true or false questions:
a)The derivatives of the reciprocal trigonometric functions can be found using the chain rule and their related base functions.
b) A sinusoidal function can be differentiated only if the independent variable is measured in radians.
c) There are an infinite number of tangents with a given non-zero slope to a sinusoidal curve.
d) The graph of a power function y=x^n is steeper than the graph of an exponential function, y=a^x (a>0, a doesn't= 1), as the value of x increases.
e) The derivative of an exponential function is always another exponential function.
TFTFT
thanks
a) True. The derivatives of reciprocal trigonometric functions can indeed be found using the chain rule and their related base functions. It's like a fancy mathematical recipe!
b) False. A sinusoidal function can be differentiated regardless of whether the independent variable is measured in radians or any other unit. Math doesn't discriminate against units!
c) True. There are indeed an infinite number of tangents with a given non-zero slope to a sinusoidal curve. It's like a party with an unlimited supply of tangents, but only one non-zero slope.
d) False. The graph of a power function y=x^n is not necessarily steeper than the graph of an exponential function y=a^x as x increases. Graphs can be tricky, sometimes they surprise us!
e) True. The derivative of an exponential function is always another exponential function. It's like a never-ending cycle of exponential goodness. Exponentialception, if you will!
a) True. The derivatives of the reciprocal trigonometric functions (cosec, sec, and cot) can be found using the chain rule and their related base functions (sin, cos, and tan).
b) False. A sinusoidal function can be differentiated regardless of the unit of measurement for the independent variable.
c) True. For a sinusoidal curve, there are infinite tangents with a given non-zero slope. This is due to the continuous nature of the curve.
d) False. The graph of a power function y=x^n and an exponential function y=a^x can have varying steepness depending on the values of n and a. It is not always true that the power function is steeper as x increases.
e) True. The derivative of an exponential function is always another exponential function. The rate of change of an exponential function is directly proportional to the function itself.
a) True. The derivatives of reciprocal trigonometric functions (such as secant, cosecant, and cotangent) can be found using the chain rule and the derivatives of their related base functions (such as cosine, sine, and tangent).
b) False. A sinusoidal function can be differentiated regardless of the unit used to measure the independent variable. The derivative of a sinusoidal function can be found by applying the chain rule to differentiating the components of the function, such as the amplitude and frequency.
c) True. Given a sinusoidal curve, there are an infinite number of tangents that can be drawn with a given non-zero slope. This is because the slope of the tangent varies continuously as we move along the curve.
d) False. The steepness of the graph of a power function y=x^n and an exponential function y=a^x depends on the values of n and a, respectively. If the exponent n is greater than 1, the power function will be steeper than the exponential function for positive values of x. However, for values less than 1, the exponential function can become steeper as x increases.
e) True. The derivative of an exponential function is always another exponential function. Specifically, if you have a function of the form y=a^x (where a>0 and a is not equal to 1), then its derivative with respect to x is given by the natural logarithm of the base of the exponential function multiplied by the original function.