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This warm-up exercise reviews trigonometric functions, power rule, and derivatives, focusing on finding amplitudes, periods, domains, and ranges of functions, as well as calculating derivatives of trigonometric functions. It also covers the concept of jerk and its applications in various scenarios. The exercise concludes with finding tangent and normal lines and calculating numerical derivatives using a calculator.
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Warm Up #2: Trig Review, Power Rule Review AND pg 139 #40-45 AND pg 146 #51 1) Find the amplitude, period, domain, and range of the following: • y = sin(4x) • y = -3cos(x) • y = 3cos(x) + 5 2) Find the derivative of • y = • y = AND remember sin²(x) + cos²(x) = 1
Review #11 on hw When does the body reverse direction? When (approximately) is the body moving at a constant speed? Graph the body’s speed for 0,10 Graph the acceleration, where defined.
CH 3.5 Derivatives of Trigonometric Functions
Why are Trigonometric Functions Important? • So many of the phenomena we want to learn about are periodic (heart rhythms, earthquakes, tides, weather…) • Periodic functions can always be expressed in terms of sines and cosines, so these play a key role in describing periodic change!
Review: PreCalc Double Angle Formulas sin(A+B) = sin A cos B + cos A sin B sin(A−B) = sin A cos B − cos A sin B cos(A+B) = cos A cos B − sin A sin B cos(A−B) = cos A cos B + sin A sin B tan(A+B) = (tan A + tan B) / (1 − tan A tan B) tan(A−B) = (tan A − tan B) / (1 + tan A tan B)
What is the derivative of the sine function? = cos(x) Proof Analytically: = = = + cos(x)* = sin(x)*0 + cos(x)*1 = cos(x)
What is the derivative of each of the 6 trigonometric functions? = cos(x) = - sin(x) #24 on your hw asks you to prove = - sin(x)
EX 1: Using the Differentiation Rules Find the derivatives of the following: a) y = x²sin(x) b) y =
EX 2: A Trig Second Derivative a) Find y” if y = sec(x) b) Find y’’ if y = 7sin(x) + 1
You Try! Find the first derivative of the following: a) y = 2sin(x) - tan(x) b) y = xsec(x) c) y = 3x + xtan(x) d) y = e) y =
What is jerk? • Jerk is a sudden change in acceleration. • EX: When a ride in a car or bus is jerky, it is not the acceleration but rather the abrupt changes in acceleration that might make a person sick. • EX: Recent tests have shown that you are more likely to get sick from acceleration whose changes in magnitude or direction take us by surprise. So a driver or someone who keeps their eyes on the road will not get as sick. • Jerk is the derivative of acceleration. If a body’s position at time t is s(t), the body’s jerk at time t is j(t) =
EX 3: A Couple of Jerks • What is the jerk caused by gravity? The constant of acceleration of gravity is g = -32 ft/sec² The third derivative is 0 which is why we don’t experience sickness while just siting around! b) What is the jerk of the simple harmonic motion s = 5cos(t)? j = = 5sint
EX 4: Applications of Trig - The motion of a Weight on a Spring A weight hanging from a spring is stretched 5 units beyond its rest position (s = 0) and released at time t = 0 to bob up and down. Its position at any later time t is s = 5cos(t). a) Describe its motion. The position function s(t) = 5cos(t). This shows that the weight moves up and down between s = -5 and s = 5. There is a range of 10 units What are its velocity and acceleration at time t? v(t) = -5sin(t) it attains its greatest speed, 5, at n The speed is zero when sin(t) = 0. This occurs when s = 5cos(t) = ±5 when t = n so the endpoints of the interval of motion. a(t) = -5 cos(t) a(t) is always the exact opposite of the position value. When the weight is above the rest position, gravity is pulling it back down. When the weight is below the rest position the spring is pulling it back up.
You Try! B) A body is moving in simple harmonic motion with position function s = cos(t)-3sin(t). • Find the body’s velocity, speed, and acceleration at time t. b) Find the body’s velocity, speed, and acceleration at time t = c) Describe the motion of the body. A) A body is moving in simple harmonic motion with position function s = 1 – 4cos(t) (s in meters, t in seconds). • Find the body’s velocity, speed, and acceleration at time t. b) Find the body’s velocity, speed, and acceleration at time t = c) Describe the motion of the body.
EX 5: Finding Tangent and Normal Lines Find equations for the lines that are tangent and normal to the graph of f(x) = at x = π.
You Try! Find equations for the lines that are tangent and normal to the graph of y = sec(x) at x =
How do we calculate derivatives on a calculator? Our graphing calculator uses the symmetric difference quotient to calculate the numerical derivative of f at a point a, abbreviated as NDER f(a). (sometimes the symmetric difference quotient can yield an even better approximation than the regular difference quotient) On the calculator, press MATH and then #8, often referred to as “MATH 8” NDER f(a) = = Algebraically, let h = .001 (in most cases, .001 is more than adequate)
EX: Computing a Numerical Derivative Compute NDER(x³, 2) by hand and then use a calculator (the numerical derivative of x³ at x = 2) NDER f(a) = = = 12.000001 Algebraically, let h = .001 (in most cases, .001 is more than adequate)
EX 3: Failure of NDER • NDER is accurate up to 5 decimal places • There can be times when it incorrectly produces a result. Computer NDER(|x|, 0) (the numerical derivative |x| at x = 0) = = = = 0
* A good way to think of differentiable functions is that they are locally linear. In other words, when we zoom in very close to a, a function resembles its own tangent line.
HW 3.5pg 146 QR #1-7, EX 1-31odd, 24, 32, 34, 36
