Differentiate. f (x) = (2x + 1)3
> Refer to Fig. 3, where s(t) is the number of feet traveled by a person after t seconds of walking along a straight path. Figure 3: What is the person’s velocity at time t = 3? y 12 11 10 9 00 8 7 6 5 4 3 2 1 y = s(t) 0 1 2 3 4 5
> Refer to Fig. 3, where s(t) is the number of feet traveled by a person after t seconds of walking along a straight path. Figure 3: What is the person’s average velocity from time t = 1 to t = 4? y 12 11 10 9 00 8 7 6 5 4 3 2 1 y
> Refer to Fig. 3, where s(t) is the number of feet traveled by a person after t seconds of walking along a straight path. Figure 3: How far has the person traveled after 6 seconds? y 12 11 10 9 00 8 7 6 5 4 3 2 1 y = s(t) 0 1 2 3 4 5 6 7 Figure 3 W
> Differentiate. y = x + 1 / (x + 1)
> Each day the total output of a coal mine after t hours of operation is approximately 40t + t2 – 1 /15 t3 tons, 0 ≤ t ≤ 12. What is the rate of output (in tons of coal per hour) at t = 5 hours?
> A helicopter is rising at a rate of 32 feet per second. At a height of 128 feet the pilot drops a pair of binoculars. After t seconds, the binoculars have height s(t) = -16t2 + 32t + 128 feet from the ground. How fast will they be falling when they hit t
> In Fig. 2, the straight line is tangent to the graph of f (x) = x3. Find the value of a. Figure 2: a Figure 2 (0, 2) Y
> In Fig. 1, the straight line has slope -1 and is tangent to the graph of f (x). Find f (2) and f ‘(2). Figure 1: m = -1 Figure 1 Y 2 5 y = f(x) x
> Determine the equation of the tangent line to the curve y = (2x2 - 3x)3 at x = 2.
> Determine the equation of the tangent line to the curve y = 3x3 - 5x2 + x + 3 at x = 1.
> Find the equation of the tangent line to the curve y = x2 at the point (-2, 4). Sketch the graph of y = x2 and sketch the tangent line at (-2, 4).
> Find the equation of the tangent line to the curve y = x2 at the point (3/2 , 9/4). Sketch the graph of y = x2 and sketch the tangent line at (3/2, 9/4).
> What is the slope of the curve y = 1/(3x - 5) at x = 1? Write the equation of the line tangent to this curve at x = 1.
> What is the slope of the graph of f (x) = x3 - 4x2 + 6 at x = 2? Write the equation of the line tangent to the graph of f (x) at x = 2.
> Differentiate. y = 2 / x + 1
> Compute. d/dt (dy/dt), where y =1/3t.
> Compute. d2y/dx2, where y = 4x3/2
> Compute. d2/dP2 (3P + 2) |P=4
> Compute. d2/dt2 (t3 + 2t2 - t) |t= -1
> Compute. d2/dt2 (2√t)
> Compute. d2dx2 (5x + 1)4
> Compute. d dx (4x - 10)5 |x=3
> Compute. d/dz (z3 - 4z2 + z - 3) |z= -2
> Compute. d/dn (n-5)
> Compute. d/dP (√(1 - 3P))
> Differentiate. y = 1 / x3 + 1
> Compute. d/dt (t5/2 + 2t3/2 - t1/2)
> Compute. d/dx (x4 - 2x2)
> Find the slope of the graph of y = (4 - x)5 at x = 5.
> Find the slope of the graph of y = (3x - 1)3 - 4(3x - 1)2 at x = 0.
> If g(t) = ¼ (2t - 7)4, what is g’(3)?
> If f (x) = x5/2, what is f ‘(4)?
> If h(x) = - 1/2, find h(-2) and h’(-2).
> If g(u) = 3u - 1, find g(5) and g’(5).
> If V(r) = 15πr2, find V’(1/3 ).
> If f (t) = 3t3 - 2t2, find f ‘(2).
> Differentiate. y= (x2 + 1)2 + 3(x2 - 1)2
> Differentiate. y = 3x4
> Differentiate. f (x) = √(x + √x)
> Differentiate. h(x) = 32 x3/2 - 6x2/3
> Differentiate. g(P) = 4P0.7
> Differentiate. f (t) = 2 / t - 3t3
> Differentiate. h(t) = 3√2
> Differentiate. g(t) = 3√t – 3/√t
> Differentiate. f (t) = t10 - 10t9
> Differentiate. f (x) = [x5 - (x - 1)5]10
> Differentiate. f (x) = 5x/2 – 2/5x
> Differentiate. f (x) = 5
> Differentiate. y = 1 / 5x5
> Differentiate. f (x) = 1/4√x
> Differentiate. y = 5 / 7x2 + 1
> Differentiate. y = √(x2 + 1)
> Differentiate. y = (x3 + x2 + 1)5
> Differentiate. y = 1 / 5x - 1
> Differentiate. y = ¾ x4/3 + 4/3 x3/4
> Differentiate. y = (3x2 - 1)8
> Differentiate. y = x4 – 4/x
> Differentiate. y = 3/x
> Differentiate. y = (x - 1)3 + (x + 2)4
> Differentiate. y = x7 + 3x5 + 1
> Differentiate. y = 6√x
> Differentiate. y = 5x8
> Differentiate. y = x7 + x3
> Find the equation and sketch the graph of the following lines. The x-axis.
> Find the equation and sketch the graph of the following lines. The y-axis.
> Find the equation and sketch the graph of the following lines. Vertical and 4 units to the right of the y-axis.
> Find the equation and sketch the graph of the following lines. Horizontal with height 3 units above the x–axis.
> Find the equation and sketch the graph of the following lines. Perpendicular to 3x + 4y = 5, passing through (6, 7).
> Find the equation and sketch the graph of the following lines. Perpendicular to y = 3x + 4, passing through (1, 2).
> Differentiate. y = 2x + (x + 2)3
> Find the equation and sketch the graph of the following lines. Through (2, 1) and (5, 1).
> Find the equation and sketch the graph of the following lines. Through (-1, 4) and (3, 7).
> What can you say about the slopes of parallel lines? Perpendicular lines?
> Describe how to find an equation for a line when you know the coordinates of two points on the graph of a line.
> What is the point–slope form of the equation of a line?
> Define the slope of a nonvertical line and give a physical description.
> How do you determine the proper units for a rate of change? Give an example.
> Describe marginal cost in your own words.
> What expression involving a derivative gives an approximation to f (a + h) - f (a)?
> Explain the relationship between derivatives and velocity and acceleration.
> Differentiate. y = 2√(x + 1)
> How is an (instantaneous) rate of change related to average rates of change?
> What is meant by the average rate of change of a function over an interval?
> Give two different notations for the first derivative of f (x) at x = 2. Do the same for the second derivative.
> State the general power rule and give an example.
> In your own words, explain the meaning of “ f (x) is differentiable at x = 2.” Give an example of a function f (x) that is not differentiable at x = 2.
> In your own words, explain the meaning of “ f (x) is continuous at x = 2.” Give an example of a function f (x) that is not continuous at x = 2.
> In your own words, explain the meaning of lim x→∞ f (x) = 3. Give an example of such a function f (x). Do the same for lim x→∞ f (x) = 3.
> Give the limit definition of f ‘(2), that is, the slope of f (x) at the point (2, f (2)).
> In your own words, explain the meaning of lim x→2 f (x) = 3. Give an example of a function with this property.
> Explain how to calculate f ‘(2) as the limit of slopes of secant lines through the point (2, f (2)).
> Differentiate. y = 3 3√(2x2 + 1)
> State the power rule, the constant-multiple rule, and the sum rule, and give an example of each.
> Explain why the derivative of a constant function is 0.
> What does f ‘(2) represent?
> Give a physical description of what is meant by the slope of f (x) at the point (2, f (2)).
> Suppose that f (x) = 4x2. (a) What is the average rate of change of f (x) over each of the intervals 1 to 2, 1 to 1.5, and 1 to 1.1? (b) What is the (instantaneous) rate of change of f (x) when x = 1?
> If f (x) = 3x2 + 2, calculate the average rate of change of f (x) over the following intervals (a) 0 ≤ x ≤ .5, (b) 0 ≤ x ≤ .1, (c) 0 ≤ x ≤ .0.01.
> If f (x) = x2 + 3x, calculate the average rate of change of f (x) over the following intervals (a) 1 …≤ x ≤ 2, (b) 1 ≤ x ≤ 1.5, (c) 1 ≤ x ≤ 1.1.
> A ball thrown straight up into the air has height s(t) = 102t - 16t2 feet after t seconds. (a) Display the graphs of s(t) and s’(t) in the window [0, 7] by [-100, 200]. Use these graphs to answer the remaining questions. (b) How high is the ball after 2
