The average blood alcohol concentration (BAC) of eight male subjects was measured after consumption of 15 mL of ethanol (corresponding to one alcoholic drink). The resulting data were modeled by the concentration function C(t) = 0.0225te-0.0467t where t is measured in minutes after consumption and C is measured in mg/mL. (a) How rapidly was the BAC increasing after 10 minutes? (b) How rapidly was it decreasing half an hour later?
> Find |u × v | and determine whether u × v is directed into the page or out of the page. |v| =5 45° |u| = 4
> Find the work done by a force F = 8 i - 6 j + 9k that moves an object from the point (0, 10, 8) to the point (6, 12, 20) along a straight line. The distance is measured in meters and the force in newtons.
> (a) How is the number e defined? (b) Use a calculator to estimate the values of the limits correct to two decimal places. What can you conclude about the value of e? 2.7h – 1 lim 2.8h – 1 lim and h h
> Reduce the equation to one of the standard forms, classify the surface, and sketch it. x2 + y2 - 2x - 6y - z + 10 = 0
> Small birds like finches alternate between flapping their wings and keeping them folded while gliding (see Figure 1). In this project we analyze this phenomenon and try to determine how frequently a bird should flap its wings. Some of the principles are
> If g(x) = x/ex, find g(n)(x).
> Use logarithmic differentiation to find the derivative of the function. y = x cos x
> If f (x) = x2 / (1 + x), find f “(1).
> Use logarithmic differentiation to find the derivative of the function. y = xx
> (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 ‘’.
> (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 ‘’.
> Differentiate. y = 1 t 3 + 2 t 2 – 1
> Differentiate the function. B(y) = cy-6
> Differentiate. y = x2 + 1 / x3 + 1
> Differentiate. f (z) = (1 – ez)(z + ez)
> Find dy/dx by implicit differentiation. 2x2 + xy - y2 = 2
> Differentiate. J(v) = (v3 - 2v)(v-4 + v-2)
> Differentiate the function. g(x) = x2 (1 -2x)
> Sketch the parabolas y = x2 and y = x2 - 2x + 2. Do you think there is a line that is tangent to both curves? If so, find its equation. If not, why not?
> If c > ½, how may points through the point (0,c) are normal lines to the parabola y = x2? What if c ≤ ½ ?
> Draw a diagram showing two perpendicular lines that intersect on the y-axis and are both tangent to the parabola y = x2. Where do these lines intersect?
> A tangent line is drawn to the hyperbola xy = c at a point P. (a) Show that the midpoint of the line segment cut from this tangent line by the coordinate axes is P. (b) Show that the triangle formed by the tangent line and the coordinate axes always has
> Let Find the values of m and b that make f differentiable everywhere. -2 if x<2 f(x) = тx + b ifx>2
> 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 interva
> (a) If f (x) = ex / (2x2 + x + 1), find f ‘(x). (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f ‘.
> What is the value of c such that the line y − 2x + 3 is tangent to the parabola y = cx2?
> Use logarithmic differentiation to find the derivative of the function. y = (x2 + 2)2 (x4 + 4)4
> Differentiate the function. f(t) = 1.4t5 – 2.5t2 + 6.7
> For what values of a and b is the line 2x + y = b tangent to the parabola y = ax2 when x = 2?
> Suppose the y = x4 + ax3 + bx2 + cx + d curve 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.
> Find the parabola with y = ax2 + bx equation whose tangent line at (1, 1) has equation y = 3x - 2.
> At what numbers is the following function g differentiable? Give a formula for g’ and sketch the graphs of g and g’. 2х if x<0 9(х) — { 2х — х? if 0 <x<2 2 — х if x> 2
> Let Is f differentiable at 1? Sketch the graphs of f and f ’. (x² + 1 if x <1 f(x) x + 1 if x>1
> Find a parabola with y = ax2 + bx + c equation that has slope 4 at x = 1, slope -8 at x = -1, and passes through the point (2, 15).
> Find a cubic function y = ax3 + bx2 + cx + d whose graph has horizontal tangents at the points (-2, 6) and (2, 0).
> 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 satisfies y = Ax2 + Bx + C this equation. (Differential equations
> (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 ‘.
> Let f (x) = logα (3x2 - 2). For what value of b is f ‘(1) = 3?
> Differentiate the function. f(t) = 2t3 – 3t2 – 4t
> Use the Chain Rule and the Product Rule to give an alternative proof of the Quotient Rule.
> Find a second-degree polynomial P such that P(2) = 5, P’(2) = 3, and P’’(2)= 2.
> 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.
> 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 find the derivative in Example 5 and compare with the
> The table gives the US 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 is the fit? (b) Estimate the rates of population growth
> The flash unit on a camera operates by storing charge on a capacitor and releasing it suddenly when the flash is set off. The following data describe the charge Q remaining on the capacitor (measured in microcoulombs, mC) at time t (measured in seconds).
> Air is being pumped into a spherical weather balloon. At any time t, the volume of the balloon is V(t) and its radius is r(t). (a) What do the derivatives dV/dr and dV/dt represent? (b) Express dV/dt in terms of dr/dt.
> Differentiate the function. R(a) = (3a + 1)2
> 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?
> In Section 1.4 we modeled the world population from 1900 to 2010 with the exponential function P(t) = (1436.53) . (1.01395)t where t = 0 corresponds to the year 1900 and Pstd is measured in millions. According to this model, what was the rate of increase
> Find the nth derivative of each function by calculating the first few derivatives and observing the pattern that occurs. Ка) f(x) — х" (b) f(x) = 1/x
> 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 function. Suppose the equation of motion of a point on suc
> If the equation of motion of a particle is given by s = A cos (ωt + δ), the particle is said to undergo simple harmonic motion. (a) Find the velocity of the particle at time t. (b) When is the velocity 0?
> The displacement of a particle on a vibrating string is given by the equation s(t) = 10 + 1/4 sin(10πt) where s is measured in centimeters and t in seconds. Find the velocity of the particle after t seconds.
> Let f (x) = cx + ln (cos x). For what value of c is f ‘( π/4) = 6?
> Find the 1000th derivative of f (x)=xe-x.
> For what values of r does the function y=erx satisfy the differential equation y’’ - 4y’ + y=0?
> Show that the function y=e2x (A cos 3x + B sin 3x) satisfies the differential equation y’’ - 4y’ + 13y=0.
> If F(x) = f (x f (x f (x))), where f (1) = 2, f (2) = 3, f ‘(1) = 4, f ‘(2) = 5, and f ‘(3) = 6, find f’(1).
> If F(x) = f (3 f(4 f (x))), where f (0) = 0 and f ‘(0) = 2, find F’(0).
> If g is a twice differentiable function and f (x) = xg(x2), find f ‘’ in terms of g, g’, and g’’.
> 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).
> Let g(x) = ecx + f (x) and h(x) = ekx f (x), where f (0) = 3, f ‘(0) = 5, and f ‘’(0) = 22. (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.
> Find equations of the tangent lines to the curve y= ln x/x at the points (1, 0) and (e, 1/e). Illustrate by graphing the curve and its tangent lines.
> 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 α is a real number. Let F(x) = f (ex) and G(x) = [f(x)]α . Find expressions for (a) F’ (x) and (b) G’ (x).
> Differentiate. f (x) − x2 sin x
> If g(x) = f(x) , where the graph of f is shown, evaluate g’ (3). |f/ -1 1
> If g is a differentiable function, find an expression for the derivative of each of the following functions. (a) y = x g(x) (b) y = x / g(x) (c) y = g(x) / x
> 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) y= f(x) 1 - 이 1
> If 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) wâ€&#
> Let f and g be the functions in Exercise 63. (a) If F(x) = f (f(x)), find F’(2). (b) If G(x) = g(g(x), find G’(3). Data from Exercise 63: A table of values for f, g, f ‘, and g ‘ is
> 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). f(x) g(x) f'(x) g'(x) 1 4 1 8 5 3 7 2 7 679 3. 2.
> 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 f (x) = sin x + ln x, find f 9sxd. Check that your answer is reasonable by comparing the graphs of f and f ‘.
> Find all points on the graph of the function f (x) = 2 sin x + sin2x at which the tangent line is horizontal.
> The function f (x) = sin (x + sin 2x), 0 ≤ x ≤ π, arises in applications to frequency modulation (FM) synthesis. (a) Use a graph of f produced by a calculator to make a rough sketch of the graph of f ‘. (b) Calculate f ‘(x) and use this expression, with
> 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). F G 1
> (a) The curve y = |x |/ 2 – x 2 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) 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.
> Find an equation of the tangent line to the curve at the given point. y = sin (sin x) , (π , 0)
> Find an equation of the tangent line to the curve at the given point. y = x2 ln x
> Find an equation of the tangent line to the curve at the given point. y = 2x, (0 , 1)
> Find y’ and y’’. y = cos (sin θ)
> If f and t are the functions 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). f 1 1
> Find the derivative of the function. y = [x+(x + sin2 x)3]4
> Find the derivative of the function. g(x) = (2rarx + n)p
> Find an equation of the tangent line to the curve at the given point. y = ln (x2 – 3x + 1)
> Find the derivative of the function. f(1) = sin²(e*n*r)
> Find the derivative of the function. y = e sin 2x + sin (e2x)
> Find the derivative of the function. f(t) = tan (sec(cos t))
> Find the derivative of the function. y = cot2 (sin θ)
> If f (2) = 10 and f ‘(x)= x2 f (x) for all x, find f ‘’(2).
> Find the derivative of the function. y = x2 e-1/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
> If f (x) = cos(ln x2), find f ‘(1).
> Find the derivative of the function. G(x) = 4 C/x
> Find the derivative of the function. F(t) = e t sin 2t
> Find the derivative of the function. J (θ) = tan2 (nθ)
> If g(x) = x f (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 (x) = ln(x + ln x), find f ‘(1).
