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All Textbook Solutions for Calculus: Early Transcendentals
Let f(x)=x+x. (a) For what values of a does limxaf(x) exist? (b) At what numbers is f discontinuous?Evaluate limx1x31x1Find numbers a and b such that limx0ax+b2x=1.3PThe figure shows a point P on the parabola y = x2 and the point Q where the perpendicular bisector of OP intersects the y-axis. As P approaches the origin along the parabola, what happens to Q? Does it have a limiting position? If so, find it.Evaluate the following limits, if they exist, where x denotes the greatest integer function. (a) limx0xx (b) limx0x1/xSketch the region in the plane defined by each of the following equations. (a) x2+y2=1 (b) x2y2=3 (c) x+y2=1 (d) x+y=1Find all values of a such that f is continuo us on : f(x)={x+1ifxax2ifxaA fixed point of a function f is a number c in its domain such that f(c) = c. (The function doesn't move c; it stays fixed.) (a) Sketch the graph of a continuous function with domain [0, 1] whose range also lies in [0, 1]. Locate a fixed point of f. (b) Try to draw the graph of a continuous function with domain [0, I] and range in [0, 1] that does not have a fixed point. What is the obstacle? (c) Use the Intermediate Value Theorem to prove that any continuous function with domain [0, 1] and range in [0, 1] must have a fixed point.9P(a) The figure shows an isosceles triangle ABC with B = C. The bisector of angle B intersects the side AC at the point P. Suppose that the base BC remains fixed but the altitude |AM | or the triangle approaches 0, so A approaches the midpoint M or BC. What happens to P during this process? Does it have a limiting position? If so, find it. (b) Try to sketch the path out by P during this process. Then find an equation of this curve and use this equation to sketch the curve.11PIf f is a differentiable function and g(x) = xf(x), use the definition of a derivative to show that g'(x) = xf'(x) + f(x).Suppose f is a function that satisfies the equation f(x + y) = f(x) + f(y) + x2y + xy2 for all real numbers x and y. Suppose also that limx0f(x)x=1 (a) Find f(0). (b) Find f'(0). (c) Find f'(x).Suppose f is a function with the property that | f(x) | x2 for all x. Show that f(0) = 0. Then show that f'(0) = 0.(a) How is the number e defined? (b) Use a calculator to estimate the values of the limits limh02.7h1handlimh02.8h1h correct to two decimal places. What can you conclude about the value of e?(a) Sketch, by hand, the graph of the function f(x) = ex, paying particular attention to how the graph crosses the y-axis. What fact allows you to do this? (b) What types of functions are f(x) = ex and g(x) = xe? Compare the differentiation formulas for f and g. (c) Which of the two functions in part (b) grows more rapidly when x is large?Differentiate the function. f(x) = 240Differentiate the function. f(x) = e5Differentiate the function. f(x) = 5.2x + 2.3Differentiate the function. g(x)=74x23x+12Differentiate the function. f(t) = 2t3 3t2 4tDifferentiate the function. f(t) = 1.4t5 2.5t2+ 6.7Differentiate the function. g(x) = x2(1 2x)Differentiate the function. H(u) = (3u 1)(u + 2)Differentiate the function. g(t) = 2t3/4Differentiate the function. B(y) = cy6Differentiate the function. F(r)=5r3Differentiate the function. y = x5/3 x2/3Differentiate the function. R(a) = (3a + 1)2Differentiate the function. h(t)=t44etDifferentiate the function. S(p)=ppDifferentiate the function. y=x3(2+x)Differentiate the function. y=3ex+4x3Differentiate the function. S(R) = 4R2Differentiate the function. h(u)=Au3+Bu2+CuDifferentiate the function. y=x+xx2Differentiate the function. y=x2+4x+3xDifferentiate the function. G(t)=5t+7tDifferentiate the function. j(x) = x2.4 + e2.4Differentiate the function. k(r) = er + reDifferentiate the function. G(q) = (1 + q1)2Differentiate the function. F(z)=A+Bz+Cz2z2Differentiate the function. f(v)=v32vevvDifferentiate the function. D(t)=1+16t2(4t)3Differentiate the function. z=Ay10+BeyDifferentiate the function. y = ex + 1 + 1Find an equation of the tangent line to the curve at the given point. y = 2x3 x2 + 2, (1, 3)Find an equation of the tangent line to the curve at the given point. y = 2ex + x, (0, 2)Find an equation of the tangent line to the curve at the given point. y=x+2x,(2,3)Find an equation of the tangent line to the curve at the given point. y=x4x,(1,0)Find equations of the tangent line and normal line to the curve at the given point. y = x4 + 2ex, (0, 2)Find equations of the tangent line and normal line to the curve at the given point. y2 = x3, (1, 1)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 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=xx,(1,0)Find f'(x). Compare the graphs of f and f' and use them to explain why your answer is reasonable. f(x) = x4 2x3 + x2Find f'(x). Compare the graphs of f and f' and use them to explain why your answer is reasonable. f(x) = x5 2x3 + x 143E44EFind the first and second derivatives of the function. f(x)=0.001x50.02x3Find the first and second derivatives of the function. G(r)=r+r3Find 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)=2x5x3/4Find 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)=exx3The equation of motion of a particle is s = t3 3t, where s is in meters and t is in seconds. Find (a) the velocity and acceleration as functions of t, (b) the acceleration after 2 s, and (c) the acceleration when the velocity is 0.The equation of motion of a particle is s = t4 2t3 + t2 t, wheres is in meters and t is in seconds. (a) Find the velocity and acceleration as functions of t. (b) Find the acceleration after 1 s. (c) Graph the position, velocity, and acceleration functions on the same screen.Biologists have proposed a cubic polynomial to model the length L of Alaskan rockfish at age A: L = 0.0 155A3 0.372A2 + 3.95A + 1.21 where L is measured in inches and A in years. Calculate dLdA|A=12 and interpret your answer.The number of tree species S in a given area A in the Pasoh Forest Reserve in Malaysia has been modeled by the power function S(A) = 0.882A0.842 where A is measured in square meters. Find S'(100) and interpret your answer. Source: Adapted from K. Kochummen et al., Floristic Composition of Pasoh Forest Reserve, A Lowland Rain Forest in Peninsular Malaysia, Journal of Tropical Forest Science (1991):113.Boyles Law states that when a sample of gas is compressed at a constant temperature, the pressure P of the gas is inversely proportional to the volume V of the gas. (a) Suppose that the pressure of a sample of air that occupies 0.106 m3 at 25C is 50 kPa Write V as a function of P. (b) Calculate dV/dP when P = 50 kPa. What is the meaning of the derivative? What are its units?Find the points on the curve y = 2x3 + 3x2 12x + 1 where the tangent is horizontal.For what value of x does the graph of f(x) = ex 2x have a horizontal tangent?Show that the curve y = 2ex + 3x + 5x3 has no tangent line with slope 2.Find an equation of the tangent line to the curve y = x4 + 1 that is parallel to the line 32x y = 15.Find equations of both lines that are tangent to the curve y = x3 3x2 + 3x 3 and are parallel to the line 3x y = 15.At what point on the curve y = 1 + 2ex 3x is the tangent line parallel to the line 3x y = 5? Illustrate by graphing the curve and both lines.Find an equation of the normal line to the curve y=x that is parallel to the line 2x + y = 1.Where does the normal line to the parabola y = x2 1 at the point (1, 0) intersect the parabola a second time? Illustrate with a sketch.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 parabola.(a) Find equations of both lines through the point (2, 3) that are tangent to the parabola y = x2 + x. (b) Show that there is no line through the point (2, 7) that is tangent to the parabola. Then draw a diagram to see why.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 the nth derivative of each function by calculating the first few derivatives and observing the pattern that occurs. (a) f(x) = xn (b) f(x) = 1/xFind a second-degree polynomial P such that P(2) = 5, P'(2) = 3, and P" = (2) = 2.The equation y" + y' 2y = x2 is called a differential equation because it involves an unknown function y and its derivatives y' and y". Find constants A, B. and C such that the function y = Ax2 + Bx + C satisfies this equation. (Differential equations will he studied in detail in Chapter 9.)Find a cubic function y = ax3 + bx2 + cx + d whose graph has horizontal tangents at the points (2, 6) and (2, 0).Find a parabola with equation y = ax2 + bx + c that has slope 4 at x = 1, slope 8 at x = 1, and passes through the po.int (2, 15).Let {x2+1ifx1x+1ifx1 Is f differentiable at 1? Sketch the graphs of f and f'.At what numbers is the following function g differentiable? g(x){2xifx02xx2if0x22xifx2 Give a formula for g' and sketch the graphs of g and g'.(a) For what values of x is the function f(x) = |x2 9| differentiable? Find a formula for f'. (b) Sketch the graphs of f and f'.Where is the function h(x) = |x 1| + |x + 2| differentiable? Give a formula for h' and sketch the graphs of hand h'.Find the parabola with equation y = ax2 + bx whose tangent line at (1, 1) has equation y = 3x 2.Suppose the curve y = x4 + ax3 + bx2 + cx + d has a tangent line when x = 0 with equation y = 2x + 1 and a tangent line when x = 1 with equation y = 2 3x. Find the values of a, b, c, and d.For what values of a and b is the line 2x + y = b tangent to the parabola y = ax2 when x = 2?78EWhat is the value of c such that the line y = 2x + 3 is tangent to the parabola y = cx2?The graph of any quadratic function f(x) = ax2 + bx + c is a parabola. Prove that the average of the slopes of the tangent lines to the parabola at the endpoints of any interval [p, q] equals the slope of the tangent line at the midpoint of the interval.Let f(x){x2ifx2mx+bifx2 Find the values of m and b that make f differentiable everywhere.82E83E84EIf c12, how many lines through the point (0, c) are normal lines to the parabola y = x2? What if c12?86EFind 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 the derivative o f the function F(x)=x45x3+xx2 in two ways: by using the Quotient Rule and by simplifying first. Show that your answers are equivalent. Which method do you prefer?Differentiate. f(x) = (3x2 5x)ex4. Differentiate. g(x)=(x+2x)exDifferentiate. y=xexDifferentiate. y=ex1exDifferentiate. g(x)=1+2x34xDifferentiate. G(x)=x222x+1Differentiate. H(u)=(uu)(u+u)Differentiate. J(v) = (v3 2v)(v4 + v2)11EDifferentiate. f(z) = (1 ez)(z + ez)Differentiate. y=x2+1x3114EDifferentiate. y=t3+3tt24t+3Differentiate. y=1t3+2t21Differentiate. y=ep(p+pp)Differentiate. h(r)=aerb+erDifferentiate. y=sss220EDifferentiate. f(t)=t3t3Differentiate. V(t)=4+ttetDifferentiate. f(x)=x2exx2+ex24EDifferentiate. f(x)=xx+cxDifferentiate. f(x)=ax+bcx+dFind f'(x) and f"(x). f(x) = (x3 + 1)exFind f'(x) and f"(x). f(x)=xexFind f'(x) and f"(x). f(x)=x21+exFind f'(x) and f"(x). f(x)=xx21Find an equation of the tangent line to the given curve at the specified point. y=x21x2+x+1,(1,0)Find an equation of the tangent line to the given curve at the specified point. y=1+x1+ex,(0,12)Find equations of the tangent line and normal line to the given curve at the specified point. y = 2xex, (0, 0)Find equations of the tangent line and normal line to the given curve at the specified point. y=2xx2+1,(1,1)(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 (1,12). (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.(a) The curve y = x/(1 + x2) is called a serpentine. Find an equation of the tangent line to this curve at the point (3, 0.3). (b) Illustrate pan (a) by graphing the curve and the tangent line on the same screen.(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 und f'.38E(a) If f(x) = (x2 1)/(x2 + 1), find f'(x) and f"(x). (b) Check to see that your answers to part (a) ore reasonable by comparing the graphs of f, f' and f".(a) If f(x) = (x2 1)ex, 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/(l + x), find f"(1).If g(x) = x/ex. find g(n)(x).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)Suppose that f(4) = 2, g(4) = 5, f'(4) = 6. and g'(4) = 3. Find h'(4). (a) h(x) = 3f(x) + 8g(x) (b) h(x) = f(x)g(x) (c) h(x)=f(x)g(x) (d) h(x)=g(x)f(x)+g(x)If f(x) = exg(x), where g(0) = 2 and g'(0) = 5, find f'(0).If h(2) = 4 and h'(2) = 3, find ddx(h(x)x)|x=2If 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(2) = 10 and f'(x) = x2f(x) for all x, find f"(2).If f and g are the functions whose graphs are shown, let u(x) = f(x)g(x) and v(x) = f(x)/g(x). (a) Find u'(l). (b) Find v'(5).Let P(x) = F(x)G(x) and Q(x) = F(x)/G(x), where F and G are the functions whose graphs are shown. (a) Find P'(2). (b) Find Q'(7).If g is a differentiable function, find an expression for the derivative of each of the following functions. (a) y = xg(x) (b) y=xg(x) (c) y=g(x)xIf f is a differentiable function, find an expression for the derivative of each of the following functions. (a) y = x2f(x) (b) y=f(x)x2 (c) y=x2f(x) (d) y=1+xf(x)xHow 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 equations of the tangent lines to the curve y=x1x+1 that are parallel to the line x 2y = 2.Find R'(0), where R(x)=x3x3+5x51+3x3+6x6+9x9 Hint: Instead of finding R'(x) first, let f(x) be the numerator and g(x) the denominator of R(x) and compute R'(0) from f(0), f'(0), g(0), and g'(0).Use the method of Exercise 55 to compute Q'(0), where Q(x)=1+x+x2+xex1x+x2xexIn 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 population was increasing at roughly 9200 people per year. The average annual income was 30,593 per capita, and this average was increasing at about 1400 per year (a little above the national average of about 1225 yearly). Use the Product Rule and these figures to estimate the rate at which total personal income was rising in the Richmond-Petersburg area in 1999. Explain the meaning of each term in the Product Rule.A manufacturer produces bolts of a fabric with a fixed width. The quantity q of this fabric (measured in yards) that is sold is a function of the selling price p (in dollars per yard), so we can write q = f(p). Then the total revenue earned with selling price p is R(p) = pf(p). (a) What does it mean to say that f(20) = 10,000 and f'(20) = 350? (b) Assuming the values in part (a), find R'(20) and interpret your answer.The Michaelis-Menten equation for the enzyme chymotrypsin is v=0.14[S]0.015+[S] where v is the rate of an enzymatic reaction and [S] is the concentration of a substrate S. Calculate dv/d[S] and interpret it.60E(a) Use the Product Rule twice to prove that if f, g, and h are differentiable, then (fgh)' = f'gh + fg'h + fgh'. (b) Taking f = g = h in part (a), show that dx[f(x)]3=3[f(x)]2f(x) (c) Use part (b) to differentiate y = e3x.(a) If F(x) = f(x) g(x), where f and g have derivatives of all orders, show that F" = f"g + 2f'g' + fg". (b) Find similar formulas for F"' and F(4). (c) Guess a formula for F(n).63E(a) If g is differentiable, the Reciprocal Rule says that ddx[1g(x)]=g(x)[g(x)]2 Use the Quotient Rule to prove the Reciprocal Rule. (b) Use the Reciprocal Rule to differentiate the function in Exercise 16. (c) Use the Reciprocal Rule to verify that the Power Rule is valid for negative integers, that is, ddx(xn)=nxn1 for all positive integers n.Differentiate. f(x) = x2 sin xDifferentiate. f(x) = x cos x + 2 tan xDifferentiate. f(x) = ex cos xDifferentiate. y = 2 sec x csc xDifferentiate. y = sec tanDifferentiate. g() = e(tan )Differentiate. y = c cos t + t2 sin tDifferentiate. f(t)=cottet9EDifferentiate. y = sin cosDifferentiate f()=sin1+cosDifferentiate. y=cosx1sinxDifferentiate. y=tsint1+tDifferentiate. y=sint1+tantDifferentiate. f() = cos sinDifferentiate. f(t) = tet cot tProve that ddx(cscx)=cscxcotx.Prove that ddx(secx)=secxtanxProve that ddx(cotx)=csc2x.Prove, using the definition of derivative. that if f(x) = cos x, then f'(x) = sin x.Find an equation of the tangent line to the curve at the given point. y = sin x + cos x, (0, 1)Find an equation of the tangent line to the curve at the given point. y = ex cos x, (0, 1)Find an equation of the tangent line to the curve at the given point. y = cos x sin x, (, 1)Find an equation of the tangent line to the curve at the given point. y = x + tan x, (, )(a) Find an equation of the tangent line to the curve y = 2x sin x at the point (/2, ). (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.26E(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| I / 2.(a) If f(x) = ex cos x, find f'(x) and f"(x). (b) Check to see that your answers to part (a) are reasonable by graphing f, f', and f".If H() = sin , find H'() and H"( ).If f(t) = sec t, find f"(/4).(a) Use the Quotient Rule to differentiate the function f(x)=tanx1secx (b) Simplify the expression for f(x) by writing it in terms of sin x and cos x, and then find .f'(x). (c) Show that your answers to parts (a) and (b) are equivalent.Suppose f(/3) = 4 and f'(/3) = 2, and let g(x) = f(x) sin x and h(x) = (cos x)/f(x). Find (a) g'(/3) (b) h'(/3)For what values of x does the graph of f have a horizontal tangent? f(x) = x + 2 sin xFor what values of x does the graph of f have a horizontal tangent? f(x) = ex cos xA 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 t is in seconds and x in centimeters. (a) Find the velocity and acceleration at time t. (b) Find the position, velocity, and acceleration of the mass at time t = 2/3. In what direction is it moving at that time?An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the mass is pulled downward and then released, it vibrates vertically. The equation of motion is s = 2 cos t + 3 sin t, t 0, where s is measured in centimeters and t in seconds. (Take the positive direction to be downward.) (a) Find the velocity and acceleration at time t. (b) Graph the velocity and acceleration functions. (c) When does the mass pass through the equilibrium position for the first time? (d) How far from its equilibrium position does the mass travel? (e) When is the speed the greatest?A ladder 10 ft long rests against a vertical wall. Let be the angle between the top of the ladder and the wall and let x be the distance from the bottom of the ladder to the wall . If the bottom of the ladder slides away from the wall, how fast does x change with respect to when = /3?An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle with the plane, then the magnitude of the force is F=Wsin+cos where is a constant called the coefficient of friction. (a) Find the rate of change of F with respect to . (b) When is this rate of change equal to 0? (c) If W = 50 lb and = 0.6, draw the graph of F as a function of and use it to locate the value of for which dF/d = 0. Is the value consistent with your answer to part (b)?Find the limit. limx0sin5x3xFind the limit. limx0sinxsinxFind the limit. limt0tan6tsin2t42EFind the limit. limx0sin3x5x34xFind the limit. limx0sin3xsin5xx245EFind the limit. limx0cscxsin(sinx)47EFind the limit. limx0sin(x2)x49E50EFind the given derivative by finding the first few derivatives and observing the pattern that occurs. d99dx99(sinx)Find the given derivative by finding the first few derivatives and observing the pattern that occurs. d35dx35(xsinx)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.(a) Evaluate limxxsin1x. (b) Evaluate limx0xsin1x. (c) Illustrate parts (a) and (b) by graphing y = xsin( 1/x).Differentiate each trigonometric identity to obtain a new (or familiar) identity. (a) tanx=sinxcosx (b) secx=1cosx (c) sinx+cosx=1+cotxcscxA semicircle with diameter PQ sits on an isosceles triangle PQR to form a region shaped like a two-dimensional ice-cream cone, as shown in the figure. If A() is the area of the semicircle and B() is the area of the triangle, find. lim0+A()B()The figure shows a circular arc of length s and a chord of length d, both subtended by a central angle . Find lim0+sd58EWrite 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. y=1+4x3Write 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. y = (2x3 + 5)4Write 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. y = tan xWrite 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. y = sin( cot x)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. y=exWrite 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. y=2exFind the derivative of the function. F(x) = (5x6 + 2x3)4Find the derivative of the function. F (x) = (1 + x + x2)99Find the derivative of the function. f(x)=5x+1Find the derivative of the function. f(x)=1x213Find the derivative of the function. f() = cos(2)Find the derivative of the function. g() = cos2Find the derivative of the function. y = x2e3xFind the derivative of the function. f(t) = t sin tFind the derivative of the function. f(t) = eat sin btFind the derivative of the function. g(x)=ex2xFind the derivative of the function. f(x) = (2x 3)4(x2 + x + 1)5Find the derivative of the function. g(x) = (x2 + 1)3(x2 + 2)6Find the derivative of the function. h(t) = (t + 1)2/3 (2t2 1)3Find the derivative of the function. F(t) = (3t 1)4(2t + 1)3Find the derivative of the function. y=xx+1Find the derivative of the function. y=(x+1x)5Find the derivative of the function. y = e tanFind the derivative of the function. f(t)2t3Find the derivative of the function. g(u)=(u31u3+1)8Find the derivative of the function. s(t)=1+sint1+costFind the derivative of the function. r(t)=10t2Find the derivative of the function. f(z) = ez/(z1)Find the derivative of the function. H(r)=(r21)3(2r+1)5Find the derivative of the function. J() = tan2(n)Find the derivative of the function. F(t) = et sin 2tFind the derivative of the function. F(t)=t2t3+1Find the derivative of the function. G(x) = 4C/x34EFind the derivative of the function. y=cos(1e2x1+e2x)36EFind the derivative of the function. y = cot2(sin )38EFind the derivative of the function. f(t) = tan(sec(cos t))Find the derivative of the function. y = esin 2x + sin(e2x)Find the derivative of the function. f(t)=sin2(esin2t)Find the derivative of the function. y=x+x+xFind the derivative of the function. g(x) = (2 rarx + n)PFind the derivative of the function. y=234xFind the derivative of the function. y=cossin(tanx)Find the derivative of the function. y = [x + (x + sin2x)3]4Find y and y. y = cos(sin 3)Find y and y. y=1(1+tanx)2Find y and y. y=1sectFind y and y. y=eexFind an equation of the tangent line to the curve at the given point. y = 2x, (0. 1)Find an equation of the tangent line to the curve at the given point. y=1+x3,(2,3)Find an equation of the tangent line to the curve at the given point. y = sin(sin x), (, 0)Find an equation of the tangent line to the curve at the given point. y=xex2,(0,0)55E(a) The curve y=|x|/2x2 is called a bullet-nose curve. Find an equation of the tangent line to this curve at the point (1, 1). (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.(a) If f(x)=2x2x, find f(x). (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f.58EFind all points on the graph of the function f(x) = 2 sin x + sin2x at which the tangent line is horizontal.At what point on the curve y=1+2x is the tangent line perpendicular to the line 6x + 2y = 1?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).If h(x)=4+3f(x), where f(1) = 7andf(1) = 4, find h(1).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).64EIf f and g 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) v(1) (c) w(1)If f is the function whose graph is shown, let h(x) = f(f(x)) and g(x) = f(x2). Use the graph of f to estimate the value of each derivative. (a) h(2) (b) g(2)If g(x)=f(x), where the graph off is shown, evaluate g(3).68ESuppose f is differentiable on . Let F(x) = f(ex) and G(x) = ef(x). Find expressions for (a) F(x) and (b) G(x).Let g(x) = ecx + f(x) and h(x) = ekxf(x), where f(0) = 3, f(0) = 5, and f(0) = 2. (a) Find g(0) and g(0) in terms of c. (b) In terms of k, find an equation of the tangent line to the graph of h at the point where x = 0.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).If g is a twice differentiable function and f(x) = xg(x2), find f in terms of g, g, and g.73E74EShow that the function y = e2x (A cos 3x + B sin 3x) satisfies the differential equation y 4y + 13y = 0.For what values of r does the function y = erx satisfy the differential equation y 4y + y = 0?Find the 50th derivative of y = cos 2x.78E