1 Functions And Models 2 Limits And Derivatives 3 Differentiation Rules 4 Applications Of Differentiation 5 Integrals 6 Applications Of Integration 7 Techniques Of Integration 8 Further Applications Of Integration 9 Differential Equations 10 Parametric Equations And Polar Coordinates 11 Sequences, Series, And Power Series 12 Vectors And The Geometry Of Space 13 Vector Functions 14 Partial Derivatives 15 Multiple Integrals 16 Vector Calculus A Numbers, Inequalities, And Absolute Values B Coordinate Geometry And Lines C Graphs Of Second-degree Equations D Trigonometry E Sigma Notation F Proofs Of Theorems G The Logarithm Defined As An Integral expand_more
3.1 Derivatives Of Polynomials And Exponential Functions 3.2 The Product And Quotient Rules 3.3 Derivatives Of Trigonometric Functions 3.4 The Chain Rule 3.5 Implicit Differentiation 3.6 Derivatives Of Logarithmic And Inverse Trigonometric Functions 3.7 Rates Of Change In The Natural And Social Sciences 3.8 Exponential Growth And Decay 3.9 Related Rates 3.10 Linear Approximations And Differentials 3.11 Hyperbolic Functions Chapter Questions expand_more
Problem 1E: (a) How is the number e defined? (b) Use a calculator to estimate the values of the limits... Problem 2E: (a) Sketch, by hand, the graph of the function f(x) = ex, paying particular attention to how the... Problem 3E: Differentiate the function. 3. g(x)=4x+7 Problem 4E: Differentiate the function. 4. g(t)=5t+4t2 Problem 5E: Differentiate the function. 5. f(x)=x75x+3 Problem 6E: Differentiate the function. g(x)=74x23x+12 Problem 7E: Differentiate the function. 7. f(t)=2et Problem 8E: Differentiate the function. 8. F(t)=t3+e3 Problem 9E: Differentiate the function. 9. W()=1.83 Problem 10E: Differentiate the function. 10. r(z)=z5z1/2 Problem 11E: Differentiate the function. 11. f(x)=x3/2+x3 Problem 12E: Differentiate the function. 12. V(t)=t3/5+t4 Problem 13E: Differentiate the function. 13. s(t)=1t+1t2 Problem 14E: Differentiate the function. 14. r(t)=at2+bt4 Problem 15E: Differentiate the function. 15. ] y=2x+x Problem 16E: Differentiate the function. 16. h(w)=2w2 Problem 17E: Differentiate the function. 17. g(x)=1x+x4 Problem 18E: Differentiate the function. 18. W(t)=t2et Problem 19E: Differentiate the function. 19. f(x)=x3(x+3) Problem 20E: Differentiate the function. 20. F(t)=(2t3)2 Problem 21E: Differentiate the function. 21. y=3ex+4x3 Problem 22E Problem 23E: Differentiate the function. 23. f(x)=3x2+x3x Problem 24E: Differentiate the function. 24. y=x+xx2 Problem 25E: Differentiate the function. 25. G(r)=3r3/2+r5/2r Problem 26E: Differentiate the function. 26. G(t)=5t+7t Problem 27E: Differentiate the function. 27. j(x)=x2.4+e2.4 Problem 28E: Differentiate the function. 28. k(r)=er+re Problem 29E: Differentiate the function. 29. F(z)=A+Bz+Cz2z2 Problem 30E Problem 31E Problem 32E: Differentiate the function. 32. f()=32ev Problem 33E: Differentiate the function. 33. P(w)=2w2w+4w Problem 34E: Differentiate the function. y = ex + 1 + 1 Problem 35E Problem 36E Problem 37E: Find an equation of the tangent line to the curve at the given point. y = 2x3 x2 + 2, (1, 3) Problem 38E: Find an equation of the tangent line to the curve at the given point. y = 2ex + x, (0, 2) Problem 39E: Find an equation of the tangent line to the curve at the given point. y=x+2x,(2,3) Problem 40E: Find an equation of the tangent line to the curve at the given point. y=x4x,(1,0) Problem 41E: Find equations of the tangent line and normal line to the curve at the given point. y = x4 + 2ex,... Problem 42E Problem 43E: Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the... Problem 44E: Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the... Problem 45E: Find f'(x). Compare the graphs of f and f' and use them to explain why your answer is reasonable.... Problem 46E: Find f'(x). Compare the graphs of f and f' and use them to explain why your answer is reasonable.... Problem 47E Problem 48E Problem 49E: Find the first and second derivatives of the function. f(x)=0.001x50.02x3 Problem 50E: Find the first and second derivatives of the function. G(r)=r+r3 Problem 51E: Find the first and second derivatives of the function. Check to see that your answers are reasonable... Problem 52E: Find the first and second derivatives of the function. Check to see that your answers are reasonable... Problem 53E: The equation of motion of a particle is s = t3 3t, where s is in meters and t is in seconds. Find... Problem 54E: The equation of motion of a particle is s = t4 2t3 + t2 t, wheres is in meters and t is in... Problem 55E: Biologists have proposed a cubic polynomial to model the length L of Alaskan rockfish at age A: L =... Problem 56E: The number of tree species S in a given area A in the Pasoh Forest Reserve in Malaysia has been... Problem 57E: Boyles Law states that when a sample of gas is compressed at a constant temperature, the pressure P... Problem 58E Problem 59E Problem 60E: For what value of x does the graph of f(x) = ex 2x have a horizontal tangent? Problem 61E: Show that the curve y = 2ex + 3x + 5x3 has no tangent line with slope 2. Problem 62E: Find an equation of the tangent line to the curve y = x4 + 1 that is parallel to the line 32x y =... Problem 63E: Find equations of both lines that are tangent to the curve y = x3 3x2 + 3x 3 and are parallel to... Problem 64E: At what point on the curve y = 1 + 2ex 3x is the tangent line parallel to the line 3x y = 5?... Problem 65E: Find an equation of the normal line to the curve y=x that is parallel to the line 2x + y = 1. Problem 66E: Where does the normal line to the parabola y = x2 1 at the point (1, 0) intersect the parabola a... Problem 67E: Draw a diagram to show that there are two tangent lines to the parabola y = x2 that pass through the... Problem 68E: (a) Find equations of both lines through the point (2, 3) that are tangent to the parabola y = x2 +... Problem 69E: Use the definition of a derivative to show that if f(x) = 1/x, then f'(x) = 1/x2. (This proves the... Problem 70E: Find the nth derivative of each function by calculating the first few derivatives and observing the... Problem 71E: Find a second-degree polynomial P such that P(2) = 5, P'(2) = 3, and P" = (2) = 2. Problem 72E: The equation y" + y' 2y = x2 is called a differential equation because it involves an unknown... Problem 73E: Find a cubic function y = ax3 + bx2 + cx + d whose graph has horizontal tangents at the points (2,... Problem 74E: Find a parabola with equation y = ax2 + bx + c that has slope 4 at x = 1, slope 8 at x = 1, and... Problem 75E: Let {x2+1ifx1x+1ifx1 Is f differentiable at 1? Sketch the graphs of f and f'. Problem 76E: At what numbers is the following function g differentiable? g(x){2xifx02xx2if0x22xifx2 Give a... Problem 77E: (a) For what values of x is the function f(x) = |x2 9| differentiable? Find a formula for f'. (b)... Problem 78E: Where is the function h(x) = |x 1| + |x + 2| differentiable? Give a formula for h' and sketch the... Problem 79E: Find the parabola with equation y = ax2 + bx whose tangent line at (1, 1) has equation y = 3x 2. Problem 80E: Suppose the curve y = x4 + ax3 + bx2 + cx + d has a tangent line when x = 0 with equation y = 2x + 1... Problem 81E: For what values of a and b is the line 2x + y = b tangent to the parabola y = ax2 when x = 2? Problem 82E Problem 83E: What is the value of c such that the line y = 2x + 3 is tangent to the parabola y = cx2? Problem 84E: The graph of any quadratic function f(x) = ax2 + bx + c is a parabola. Prove that the average of the... Problem 85E: Let f(x){x2ifx2mx+bifx2 Find the values of m and b that make f differentiable everywhere. Problem 86E: Find numbers a and b such that the given function g is differentiable at 1. g(x)=ax33xifx1bx2+2ifx1 Problem 87E Problem 88E Problem 89E Problem 90E Problem 91E: If c12, how many lines through the point (0, c) are normal lines to the parabola y = x2? What if... Problem 1AP Problem 2AP format_list_bulleted
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