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Chris Seo, Nathaniel Wichman, Sean Moore,Tyler Pritchard IB Calculus Mr. Wahbeh Chapter 3 Practice Quiz 3.1 - Derivative of a Function Find the derivative of the given function at the indicated point for the following questions: 1. f(x)=2x, a=2 2. f(x)=x2+4, a=1 3. f(x)=3-x2, a=-1 4. f(x)=x2+x, a=0 Find... 1. ddx(x3) 2. ddx(x3+3x2+4) 3. ddxf(x) if f(x)=4x4 4. ddx(12x5+45x3+❤) 3.2 - Differentiability True or False 1. If f has a derivative at x=a, then f is continuous at x=a. 2. If f is continuous at x=a, then f has a derivative at x=a. The function fails to be differentiable at x=0. Tell whether it is a corner, a usp, a vertical tangent, or a discontinuity 1. y=x4/5 2. y=3-3x 3.3 - Rules of Differentiation Find the dy/dx 1. y=-x3+4 2. y=x2+x+1 3. y=x33-x Find the horizontal tangents of the curve 1. y=x3+x2-x 2. x3-2x2+x+1 Find the equation for the line tangent to the curve at the given point 1. y=x3+12x, x=1 3.4 - Velocity and Other Rates of Change 1. the figure below depicts the velocity v = f(t) of a dangerous, out of control freight train a. when is the train moving backwards? b. when is the train speeding up? c. when does the train have a negative acceleration? d. when does the train change directions? 2. find the instantaneous velocity at t=4 of a dangerous, out of control train whose position is given by the function s(t) = t2 – 3t + 2, where s is measured in kilometers and t is measured in nanoseconds 3.5 - Derivatives of Trigonometric Functions 1. find the y’, y’’, y’’’ of y=sin(x) 2. find the derivative of... a. tan(5x) b. sec(3x) c. cot(7x) d. csc(9x) 3.6 - Chain Rule Use the chain rule to find the derivatives of the following equations. 1. y = sin (3x +1) 2. y = cos (x3) 3. y = sin3x tan 4x 4. y = 32x + 1 Find the tangent line of (f o g) at the given value of x. 1. f(x)= x5+ 1, g(x)=x, x=1 2. f(x)=cotx10, g(x)=5x, x=1 Multiple Choice 1. Which of the following is dy/dx if y=tan(4x)? a. 4 sec (4x) tan (4x) b. 4 sec2(4x) c. 4 cot (4x) 2. Which of the following is y’ if y=cos2(x3+x2) a. -2(3x2+2x) cos (x3+x2) sin (x3+x2) b. 2(3x2+2x) cos (x3+x2) sin (x3+x2) c. -2(3x2+2x) 3.7 - Implicit Differentiation Find the derivatives 1. x2y – xy2 = 6 2. x + tan (xy) = 0 Find the equation of lines that are (a) tangent and (b) normal to the curve at the given point 3. x2y2 = 9, (-1,3) 3.8 - Derivatives of Inverse Trigonometric Function Use the derivatives of inverse trig. functions to find the value of y’. 1. y=cos-1(x2) 2. y=csc-1(x2+1) 3. y=sin-1(3x2) 4. y=cot-1x-1 Find the tangent line for the graph of y at the following indicated point. 1. y=sin-1(x4), x=2 2. y=tan-1(x2), x=1 Multiple Choice 1. Which of the following is the derivative of y=tan-1(3x) a. y'=-31+9x2 b. y'=31-9x2 c. y'=31+9x2 2. Which of the following is the derivative of y=sec-1(x2) a. y'=2x1-x4 b. y'=2x2x4-1 c. y'=2x2x2-1 3.9 - Derivatives of Exponential and Logarithmic Function 1. What is the derivative e❤? a. e❤❤ b. e❤❤’ c. 1❤❤’ 2. What is the derivative of ❤❤? . ❤❤ln(❤+1) a. ❤❤log(❤+1) b. ❤❤log(❤) 3. Find the derivatives of the following? . y= ln(6x+9) a. y= log6(3x2) Answer Key 3.1 Find the derivative of the given function 1. 1/2 2. 2 3. 2 4. 1 Find... 1. 3x2 2. 3x2+6x 3. 16x3 4. 60x4+135x2 3.2 True or False 1. True 2. False Tell whether the problem is a corner, a cusp, a vertical tangent, or a discontinuity 1. Cusp 2. Vertical Tangent 3.3 Find dy/dx 1. -3x 2. 2x+1 3. x2-1 Find the horizontal tangents of the curve 1. x=13, x=-1 2. x=1, x=13 Find the equation for the line tangent to the curve at the given point 1. y-1=12(x-1) 3.4 1. velocity of Thomas the Tank Engine a. moving backwards at 1<t<5 b. speeds up at 1<t<2 and 5<t<6 c. has a negative acceleration at 0≤t<2 and 6<t<7 d. 3.5 changes direction at t=1 and t=5 2. 5 km/ns 1. y’=cos(x) y’’=-sin(x) y’’’=-cos(x) 2. a. 5sec2(5x) b. 3sec(3x)tan(3x) c.-7csc2(7x) d.-9csc(9x)cot(9x) 3.6 Finding derivatives 1. y'=3 cos (3x+1) 2. y'=-3 sin (x3) 3. y'=4 sin3x sec24x+3 sin2x cos x tan 4x 4. y'=-3(2x+1)-32 Finding the tangent line of (f o g) 1. y-2=52(x-1) 2. y-2=-4(x-1) Multiple Choice 1. B 2. A 3.7 Finding derivatives 1. y’ =-2xy + y22xy + x2 2. y’ = -1xcos2(xy)-yx 3. a. y = 3x + 6 b. y = -13x+83 3.8 Finding Derivatives 1. y'=-2x1-x4 2. y'=-2(x2+1)x2+2 3. y'=-6xx4-9 4. y'=-12xx-1 Finding the tangent of the line 1. y=0.378x+0.286 2. y=0.118x+0.668 Multiple Choice 1. C 2. B 3.9 1. B 2. A 3. a. y’=63x+9 b. y’=6xln(6)3x