table of values for f, g, f', and g' is given.(a) If h(x) = f(g (x)), find h'(1). (b) If H(x) = g(f(x)). Find H'(1).
If f and are the functions whose graphs are shown, let u(x) = f(g (x)), v(x) = g(f (x)), and w(x) = g(g (x)). Find each derivative, if it exists. If it does not exist, explain why.(a) u'(1)(b)
If f(x) = ˆšf(X), where the graph of is shown, evaluate g'(3).
Suppose f is differentiable on R. Let F(x) = f(ex) and G(x) = ef(x). Find expressions for (a) F'(x) (b) G'(x).
Find the derivative of the function. (a) F(x) = (x4 + 3x2 - 2)5 (b) F(x) √1 - 2x (c) f(z) = 1/z2 + 1
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).

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